https://padlet.com/catherine_latham/nkndg2tg9ivc How can we solve the problems associated with Waste, Transport, Energy and Water? en-us 2017-12-11 22:25:46 UTC 2023-02-11 14:22:21 UTC hello@padlet.com https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245067 We can solve the problem associated with energy  by using more sustainable resources. An example of this is by using solar energy.  We can use less materials that don't recycle so we will have more rubbish to recycle to make new things. an example of this is using paper bags instead of plastic.


]]> 2017-12-11 22:43:52 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245067 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245104 we can solve the problems with waste by burning watse]]> 2017-12-11 22:44:03 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245104 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245208 Instead of buying everything you want that you think will "help you live your life", ask yourself, do you NEED it? Or do you WANT it? Because as you use more, more products are being decomposed, and some products just don't decompose which leads to lots of waste.

 ]]> 2017-12-11 22:44:40 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245208 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245256 Transport - People can use buses and trains more often then using their personal transportation like cars]]> 2017-12-11 22:44:49 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245256 taliesin_tirado https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245381 Focus on solar energy evolution]]> 2017-12-11 22:45:02 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245381 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245386 -Use environmentally friendly (pollution free) buses
Recycle your plastic bags at your local grocery plastic bag recycling bin
 

 ]]> 2017-12-11 22:45:04 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245386 keon_sano https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245451 we can bring our own bag instead of the plastic bag]]> 2017-12-11 22:45:16 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245451 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245493 think before buying ]]> 2017-12-11 22:45:22 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245493 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245528 to sustain energy wisely 
and use our resources not over the limit.]]> 2017-12-11 22:45:35 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245528 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245631  

look how pretty it is when we dump rubbish in space]]> 2017-12-11 22:46:15 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245631 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245634 Energy, turn the lights off when you are not using them, solar panels]]> 2017-12-11 22:46:15 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245634 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245669 try using public transport ]]> 2017-12-11 22:46:27 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245669 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245752 THROW IT INTO THE CORE OF THE EARTH (HELL)

]]> 2017-12-11 22:47:11 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245752 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245930  

 ]]> 2017-12-11 22:48:17 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215245930 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246065 2017-12-11 22:49:15 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246065 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246219  

 ]]> 2017-12-11 22:50:06 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246219 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246297 2017-12-11 22:50:43 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246297 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246325 https://lh3.googleusercontent.com/-I_fLwJCbLGs/VkWq4XJfmoI/AAAAAAAAAKI/Xrv3DIHujC8/w506-h750/Dec%2B19%252C%2B2014%2B7%253A38%253A17%2BAM.jpg%20]]> 2017-12-11 22:50:51 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246325 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246393 2017-12-11 22:51:16 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246393 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246399 2017-12-11 22:51:18 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246399 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246470 2017-12-11 22:51:52 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246470 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246567 2017-12-11 22:52:27 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246567 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246622 2017-12-11 22:52:51 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246622 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246631 2017-12-11 22:52:56 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246631 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246633 2017-12-11 22:52:57 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246633 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246639 2017-12-11 22:52:59 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246639 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246648 2017-12-11 22:53:02 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246648 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246652 2017-12-11 22:53:03 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246652 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246664 2017-12-11 22:53:09 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246664 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246666 2017-12-11 22:53:10 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246666 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246692 2017-12-11 22:53:20 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246692 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246708 2017-12-11 22:53:28 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246708 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246713 2017-12-11 22:53:30 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246713 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246721 2017-12-11 22:53:32 UTC https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246721 https://padlet.com/catherine_latham/nkndg2tg9ivc/wish/215246735 >> Additional resources online
STUART PALMER | DAVID GREENWOOD
BRYN HUMBERSTONE | JUSTIN ROBINSON
JENNY GOODMAN | JENNIFER VAUGHAN
NSW SYLLABUS FOR THE AUSTRALIAN CURRICULUM
CambridgeMATHS
YEAR7
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
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It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning and research at the highest international levels of excellence.
www.cambridge.edu.au
Information on this title: www.cambridge.org/9781107626973
© Stuart Palmer, David Greenwood, Bryn Humberstone,
Justin Robinson, Jenny Goodman, Jennifer Vaughan 2013
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
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First published 2013
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ISBN 978-1-107-62697-3 Paperback
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Reproduction and communication for educational purposes
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ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
Table of Contents
iii
1 Number and Algebra
Computation with Integers
MA4–4NA
Strand and substrand
2 Measurement and Geometry
Angle Relationships
MA4–18MG
About the authors viii
Introduction and guide to this book ix
Acknowledgements xii
Computation with positive integers 2
Pre-test 4
1A Place value in ancient number systems FRINGE 5
1B Place value in Hindu-Arabic numbers REVISION 10
1C Adding and subtracting positive integers 14
1D Algorithms for adding and subtracting 19
1E Multiplying small positive integers 23
1F Multiplying large positive integers 27
1G Dividing positive integers and dealing with remainders 31
1H Estimating and rounding positive integers 36
1I Order of operations with positive integers 40
Investigation 44
Puzzles and challenges 46
Review: Chapter summary 47
Multiple-choice questions 48
Short-answer questions 49
Extended-response questions 50
Angle relationships 52
Pre-test 54
2A Points, lines, intervals and angles 55
2B Measuring and classifying angles REVISION 60
2C Adjacent angles and vertically opposite angles 66
2D Transversal lines and parallel lines 71
2E Solving geometry problems 78
2F Circles and constructions with ruler and compasses FRINGE 83
2G Constructions with dynamic geometry software EXTENSION 87
Investigation 90
Puzzles and challenges 91
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
iv
3 Number and Algebra
Computation with Integers
MA4–4NA
4 Number and Algebra
Fractions, Decimals and
Percentages
MA4–5NA
Review: Chapter summary 92
Multiple-choice questions 93
Short-answer questions 94
Extended-response questions 97
Computation with positive and negative integers 98
Pre-test 100
3A Working with negative integers 101
3B Adding or subtracting a positive integer 105
3C Adding or subtracting a negative integer 109
3D Multiplying or dividing by an integer 114
3E Order of operations with positive and negative integers 118
3F The Cartesian plane 122
Investigation 126
Puzzles and challenges 127
Review: Chapter summary 128
Multiple-choice questions 129
Short-answer questions 130
Extended-response questions 131
Understanding fractions, decimals and percentages 132
Pre-test 134
4A Factors and multiples 135
4B Highest common factor and lowest common multiple 140
4C What are fractions? 144
4D Equivalent fractions and simplifi ed fractions 150
4E Mixed numerals and improper fractions 156
4F Ordering positive and negative fractions 162
4G Place value in decimals and ordering decimals 168
4H Rounding decimals 172
4I Decimal and fraction conversions 176
4J Connecting percentages with fractions and decimals 181
4K Decimal and percentage conversions 188
4L Fraction and percentage conversions 193
4M Percentage of a quantity 198
4N Using fractions and percentages to compare two quantities 203
Investigation 208
Puzzles and challenges 210
Review: Chapter summary 212
Multiple-choice questions 213
Short-answer questions 214
Extended-response questions 216
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
v
7 Measurement and Geometry
6 Number and Algebra
Fractions, Decimals and
Percentages
MA4–5NA
Time
MA4–13MG
5
Probability
MA4–21SP
Statistics Probability 218 and Probability
Pre-test 220
5A Describing probability 221
5B Theoretical probability in single-step experiments 228
5C Experimental probability in single-step experiments 233
5D Compound events in single-step experiments 238
5E Venn diagrams and two-way tables 243
5F Probability in two-step experiments EXTENSION 250
Investigation 255
Puzzles and challenges 257
Review: Chapter summary 259
Multiple-choice questions 260
Short-answer questions 261
Extended-response questions 262
Computation with decimals and fractions 264
Pre-test 266
6A Adding and subtracting decimals 267
6B Adding fractions 271
6C Subtracting fractions 278
6D Multiplying fractions 284
6E Multiplying and dividing decimals by 10, 100, 1000 etc. 292
6F Multiplying by a decimal 297
6G Dividing fractions 301
6H Dividing decimals 307
6I Computation with negative fractions EXTENSION 312
Investigation 318
Puzzles and challenges 321
Review: Chapter summary 322
Multiple-choice questions 323
Short-answer questions 323
Extended-response questions 325
Semester review 1 327
Time 336
Pre-test 338
7A Units of time 339
7B Working with time 344
7C Using time zones 349
Puzzles and challenges 355
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
vi
8 Number and Algebra
9 Number and Algebra
Review: Chapter summary 356
Multiple-choice questions 357
Short-answer questions 357
Extended-response question 359
Algebraic techniques 1 360
Pre-test 362
8A Introduction to formal algebra 363
8B Substituting positive numbers into algebraic expressions 368
8C Equivalent algebraic expressions 373
8D Like terms 377
8E Multiplying, dividing and mixed operations 381
8F Expanding brackets 385
8G Applying algebra EXTENSION 390
8H Substitution involving negative numbers and mixed operations 395
8I Number patterns EXTENSION 398
8J Spatial patterns EXTENSION 403
8K Tables and rules EXTENSION 411
8L The Cartesian plane and graphs EXTENSION 417
Investigation 424
Puzzles and challenges 426
Review: Chapter summary 427
Multiple-choice questions 428
Short-answer questions 429
Extended-response questions 430
Equations 1 432
Pre-test 434
9A Introduction to equations 435
9B Solving equations by inspection 440
9C Equivalent equations 444
9D Solving equations systematically 449
9E Equations with fractions 456
9F Equations with brackets 461
9G Formulas and relationships EXTENSION 465
9H Using equations to solve problems EXTENSION 469
Investigation 474
Puzzles and challenges 475
Review: Chapter summary 476
Multiple-choice questions 477
Short-answer questions 478
Extended-response questions 479
Algebraic Techniques
MA4–8NA
Equations
MA4–10NA
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
vii
10 Measurement and Geometry
11 Number and Algebra
Measurement and computation of length, perimeter and area 480
Pre-test 482
10A Measurement systems of the past and present FRINGE 483
10B Using and converting units of length REVISION 489
10C Perimeter of rectilinear fi gures 493
10D Pi and circumference of circles 498
10E Arc length and perimeter of sectors and composite fi gures 503
10F Units of area and area of rectangles 509
10G Area of triangles 515
10H Area of parallelograms 521
10I Area of composite fi gures 525
10J Mass and temperature REVISION 530
Investigation 536
Puzzles and challenges 537
Review: Chapter summary 538
Multiple-choice questions 539
Short-answer questions 540
Extended-response questions 543
Introducing indices 544
Pre-test 546
11A Divisibility tests 547
11B Prime numbers 552
11C Using indices 555
11D Prime decomposition 560
11E Squares, square roots, cubes and cube roots 564
11F The zero index and index laws 569
Investigation 574
Puzzles and challenges 575
Review: Chapter summary 577
Multiple-choice questions 578
Short-answer questions 579
Extended-response question 580
Semester review 2 581
Answers 589
Index 645
Length and Area
MA4–12MG, MA4–13MG
Indices
MA4–9NA
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
Table of Contents
viii
About the authors
Stuart Palmer was born and educated in NSW. He is a high school mathematics teacher
with more than 25 years’ experience teaching students from all walks of life in a variety of
schools. Stuart has taught all the current NSW Mathematics courses in Stages 4, 5 and 6
numerous times. He has been a head of department in two schools and is now an
educational consultant who conducts professional development workshops for teachers all
over NSW and beyond. He also works with pre-service teachers at The University of
Sydney and The University of Western Sydney.
David Greenwood is the Head of Mathematics at Trinity Grammar School in Melbourne
and has 19 years’ experience teaching mathematics from Years 7 to 12. He has run
numerous workshops within Australia and overseas regarding the implementation of the
Australian Curriculum and the use of technology for the teaching of mathematics. He
has written more than 20 mathematics titles and has a particular interest in the sequencing
of curriculum content and working with the Australian Curriculum profi ciency strands.
Bryn Humberstone graduated from University of Melbourne with an Honours degree in
Pure Mathematics, and is currently teaching both junior and senior mathematics in
Victoria. Bryn is particularly passionate about writing engaging mathematical
investigations and effective assessment tasks for students with a variety of backgrounds
and ability levels.
Justin Robinson is Head of Positive Education and a mathematics teacher at Geelong
Grammar School. Prior to this, he spent 20 years teaching mathematics and was a key writer
of in-house maths material. He has a keen interest in engaging all students through a wide
variety of effective teaching methods and materials.
Jenny Goodman has worked for 20 years in comprehensive State and selective high schools
in NSW and has a keen interest in teaching students of differing ability levels. She was
awarded the Jones medal for education at Sydney University and the Bourke prize for
Mathematics. She has written for Cambridge NSW and was involved in the Spectrum and
Spectrum Gold series.
Jennifer Vaughan has taught secondary mathematics for over 30 years in NSW, WA,
Queensland and New Zealand and has tutored and lectured in mathematics at Queensland
University of Technology. She is passionate about providing students of all ability levels with
opportunities to understand and to have success in using mathematics. She has taught special
needs students and has had extensive experience in developing resources that make
mathematical concepts more accessible; hence, facilitating student confi dence, achievement
and an enjoyment of maths.
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
Table of Contents
ix
Introduction and
guide to this book
This resource has been developed from an analysis of the NSW Syllabus for the Australian Curriculum
and the Australian Curriculum: Mathematics. It is structured on a detailed teaching program for the
implementation of the NSW Syllabus, and a comprehensive copy of the teaching program can be found
on the companion website.
The chapters are based on a logical teaching and learning sequence for the syllabus topic concerned,
so that chapter sections can be used as ready-prepared lessons. Exercises have questions graded by
level of diffi culty, as indicated in the teaching program, and are grouped according to the Working
Mathematically components of the NSW Syllabus, as indicated by badges in the margin of the exercises.
This facilitates the management of differentiated learning and reporting on students’ achievement.
For certain topics the prerequisite knowledge has been given in sections marked as REVISION,
whereas EXTENSION marks a few sections that go beyond the Syllabus. Similarly, the word FRINGE
is used to mark a few topics treated in a way that lies at the edge of the Syllabus requirements, but which
provide variety and stimulus. Apart from these, all topics are aligned exactly to the NSW Syllabus, as
indicated at the start of each chapter and in the teaching program.
In Stage 5, separate textbooks are provided for 5.1/5.2 and 5.1/5.2/5.3. In addition the NSW Syllabus
allocates topics to certain pathways, which are designated 5.2, 5.3§ and 5.3#. These categories, together
with Stage 4, 5.1, 5.2 and 5.3, are indicated for each chapter section by ‘ladder icons’ in the Year 9 and 10
textbooks and teaching programs.
Guide to this book
Features:
NSW Syllabus for the Australian
Curriculum: strands, substrands and
content outcomes for chapter (see
teaching program for more detail)
What you will learn: an
overview of chapter contents
Chapter introduction: use
to set a context for students
Chapter
Computation with
positive integers 1
What you will learn
Place value in ancient number systems
Place value in Hindu-Arabic numbers
Adding and subtracting positive integers
Algorithms for adding and subtracting
Multiplying small positive integers
Multiplying large positive integers
Dividing positive integers and dealing with remainders
Estimating and rounding positive integers
Order of operations with positive integers
1A
1B
1C
1D
1E
1F
1G
1H
1I
Whole numbers in the ancient
world and now
Whole numbers and number systems have been used
for thousands of years to help count objects and
record information.
In ancient Egypt in about 3000 BCE, if one
hundred and twenty-four blocks of stone were needed
to build a monument, this number would have been
represented as
Today, we use whole numbers to help deal with all
sorts of situations. Here are some examples:
• Recording the number of points in a game
• Calculating the number of pavers required for a
terrace
• Counting the number of items purchased at a shop
• Tallying the number of votes in an election
• Calculating the approximate distance between
two towns.
NSW Syllabus
for the Australian
Curriculum
Strand: Number and Algebra
Substrand: COMPUTATION WITH INTEGERS
Outcome: A student compares, orders,
and calculates with integers, applying a
range of strategies to aid computation
(MA4 — 4NA)
3
Pre-test: establishes prior knowledge
(also available as a printable worksheet)
Chapter 1 Computation with 4 positive integers
1 For each of the following, match the word with the symbol.
a add A −
b subtract B ÷
c multiply C +
d divide D ×
2 Write each of the following as numbers.
a fifty-seven b one hundred and sixteen
c two thousand and forty-four d eleven thousand and two
3 Answer which number is:
a 2 more than 11 b 5 less than 42
c 1 less than 1000 d 3 more than 7997
Pre-test
ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
© David Greenwood et al. 2013 Cambridge University Press
x
Guide to this book (continued)
Topic introduction: use to relate the topic
to mathematics in the wider world
HOTmaths icons: links to interactive
online content via the topic number,
1C in this case (see next page for more)
Let’s start: an activity (which can
often be done in groups) to start the lesson
Key ideas: summarises the knowledge and
skills for the lesson
Examples: solutions with explanations and
descriptive titles to aid searches (digital
versions also available for use with IWB)
Exercise questions categorised by
the working mathematically
components and enrichment
(see next page)
Questions are linked to examples
Investigations:
inquiry-based activities Puzzles and challenges
Chapter summary: mind map of
key concepts & interconnections 2 Semester reviews per book
Chapter reviews with multiple-choice, short-answer and extended-response questions
Chapter 1 Computation with 14 positive integers
Adding and subtracting positive integers
The process of finding the total value of two or more numbers is called
addition. The words ‘plus’, ‘add’ and ‘sum’ are also used to describe
addition.
The process for finding the difference between two numbers is called
subtraction. The words ‘minus’, ‘subtract’ and ‘take away’ are also used
to describe subtraction.
Let’s start: Your mental strategy
Many problems that involve addition and subtraction can be solved
mentally without the use of a calculator or complicated written working.
Consider 98 + 22 − 31 + 29
How would you work this out? What are the different ways it could be
done mentally? Explain your method.
1C
What’s the difference in our heights?
■■ The symbol + is used to show addition or find a sum.
e.g. 4 + 3 = 7
■■ a + b = b + a e.g. 4 + 3 = 3 + 4
– This is the commutative law for addition, meaning that
the order does not matter.
■■ a + (b + c) = (a + b) + c e.g. 4 + (11 + 3) = (4 + 11) + 3
– This is the associative law for addition, meaning it does not
matter which pair is added first.
■■ a − (b − c) ≠ (a − b) − c e.g. 8 − (4 − 2) = (8 − 2) − 2
■■ The symbol − is used to show subtraction or find a difference.
e.g. 7 − 2 = 5
■■ a − b ≠ b − a (in general...) e.g. 4 − 3 ≠ 3 − 4
■■ Mental addition and subtraction can be done using different strategies.
– Partitioning (Grouping digits in the same position)
171 + 23 = 100 + (70 + 20) + (1 + 3)
= 194
– Compensating (Making a 10, 100 etc. and then adjusting or compensating by adding or
subtracting)
46 + 9 = 46 + 10 − 1
= 55
– Doubling or halving (Making a double or half and then adjusting with addition or
subtraction)
75 + 78 = 75 + 75 + 3 124 − 61 = 124 − 62 + 1
= 150 + 3 = 62 + 1
= 153 = 63
5 6 7
+3
3 4 8
6 7
−2
4 5 8
6 7
−2
4 5 8
Key ideas
Stage
5.3#
5.3
5.3§
5.2
5.2◊
5.1
4
EMAC_NSW_7_txtdesign.indd 14 19/02/13 6:23 PM
14 Chapter 1 Computation with positive integers
Adding and subtracting positive integers
The process of finding the total value of two or more numbers is called
addition. The words ‘plus’, ‘add’ and ‘sum’ are also used to describe
addition.
The process for finding the difference between two numbers is called
subtraction. The words ‘minus’, ‘subtract’ and ‘take away’ are also used
to describe subtraction.
Let’s start: Your mental strategy
Many problems that involve addition and subtraction can be solved
mentally without the use of a calculator or complicated written working.
Consider 98 + 22 − 31 + 29
How would you work this out? What are the different ways it could be
done mentally? Explain your method.
1C
What’s the difference in our heights?
■■ The symbol + is used to show addition or find a sum.
e.g. 4 + 3 = 7
■■ a + b = b + a e.g. 4 + 3 = 3 + 4
– This is the commutative law for addition, meaning that
the order does not matter.
■■ a + (b + c) = (a + b) + c e.g. 4 + (11 + 3) = (4 + 11) + 3
– This is the associative law for addition, meaning it does not
matter which pair is added first.
■■ a − (b − c) ≠ (a − b) − c e.g. 8 − (4 − 2) = (8 − 2) − 2
■■ The symbol − is used to show subtraction or find a difference.
e.g. 7 − 2 = 5
■■ a − b ≠ b − a (in general...) e.g. 4 − 3 ≠ 3 − 4
■■ Mental addition and subtraction can be done using different strategies.
– Partitioning (Grouping digits in the same position)
171 + 23 = 100 + (70 + 20) + (1 + 3)
= 194
– Compensating (Making a 10, 100 etc. and then adjusting or compensating by adding or
subtracting)
46 + 9 = 46 + 10 − 1
= 55
– Doubling or halving (Making a double or half and then adjusting with addition or
subtraction)
75 + 78 = 75 + 75 + 3 124 − 61 = 124 − 62 + 1
= 150 + 3 = 62 + 1
= 153 = 63
5 6 7
+3
3 4 8
6 7
−2
4 5 8
6 7
−2
4 5 8
Key ideas
Stage
5.3#
5.3
5.3§
5.2
5.2◊
5.1
4
EMAC_NSW_7_txtdesign.indd 14 19/02/13 6:23 PM
Number and Algebra 15
Example 4 Mental addition and subtraction
Use the suggested strategy to mentally work out the answer.
a 132 + 156 (partitioning) b 25 + 19 (compensating)
c 56 – 18 (compensating) d 35 + 36 (doubling or halving)
SOLUTION EXPLANATION
a 132 + 156 = 288 100 30 2
100 50 6
200 80 8
+ +
+ +
+ +
b 25 + 19 = 44 25 + 19 = 25 + 20 – 1
= 45 – 1
= 44
c 56 – 18 = 38 56 – 18 = 56 – 20 + 2
= 36 + 2
= 38
d 35 + 36 = 71 35 + 36 = 35 + 35 + 1
= 70 + 1
= 71
1 a List three words that mean addition.
b List three words that mean subtraction.
2 Write the number which is:
a 3 more than 7 b 58 more than 11
c 7 less than 19 d 137 less than 157
3 a Find the sum of these pairs of numbers.
i 2 and 6 ii 19 and 8 iii 62 and 70
b Find the difference between these pairs of numbers.
i 11 and 5 ii 29 and 13 iii 101 and 93
4 State whether each of these statements is true or false.
a 4 + 3 > 6 b 11 + 19 ≥ 30 c 13 − 9 < 8
d 26 − 15 ≤ 10 e 1 + 7 − 4 ≥ 4 f 50 − 21 + 6 < 35
5 Give the result for each of these computations.
a 7 plus 11 b 22 minus 3 c the sum of 11 and 21
d 128 add 12 e 36 take away 15
f the difference between 13 and 4
Exercise 1C
WORKING
MATHEMATICALLY
U F
R PS
C
16 Chapter 1 Computation with positive integers 1C
6 Mentally find the answers to these sums. Hint: Use the partitioning strategy.
a 23 + 41 b 71 + 26 c 138 + 441
d 246 + 502 e 937 + 11 f 1304 + 4293
g 140 273 + 238 410 h 390 447 + 201 132 i 100 001 + 101 010
7 Mentally find the answers to these differences. Hint: Use the partitioning strategy.
a 29 − 18 b 57 − 21 c 249 − 137
d 1045 − 1041 e 4396 − 1285 f 10 101 − 100
8 Mentally find the answers to these sums. Hint: Use the compensating strategy.
a 15 + 9 b 64 + 11 c 19 + 76
d 18 + 115 e 31 + 136 f 245 + 52
9 Mentally find the answers to these differences. Hint: Use the compensating strategy.
a 35 − 11 b 45 − 19 c 156 − 48
d 244 − 22 e 376 − 59 f 5216 − 199
10 Mentally find the answers to these sums and differences. Hint: Use the doubling or
halving strategy.
a 25 + 26 b 65 + 63 c 121 + 123
d 240 − 121 e 482 − 240 f 1006 − 504
11 Mentally find the answers to these computations.
a 11 + 18 − 17 b 37 − 19 + 9 c 101 − 15 + 21
d 136 + 12 − 15 e 28 − 10 − 9 + 5 f 39 + 71 − 10 − 10
g 1010 − 11 + 21 − 1 h 5 − 7 + 2 i 10 − 25 + 18
13 Gary worked 7 hours on Monday, 5 hours on Tuesday, 13 hours on Wednesday, 11 hours on
Thursday and 2 hours on Friday. What is the total number of hours that Gary worked during the
week?
14 In a batting innings, Phil hit 126
runs and Mario hit 19 runs. How
many more runs did Phil hit
compared to Mario?
Example 4a
Example 4b
Example 4c
Example 4d
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
16 Chapter 1 Computation with positive integers 1C
6 Mentally find the answers to these sums. Hint: Use the partitioning strategy.
a 23 + 41 b 71 + 26 c 138 + 441
d 246 + 502 e 937 + 11 f 1304 + 4293
g 140 273 + 238 410 h 390 447 + 201 132 i 100 001 + 101 010
7 Mentally find the answers to these differences. Hint: Use the partitioning strategy.
a 29 − 18 b 57 − 21 c 249 − 137
d 1045 − 1041 e 4396 − 1285 f 10 101 − 100
8 Mentally find the answers to these sums. Hint: Use the compensating strategy.
a 15 + 9 b 64 + 11 c 19 + 76
d 18 + 115 e 31 + 136 f 245 + 52
9 Mentally find the answers to these differences. Hint: Use the compensating strategy.
a 35 − 11 b 45 − 19 c 156 − 48
d 244 − 22 e 376 − 59 f 5216 − 199
10 Mentally find the answers to these sums and differences. Hint: Use the doubling or
halving strategy.
a 25 + 26 b 65 + 63 c 121 + 123
d 240 − 121 e 482 − 240 f 1006 − 504
11 Mentally find the answers to these computations.
a 11 + 18 − 17 b 37 − 19 + 9 c 101 − 15 + 21
d 136 + 12 − 15 e 28 − 10 − 9 + 5 f 39 + 71 − 10 − 10
g 1010 − 11 + 21 − 1 h 5 − 7 + 2 i 10 − 25 + 18
13 Gary worked 7 hours on Monday, 5 hours on Tuesday, 13 hours on Wednesday, 11 hours on
Thursday and 2 hours on Friday. What is the total number of hours that Gary worked during the
week?
14 In a batting innings, Phil hit 126
runs and Mario hit 19 runs. How
many more runs did Phil hit
compared to Mario?
Example 4a
Example 4b
Example 4c
Example 4d
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
18 Chapter 1 Computation with positive integers 1DC
19 a The mental strategy of partitioning is easy to apply for 23 + 54 but harder for 23 + 59.
Explain why.
b The mental strategy of partitioning is easy to apply for 158 − 46 but harder for 151 − 46.
Explain why.
20 Complete these number sentences if the letters a, b and c represent numbers.
a a + b = c so c − __ = a b a + c = b so b − a = __
21 This magic triangle uses the digits 1 to 6, and has each side adding to the same
total. This example shows a side total of 9.
a How many other different side totals are possible using the same digits?
b Explain your method.
Enrichment: Magic squares
22 A magic square has every row, column and main diagonal adding to the
same number, called the magic sum. For example, this magic square
has a magic sum of 15.
Find the magic sums for these squares, then fill in the
missing numbers.
a
6
7 5
2
b
10
11 13
12
c
15 20
14
19
d
1 15 4
6 9
11
13 2 16
23 The sum of two numbers is 87 and their difference is 29. What are the two numbers?
6 5
2 4 3
1
4 9 2 15
3 5 7 15
8 1 6 15
15 15 15 15 15
This magic square was known in ancient China as a ‘Lo Shu’ square and uses only the numbers 1 to 9. It is shown in
the middle of this ancient design as symbols on a turtle shell, surrounded by the animals which represent the traditional
Chinese names for the years.
WORKING
MATHEMATICALLY
U F
R PS
C
44 Chapter 1 Computation with positive integers
The abacus
The abacus is a counting device that has been used for thousands
of years. They were used extensively by merchants, traders, tax
collectors and clerks before modern-day numerals systems were
developed. Counting boards called Abax date back to 500 BCE.
These were wood or stone tablets with grooves, which would hold
beans or pebbles.
The modern abacus is said to have originated in China in about
the 13th century and includes beads on wires held in a wooden
frame.
A modern abacus with thirteen wires
There are 5 beads on one side of a modern abacus
worth 1 each and 2 beads on the opposite side worth
5 each.
■■ Each wire represents a different unit, e.g. ones,
tens, hundreds etc.
■■ Beads are counted only when they are pushed
toward the centre.
ones
tens
hundreds
thousands
Here is a diagram showing the number 5716.
Investigation
A German woodcut from 1508 showing an abacus in use by gentleman on right,
while a mathematician (at left) writes algorithms.
EMAC_NSW_7_txtdesign.indd 44 19/02/13 6:23 PM
46 Chapter 1 Computation with positive integers
Puzzles and challenges
1 The extra dollar. The cost of dinner for two people is $45 and they both give the waiter $25 each.
Of the extra $5 the waiter is allowed to keep $3 as a tip and returns $1 to each person.
So the two people paid $24 each, making a total of $48, and the waiter has $3. The total is therefore
$48 + $3 = $51. Where did the extra $1 come from?
2 The sum along each line is 15. Can you place each of the digits 1, 2, 3, 4, 5,
6, 7, 8 and 9 to make this true?
3 The sum along each side of this triangle is 17. Can you place each
of the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make this true?
4 Make the total of 100 out of all the numbers 2, 3, 4, 7 and 11, using each number only once. You
can use any of the operations (+, –, ×, ÷), as well as brackets.
5 Sudoku is a popular logic number puzzle made up of a 9 by 9 square, where each column and row
can use the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once. Also, each digit is to be used only once in
each 3 by 3 square. Solve these puzzles.
7 6 9 3
4 1 8 7
8 2 9 1
3 1
2 8 5 3
5 6 9 2
3 9 5
6 8 4
5 9 7
4 2 8 7
2 8 7 9 1
6 3 5
3 7 2 8
6 5 4 7 8 2
2 6
7 5
8 3 9 2 7
EMAC_NSW_7_txtdesign.indd 46 19/02/13 6:23 PM
Number and Algebra 47
Chapter summary
Order of Operations
Brackets first, then ×
and ÷, then + and –
from left to right.
2 + 3 × 4 ÷ (9 ÷ 3)
= 2 + 12 ÷ 3
= 2 + 4
= 6
Place value
Addition and
Subtraction
Ancient
Number Systems
Estimation
Roman
955 to the nearest 10 is 960
950 to the nearest 100 is 1000
371
_+_ _8_43_
1214
29
× 13 ____
87
__2_9_0
377
937
_–_ _6_4_3
294
68
3 205
with 1
remainder
Leading digit approximation
39 × 326 ≈ 40 × 300 = 12 000
Mental strategies
172 + 216 = 300 + 80 + 8
= 388
98 – 19 = 98 – 20 + 1
= 79
Mental strategies
7 × 31 = 7 × 30 + 7 × 1 = 217
5 × 14 = 10 × 7 = 70
64 ÷ 8 = 32 ÷ 4 = 16 ÷ 2 = 8
156 ÷ 4 = 160 ÷ 4 – 4 ÷ 4
= 40 – 1
= 39
Multiplying by 10, 100, ……
38 × 100 = 3800
38 × 700 = 38 × 7 × 100
= 26 600
The place value of 3 in
1327 is 300.
2 × 100 + 7 × 10 + 3 × 1
is the expanded form of 273.
Algorithms
Algorithms
Multiplication
and Division
Whole numbers
Egyptian
Babylonian
is 71
is 23
1 1
2
2
8
LXXVI is 76
XCIV is 94
is 143
is 21
EMAC_NSW_7_txtdesign.indd 47 19/02/13 6:23 PM
48 Chapter 1 Computation with positive integers
Multiple-choice questions
1 The correct Roman numerals for the number 24 is:
A XXIII B XXIV C XXXLIV
D IVXX E IXXV
2 3 × 1000 + 9 × 10 + 2 × 1 is the expanded form of:
A 3920 B 392 C 3092
D 3902 E 329
3 Which of the following is not true?
A 2 + 3 = 3 + 2 B 2 × 3 = 3 × 2 C (2 × 3) × 4 = 2 × (3 × 4)
D 5 ÷ 2 ≠ 2 ÷ 5 E 7 − 2 = 2 − 7
4 The sum of 198 and 103 is:
A 301 B 304 C 299
D 199 E 95
5 The difference between 378 and 81 is:
A 459 B 297 C 303
D 317 E 299
6 The product of 7 and 21 is:
A 147 B 141 C 21
D 140 E 207
7 The missing digit in this division is:
A 6 B 1 C 9
D 8 E 7
8 The remainder when 317 is divided by 9 is:
A 7 B 5 C 2
D 1 E 0
9 458 rounded to the nearest 100 is:
A 400 B 500 C 460
D 450 E 1000
10 The answer to [2 + 3 × (7 – 4)] ÷ 11 is:
A 1 B 5 C 11
D 121 E 0
7) 2 6
1 1 8
1 5
EMAC_NSW_7_txtdesign.indd 48 19/02/13 6:23 PM
Semester review 1
350
Semester review 1
Whole numbers
Multiple-choice questions
1 Using numerals, thirty-fi ve thousand, two hundred and six is:
A 350 260 B 35 260 C 35 000 206 D 3526 E 35 206
2 The place value of 8 in 2 581 093 is:
A 8 thousand B 80 thousand C 8 hundred D 8 tens E 8 ones
3 The remainder when 23 650 is divided by 4 is:
A 0 B 4 C 1 D 2 E 3
4 18 − 3 × 4 + 5 simplifi es to:
A 65 B 135 C 11 D 1 E 20
5 800 ÷ 5 × 4 is the same as:
A 160 × 4 B 800 ÷ 20 C 800 ÷ 4 × 5 D 40 E 4 × 5 ÷ 800
Short-answer questions
1 Write the number seventy-four in:
a Babylonian numerals
b Roman numerals
c Egyptian numerals
2 Write the numeral for:
a 6 × 10 000 + 7 × 1000 + 8 × 100 + 4 × 10 + 9 × 1
b 7 × 100 000 + 8 × 100 + 5 × 10
3 Calculate:
a 96 481 + 2760 + 82 b 10 963 − 4096 c 147 × 3
d 980 × 200 e 4932 ÷ 3 f 9177 ÷ 12
4 State whether each of the following is true or false.
a 18 < 20 − 2 × 3 b 9 × 6 > 45 c 23 = 40 ÷ 2 + 3
5 How much more than 17 × 18 is 18 × 19?
6 Calculate:
a 7 × 6 − 4 × 3 b 8 × 8 − 16 ÷ 2 c 12 × (6 − 2)
d 16 × [14 − (6 − 2)] e 24 ÷ 6 × 4 f 56 − (7 − 5) × 7
7 State whether each of the following is true or false.
a 4 × 25 × 0 = 1000 b 0 ÷ 10 = 0 c 8 ÷ 0 = 0
d 8 × 7 = 7 × 8 e 20 ÷ 4 = 20 ÷ 2 ÷ 2 f 8 + 5 + 4 = 8 + 9
8 Insert brackets to make 18 × 7 + 3 = 18 × 7 + 18 × 3 true.
9 How many times can 15 be subtracted from 135 before an answer of zero occurs?
CUAU093-SR-1.indd 350 9/12/10 12:07:22 PM
Textbooks also include:
 Complete answers
 Index
 Using technology
activities
18 Chapter 1 Computation with positive integers 1C
18 a The mental strategy of partitioning is easy to apply for 23 + 54 but harder for 23 + 59.
Explain why.
b The mental strategy of partitioning is easy to apply for 158 − 46 but harder for 151 − 46.
Explain why.
19 Complete these number sentences if the letters a, b and c represent numbers.
a a + b = c so c − __ = a b a + c = b so b − a = __
20 This magic triangle uses the digits 1 to 6, and has each side adding to the same
total. This example shows a side total of 9.
a How many other different side totals are possible using the same digits?
b Explain your method.
Enrichment: Magic squares
21 A magic square has every row, column and main diagonal adding to the
same number, called the magic sum. For example, this magic square
has a magic sum of 15.
Find the magic sums for these squares, then fi ll in the
missing numbers.
a
6
7 5
2
b
10
11 13
12
c
15 20
14
19
d
1 15 4
6 9
11
13 2 16
22 The sum of two numbers is 87 and their difference is 29. What are the two numbers?
6 5
2 4 3
1
4 9 2 15
3 5 7 15
8 1 6 15
15 15 15 15 15
This magic square was known in ancient China as a ‘Lo Shu’ square and uses only the numbers 1 to 9. It is shown in
the middle of this ancient design as symbols on a turtle shell, surrounded by the animals which represent the traditional
Chinese names for the years.
WORKING
MATHEMATICALLY
U F
R PS
C Algorithms It is not always practical complex problems Such a procedure is For the addition sum can be carried For the subtraction the next column to let’s start: The Discuss what numbers Give reasons for your 97805211XXXXXc1_p02-51.indd 18 4/10/13 2:34 PM
Number and Algebra Example 4 Mental addition and subtraction
Use the suggested strategy to mentally work out the answer.
a 132 + 156 (partitioning) b 25 + 19 (compensating)
c 56 – 18 (compensating) d 35 + 36 (doubling or halving)
SOLUTION EXPLANATION
a 132 + 156 = 288 100 30 2
100 50 6
200 80 8
+ +
+ +
+ +
b 25 + 19 = 44 25 + 19 = 25 + 20 – 1
= 45 – 1
= 44
c 56 – 18 = 38 56 – 18 = 56 – 20 + 2
= 36 + 2
= 38
d 35 + 36 = 71 35 + 36 = 35 + 35 + 1
= 70 + 1
= 71
1 a List three words that mean addition.
b List three words that mean subtraction.
2 Write the number which is:
a 3 more than 7 b 58 more than 11
c 7 less than 19 d 137 less than 157
3 a Find the sum of these pairs of numbers.
i 2 and 6 ii 19 and 8 iii 62 and 70
b Find the difference between these pairs of numbers.
i 11 and 5 ii 29 and 13 iii 101 and 93
4 State whether each of these statements is true or false.
a 4 + 3 > 6 b 11 + 19 ≥ 30 c 13 − 9 < 8
d 26 − 15 ≤ 10 e 1 + 7 − 4 ≥ 4 f 50 − 21 + 6 < 35
5 Give the result for each of these computations.
a 7 plus 11 b 22 minus 3 c the sum of 11 and 21
d 128 add 12 e 36 take away 15
f the difference between 13 and 4
Exercise 1C
WORKING
MATHEMATICALLY
U F
R PS
C
EMAC_NSW_7_txtdesign.indd 15 19/02/13 ISBN: 9781107626973
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Working Mathematically badges
All exercises are divided into sections marked by Working Mathematically badges, such as this example:
Understanding &
Communicating
WORKING
MATHEMATICALLY
U F
R PS
C
Fluency &
Communicating
WORKING
MATHEMATICALLY
U F
R PS
C
Problem-solving &
Communicating
WORKING
MATHEMATICALLY
U F
R PS
C
Reasoning &
Communicating
WORKING
MATHEMATICALLY
U F
R PS
C
The letters U (Understanding), F (Fluency), PS (Problem-solving), R (Reasoning) and C (Communication) are
highlighted in colour to indicate which of these components apply mainly to the questions in that section. Naturally
there is some overlap between the components.
Additional resources online
TM HOTmaths interactive maths online
The CambridgeMATHS/HOTmaths integrated program for the Australian Curriculum offers the
best of textbook and interactive online resources. It can be used with a HOTmaths account
(eg for class demos via IWBs or data projectors) or with student accounts, which enable
access to the full range of features, including use at home.
The integrated program is linked from icons and the topic
numbers in the textbooks, as detailed in a document available
(free) from the Cambridge website.
Interactive
simulations &
programs
Practice quizzes with competitive scoring option
Tests with learning management
system
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content
Interactive
examples
Worksheets
(also available as
workbooks)
All HOTmaths features are included in the program, eg:
xi
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Table of Contents
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Acknowledgements
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Victorovich, p.316 / Rafa Irusta, p.318(l) / BZH22, p.318(r) / Lisa S., p.319 / panyajam patong, p.325 /
Daboost, p.326 / VLADJ55, pp.336–337 / Khakimullin Aleksandr, p.338 / Valerie Potapova, p.339 /
Tupungato, pp.342, 470 / Daniel M. Silva, p.343(r) / alessandro0770, p.343(l) / Lev Kropotov, p.344(t) /
sculpies, p.344(b) / Gordon Bell, p.347(t) / alysta, p.347(c) / Matt Gibson, p.349(t) / SF photo, p.349 (c) /
beboy, p.349(b) / Ahmad Faizal Yahya, p.354 / Patrick Foto, p.355(t&b) / WDG Photo, p.358 / Blaz
Kure, pp.360–361 / jgp-nyc, p.363 / Marc Dietrich, p.366 / Edw, p.367 / Vlue, p.372 / Gary Paul Lewis,
p.380 / Picsfi ve, p.384 / Olga Gabay, p.385 / Serg64, p.389 / Stavchansky Yakov, p.392(t) / Lisa F. Young,
p.392(b) / Elena Elisseeva, p.394 / Steven Coling, p.397 / visi.stock, p.398(l) / Alexander Raths, p.398(t-r) /
Andresr, pp.401, 443 / V.J. Matthew, p.410 / Malysh Falko, p.411 / S.john, p.413 / Rob Marmion, p.417 /
Sinitar, p.415 / Lars Hallstrom, p.422 / Weldon Scloneger, p.424 / Michael William, p.430 / yampi, p.431 /
Jaroslav Machacek, p.435 / terekhov igor, p.438 / MaszaS, p.440 / Marko Poplasen, p.442 / anweber,
p.448 / Darren Whitt, p.455 / Serghei Starus, p.459 / hfng, p.465 / vblinov, p.469 / theromb, p.473 /
Katherine Welles, pp.480-481 / 7505811966, p.486(t) / RCPPHOTO, p.486(a) / Evgeniy Ayupov, p.486(b) /
Ashley Whitworth, p.486(c) / Brad Thomson, p.486(d) / Nicola Gavin, p.486(e) / Péter Gudella, p.486(f) /
stephen mulcahey, p.488 / minik, p.489 / Noel Powell Schaumburg, p.492 / kwest, p.493 / Bork, p.501 /
aopsan, p.503 / Carlos Neto, p.580 / Terrance Emerson, p.509 / Fedorov Oleksiy, p.514 / haak78, p.515 /
C Berry Ottaway, p.518 / Salvador Garcia Gil, p.519 / Ljupco Smokovski, p.524 / Helen and Vlad Filatov,
p.525 / Gyukli Gyula, p.530(t) / Sebastian Duda, p.530(b) / Supri Suharjoto, p.533 / Andraž Cerar,
p.534(b) / darkgreenwolf, p.535 / Joern, p.536 / Tim Roberts Photography, p.542 / spirit of america,
p.543(t) / Edyta Pawlowska, p.543(b) / PL Petr Lerch, pp.544–545 / Konstantin Sutyagin, p.551 /
forestpath, p.552 / Ali Ender Birer, p.559 / fotohunter, p.563 / Marijus Auruskevicius, p.564 / Ryan Carter,
p.568 / Kuzma, p.572 / emin kuliyev, p.573 / Eric Isselée, p.574(t) / Morena Valente, p.575(4&9) / Anna
Tsekhmister, p.575(6); Wikimedia Commons. Public domain, pp.10, 18, 44(t), 101, 122, 439.
All curriculum material taken from NSW Mathematics 7-10 Syllabus © Board of Studies NSW for and on
behalf of the Crown in right of the State of New South Wales, 2012.
Every effort has been made to trace and acknowledge copyright. The publisher apologises for any
accidental infringement and welcomes information that would redress this situation.
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Chapter
Computation with
positive integers 1
What you will learn
Place value in ancient number systems FRINGE
Place value in Hindu-Arabic numbers REVISION
Adding and subtracting positive integers
Algorithms for adding and subtracting
Multiplying small positive integers
Multiplying large positive integers
Dividing positive integers and dealing with remainders
Estimating and rounding positive integers
Order of operations with positive integers
1A
1B
1C
1D
1E
1F
1G
1H
1I
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Whole numbers in the ancient
world and now
Whole numbers and number systems have been used
for thousands of years to help count objects and
record information.
In ancient Egypt in about 3000 bc, if one hundred
and twenty-four blocks of stone were needed to
build a monument, this number would have been
represented as
Today, we use whole numbers to help deal with all
sorts of situations. Here are some examples:
• Recording the number of points in a game
• Calculating the number of pavers required for a
terrace
• Counting the number of items purchased at a shop
• Tallying the number of votes in an election
• Calculating the approximate distance between
two towns.
NSW Syllabus
for the Australian
Curriculum
Strand: Number and Algebra
Substrand: COMputAtiON WitH iNtEGERS
Outcome
A student compares, orders and calculates
with integers, applying a range of
strategies to aid computation.
(MA4–4NA)
3
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Chapter 1 Computation with 4 positive integers
1 For each of the following, match the word with the symbol.
a add A −
b subtract B ÷
c multiply C +
d divide D ×
2 Write each of the following as numbers.
a fi fty-seven b one hundred and sixteen
c two thousand and forty-four d eleven thousand and two
3 Answer which number is:
a 2 more than 11 b 5 less than 42
c 1 less than 1000 d 3 more than 7997
e double 13 f half of 56
4 Complete these patterns, showing the next seven numbers.
a 7, 14, 21, 28, 35, __, __, __, __, __, __, __.
b 9, 18, 27, 36, 45, __, __, __, __, __, __, __.
c 11, 22, 33, 44, 55, __, __, __, __, __, __, __.
5 How many equal groups can 48 be divided into if each group is of the given size?
a 24 b 48 c 16 d 4
6 Give the result for each of these simple computations.
a 3 + 11 b 14 + 9 c 99 + 20 d 138 + 12
e 20 − 11 f 41 − 9 g 96 − 17 h 101 − 22
i 5 × 6 j 9 × 7 k 12 × 12 l 8 × 11
m 10 ÷ 2 n 30 ÷ 15 o 66 ÷ 6 p 132 ÷ 12
7 Arrange these numbers from smallest to largest.
a 37, 73, 58, 59, 62, 159
b 301, 103, 31, 310, 130
c 29 143, 24 913, 13 429, 24 319, 24 931
8 State the remainder when 3 is divided into these numbers.
a 12 b 10 c 37 d 62
e 130 f 263 g 1001 h 2704
pre-test
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Number and Algebra 5
place value in ancient number systems
Throughout the ages and in different countries, number systems
were developed and used to help people count and communicate
with numbers. From the ancient Egyptians to the modern day,
different systems have used pictures and symbols to represent
whole numbers. Some of the well-known number systems are
the Egyptian, Babylonian, Roman, modern Chinese and the
Hindu-Arabic or decimal system.
let’s start: Count like a Roman
Here are the letters used in the Roman number system for some
numbers that you know.
Number 1 2 3 4 5 6 7 8 9 10 50 100
Roman numerals I II III IV V VI VII VIII IX X L C
Note: On some watches and clocks IIII is used to represent the number 4.
• What numbers do you think XVII and XIX represent?
• Can you write the numbers 261 and 139 using Roman numerals?
1A
The Roman numerals on this stone show
the distance to a town.
F R I N G E
■■ Egyptian number system
– Records show that this number system was used from about 3000 bc.
– Hieroglyphics were used to represent numbers.
– From about 1600 bc hieroglyphics were used to represent groups of 10, 100, 1000 etc.
– Symbols of the same type were grouped in twos or threes and arranged vertically.
Number 1 10 100 1000 10 000 100 000 1 000 000
Hieroglyphic
Description stick or
staff
arch or
heel bone
coil of
rope
lotus
fl ower
bent fi nger
or reed
tadpole or
frog
genie
– Examples:
3 5 21 342
■■ Note that the hieroglyphic with the larger value is written in front (i.e. on the left).
■■ This number system did not have a symbol for 0.
■■ This number system did not use place value the way we do today.
Key ideas
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Chapter 1 Computation with 6 positive integers
■■ Babylonian number system
–– From about 1750 bc the ancient
Babylonians used a very
sophisticated number system and
its origins have been traced to
about 3000 bc.
–– Symbols called cuneiform
(wedge shapes) were used to
represent numbers.
–– The symbols were written into
clay tablets, which were then
allowed to dry in the Sun.
–– The number system is based on the number 60, but a different wedge shape was used to
represent groups of 10.
–– The system is positional in that the
position of each wedge shape helps
determine its value. So means
2 but means 62.
–– To represent zero, they used a blank
space or sometimes a small slanted
wedge shape for zeros inside a
number.
–– Examples:
5 11 72 121
■■ Roman number system
–– Some capital letters are used and are called Roman numerals.
–– The Roman number system was developed in about the third century bc and remained the
dominant system in many parts of the world until about the Middle Ages. It is still used today
in many situations.
–– A smaller letter value to the left of a larger letter value indicates subtraction.
e.g. IV means 5 − 1 = 4 and XC means 100 − 10 = 90. Only one letter can be placed to the left
for subtraction. Only the letter I can be placed before V or X.
For example, IV = 4 and IX = 9 but 99 ≠ IC.
The number 99 = 90 + 9 = XCIX.
Number 1 5 10 50 100 500 1000
Symbol I V X L C D M
–– Examples:
2 4 21 59 90
II IV or IIII XXI LIX XC
The Hanging Gardens of Babylon, built for his wife by King
Nebuchadnezzar II around 600 bc, were one of the seven wonders of
the ancient world.
Number 1 10 60
Symbol
Description
upright
wedge shape
sideways
wedge
upright
wedge shape
Key ideas
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Number and Algebra 7
Example 1 using ancient number systems
Write each of the numbers 3, 15 and 144 using the given number systems.
a Egyptian b Babylonian c Roman
SOlutiON ExplANAtiON
a 3
15
144
means 1
means 10
means 100
b 3
15
144
means 1
means 10
means 60
c 3 III
15 XV
144 CXLIV
I means 1
V means 5
X means 10
C means 100
XL means 40
IV means 4
1 Which number system uses these symbols?
a cuneiform (wedge shapes); e.g. b capital letters; e.g. V and L
c hieroglyphics (pictures); e.g. and
2 Draw the symbols used in these number systems for the given numbers.
a Egyptian
i 1 ii 10 iii 100 iv 1000
b Babylonian
i 1 ii 10 iii 60
c Roman
i 1 ii 5 iii 10 iv 50 v 100
3 In the Roman system, IV does not mean 1 + 5 to give 6. What do you think it means?
F R I N G E
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Exercise 1A F R I N G E
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Chapter 1 Computation with 8 positive integers
4 Write these numbers using the given number systems.
a Egyptian
i 3 ii 21 iii 114 iv 352
b Babylonian
i 4 ii 32 iii 61 iv 132
c Roman
i 2 ii 9 iii 24 iv 156
5 What number do these groups of symbols represent?
a Egyptian
i ii iii iv
b Babylonian
i ii iii iv
c Roman
i IV ii VIII iii XVI iv XL
6 Work out the answer to each of these problems. Write your answer using the same number
system that is given in the question.
a XIV + XXII b −
c − d DCLXIX + IX
Example 1
7 In ancient Babylon, a person adds goats to another group of .
How many goats are there in total? Write your answer using the Babylonian number system.
8 An ancient Roman counts the number of
people in three queues. The fi rst queue has
XI, the second has LXII and the third has
CXV. How many people are there in total?
Write your answer using the Roman number
system.
9 One Egyptian house is made from
stones and a second house is made from
stones.
How many more stones does the fi rst house
have? Write your answer using the Egyptian number system.
10 Which number system (Egyptian, Babylonian or Roman) uses the least number of symbols to
represent these numbers?
a 55 b 60 c 3104
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1A
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Number and Algebra 9
11 In the Roman system, the letters I, X and C are used to reduce either of the next two larger
numerals. So 9 is IX, not VIIII; and 49 is XLIX, not IL.
Also, only one numeral can be used to reduce another number. So 8 is VIII, not IIX.
Write these numbers using Roman numerals.
a 4 b 9 c 14 d 19
e 29 f 41 g 49 h 89
i 99 j 449 k 922 l 3401
12 The Egyptian system generally uses more symbols than the other systems described here.
Can you explain why? How many symbols are used for the number 999?
13 In the Babylonian system stands for 1, but because they did not use a symbol for zero at the
end of a number, it also represents 60. People would know what it meant, depending on the
situation it was used. Here is how it worked for large numbers. The dots represent empty spaces.
1 60 3600
...... ...... ......
a Write these numbers using the Babylonian system.
i 12 ii 72 iii 120
iv 191 v 3661 vi 7224
b Can you explain why ...... ...... represents 3600?
c What would ...... ...... ...... represent?
Enrichment: Other number systems
14 Other well-known number systems include:
i Mayan
ii modern Chinese
iii ancient Greek
Look up these number systems on the internet or
elsewhere. Write a brief sentence covering the
points below.
a When and where the number systems were
used.
b What symbols were used?
c Examples of numbers using these symbols.
An ancient Mayan carving.
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Chapter 1 Computation with 10 positive integers
■■ The Hindu-Arabic or decimal system uses base 10. This means powers of 10 (1, 10 or 101, 100 or
102, 1000 or 103, . . . .) are used to determine the place value of a digit in a number.
■■ The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are called digits.
■■ The value of each digit depends on its place in the number. The place value of the digit 2 in the
number 126, for example, is 20.
■■ 3 × 1000 + 2 × 100 + 5 × 10 + 4 × 1 (or 3 × 103 + 2 × 102 + 5 × 101 + 4 × 1) is said to be the
expanded form of the basic numeral 3254.
thousands hundreds tens ones
3 2 5 4 = 3 × 1000 + 2 × 100 + 5 × 10 + 4 × 1
expanded form
■■ Symbols used to compare numbers include the following.
– = (is equal to) 1 + 3 = 4 or 20 − 7 = 3 + 10
– ≠ (is not equal to) 1 + 3 ≠ 5 or 15 + 7 ≠ 16 + 8
– > (is greater than) 5 > 4 or 100 > 37
– ≥ (is greater than or equal to) 5 ≥ 4 or 4 ≥ 4
– < (is less than) 4 < 5 or 13 < 26
– ≤ (is less than or equal to) 4 ≤ 5 or 4 ≤ 4
– ≈ or =.
. (is approximately equal to) 4.02 ≈ 4 or 8997 = .
. 9000
Key ideas
place value in Hindu-Arabic numbers REVISION
The commonly used number system today,
called the decimal system or base 10, is also
called the Hindu-Arabic number system. Like
the Babylonian system, the value of the digit
depends on its place in the number, but only
one digit is used in each position. A digit
for zero is also used. The decimal system
originated in ancient India about 3000 bc and
spread throughout Europe through Arabic
texts over the next 4000 years.
let’s start: Largest and smallest
Without using decimal points, repeated digits or a zero (0) at the start of a number, see if you can use all
the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to write down:
• The largest possible number.
• The smallest possible number.
Can you explain why your numbers are, in fact, the largest or smallest possible?
1B
The famous ‘Histoire de la Mathematique’, a French document
showing the history of the Hindu-Arabic number system over
thousands of years.
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Number and Algebra 11
Example 2 Finding place value
Write down the value of the digit 4 in these numbers.
a 437 b 543 910
Solution Explana tion
a 4 × 100 = 400 4 is worth 4 × 100
3 is worth 3 × 10
7 is worth 7 × 1
b 4 × 10 000 = 40 000 5 is worth 5 × 100 000
4 is worth 4 × 10 000
3 is worth 3 × 1000
9 is worth 9 × 100
1 is worth 1 × 10
Example 3 Writing in expanded form
Write 517 in expanded form.
Solution Explana tion
517 = 5 × 100 + 1 × 10 + 7 × 1 Write each digit separately and multiply by the
appropriate power of 10.
Place value is very important, especially for a bank cheque.
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Chapter 1 Computation with 12 positive integers
1 Choose one of the words ‘ones’, ‘tens’, ‘hundreds’ or ‘thousands’ to describe each number.
a 100 b 1000 c 10 d 1
2 Write down these numbers using digits.
a two hundred and sixty-three b seven thousand four hundred and twenty-one
c thirty-six thousand and fi fteen d one hundred thousand and one
3 Which symbol (next to the capital letters) matches the given words?
A = B ≠ C > D ≥
E < F ≤ G ≈ H = .
.
a is not equal to b is less than c is greater than or equal to
d is equal to e is greater than f is less than or equal to
g is approximately equal to
4 Write down the value of the digit 7 in these numbers.
a 37 b 71 c 379 d 704
e 1712 f 7001 g 45 720 h 170 966
5 Write down the value of the digit 2 in these numbers.
a 126 b 2143 c 91 214 d 1 268 804
6 State whether each of these statements is true or false.
a 5 > 4 b 6 = 10 c 9 ≠ 99 d 1 < 12
e 22 ≤ 11 f 126 ≤ 126 g 19 ≥ 20 h 138 > 137
i 13 = 1 + 3 j 15 + 7 = 22 + 5 k 16 − 8 = 8 − 16 l 10 = 1 + 2 + 3 + 4
7 Write these numbers in expanded form.
a 17 b 281 c 935 d 20
e 4491 f 2003 g 10 001 h 55 555
8 Write these numbers, given in expanded form, as a basic numeral.
a 3 × 100 + 4 × 10 + 7 × 1 b 9 × 1000 + 4 × 100 + 1 × 10 + 6 × 1
c 7 × 1000 + 2 × 10 d 6 × 100 000 + 3 × 1
e 4 × 1 000 000 + 3 × 10 000 + 7 × 100 f 9 × 10 000 000 + 3 × 1000 + 2 × 10
9 Arrange these numbers from smallest to largest.
a 55, 45, 54, 44 b 729, 29, 92, 927, 279
c 23, 951, 136, 4 d 435, 453, 534, 345, 543, 354
e 12 345, 54 321, 34 512, 31 254 f 1010, 1001, 10 001, 1100, 10 100
Example 2
Example 3
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Exercise 1B R E V I S I O N
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Number and Algebra 13
10 How many numbers can be made using the given digits? Digits are not allowed to be used
more than once and all digits must be used.
a 2, 8 and 9 b 1, 6 and 7 c 2, 5, 6 and 7
11 Three different digits, not including zero, are chosen. How many numbers can be formed from
these three digits if the digits are allowed to be used more than once?
12 The letters used here represent the digits of a number. Write each one in expanded form.
a ab b abcd c a0000a
13 By considering some of the other number systems (Egyptian, Babylonian or Roman) explained in
the previous section, describe the main advantages of the Hindu-Arabic system.
Enrichment: Large numbers and expanded form
14 It is convenient to write very large numbers in expanded form with index notation.
Here is an example.
50 000 000 = 5 × 10 000 000 = 5 × 107
a Explain why it is convenient to write large numbers in this type of expanded form.
b 3200 can also be written in the form 32 × 102. All the non-zero digits are written down and then
multiplied by a power of 10. Similarly, write each of these numbers in the same way.
i 4100 ii 370 000 iii 21 770 000
c Write each of these numbers as basic numerals.
i 381 × 102 ii 7204 × 103 iii 1028 × 106
d Write these numbers in expanded form, just as you did in the examples above. Research them if
you do not know what they are.
i 1 million ii 1 billion iii 1 trillion
iv 1 googol v 1 googolplex
In 2008 in Zimbabwe, bank notes were issued in trillions of dollars, but soon became worthless due to inflation.
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Chapter 1 Computation with 14 positive integers
Adding and subtracting positive integers
The process of fi nding the total value of two or more numbers is called
addition. The words ‘plus’, ‘add’ and ‘sum’ are also used to describe
addition.
The process for fi nding the difference between two numbers is called
subtraction. The words ‘minus’, ‘subtract’ and ‘take away’ are also used
to describe subtraction.
let’s start: Your mental strategy
Many problems that involve addition and subtraction can be solved
mentally without the use of a calculator or complicated written working.
Consider 98 + 22 − 31 + 29
How would you work this out? What are the different ways it could be
done mentally? Explain your method.
1C
What’s the difference in our heights?
■■ The symbol + is used to show addition or fi nd a sum.
e.g. 4 + 3 = 7
■■ a + b = b + a e.g. 4 + 3 = 3 + 4
– This is the commutative law for addition, meaning that
the order does not matter.
■■ a + (b + c) = (a + b) + c e.g. 4 + (11 + 3) = (4 + 11) + 3
– This is called the associative law for addition, meaning it does not matter which pair is
added fi rst.
■■ a − (b − c) ≠ (a − b) − c (in general) e.g. 8 − (4 − 2) ≠ (8 − 4) − 2
■■ The symbol − is used to show subtraction or fi nd a difference.
e.g. 7 − 2 = 5
■■ a − b ≠ b − a (in general) e.g. 4 − 3 ≠ 3 − 4
■■ Mental addition and subtraction can be done using different strategies.
– Partitioning (Grouping digits in the same position)
171 + 23 = 100 + (70 + 20) + (1 + 3)
= 194
– Compensating (Making a 10, 100 etc. and then adjusting or compensating by adding or
subtracting)
46 + 9 = 46 + 10 − 1
= 55
– Doubling or halving (Making a double or half and then adjusting with addition or
subtraction)
75 + 78 = 75 + 75 + 3 124 − 61 = 124 − 62 + 1
= 150 + 3 = 62 + 1
= 153 = 63
5 6 7
+3
3 4 8
6 7
−2
4 5 8
Key ideas
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Number and Algebra 15
Example 4 Mental addition and subtraction
Use the suggested strategy to mentally work out the answer.
a 132 + 156 (partitioning) b 25 + 19 (compensating)
c 56 – 18 (compensating) d 35 + 36 (doubling or halving)
SOlutiON ExplANAtiON
a 132 + 156 = 288 100 30 2
100 50 6
200 80 8
+ +
+ +
+ +
b 25 + 19 = 44 25 + 19 = 25 + 20 – 1
= 45 – 1
= 44
c 56 – 18 = 38 56 – 18 = 56 – 20 + 2
= 36 + 2
= 38
d 35 + 36 = 71 35 + 36 = 35 + 35 + 1
= 70 + 1
= 71
1 a List three words that mean addition.
b List three words that mean subtraction.
2 Write the number which is:
a 3 more than 7 b 58 more than 11
c 7 less than 19 d 137 less than 157
3 a Find the sum of these pairs of numbers.
i 2 and 6 ii 19 and 8 iii 62 and 70
b Find the difference between these pairs of numbers.
i 11 and 5 ii 29 and 13 iii 101 and 93
4 State whether each of these statements is true or false.
a 4 + 3 > 6 b 11 + 19 ≥ 30 c 13 − 9 < 8
d 26 − 15 ≤ 10 e 1 + 7 − 4 ≥ 4 f 50 − 21 + 6 < 35
5 Give the result for each of these computations.
a 7 plus 11 b 22 minus 3 c the sum of 11 and 21
d 128 add 12 e 36 take away 15 f the difference between 13 and 4
Exercise 1C
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Chapter 1 Computation with 16 positive integers
6 Mentally fi nd the answers to these sums. Hint: Use the partitioning strategy.
a 23 + 41 b 71 + 26 c 138 + 441
d 246 + 502 e 937 + 11 f 1304 + 4293
g 140 273 + 238 410 h 390 447 + 201 132 i 100 001 + 101 010
7 Mentally fi nd the answers to these differences. Hint: Use the partitioning strategy.
a 29 − 18 b 57 − 21 c 249 − 137
d 1045 − 1041 e 4396 − 1285 f 10 101 − 100
8 Mentally fi nd the answers to these sums. Hint: Use the compensating strategy.
a 15 + 9 b 64 + 11 c 19 + 76
d 18 + 115 e 31 + 136 f 245 + 52
9 Mentally fi nd the answers to these differences. Hint: Use the compensating strategy.
a 35 − 11 b 45 − 19 c 156 − 48
d 244 − 22 e 376 − 59 f 5216 − 199
10 Mentally fi nd the answers to these sums and differences. Hint: Use the doubling or
halving strategy.
a 25 + 26 b 65 + 63 c 121 + 123
d 240 − 121 e 482 − 240 f 1006 − 504
11 Mentally fi nd the answers to these computations.
a 11 + 18 − 17 b 37 − 19 + 9 c 101 − 15 + 21
d 136 + 12 − 15 e 28 − 10 − 9 + 5 f 39 + 71 − 10 − 10
g 1010 − 11 + 21 − 1 h 5 − 7 + 2 i 10 − 25 + 18
12 Gary worked 7 hours on Monday, 5 hours on Tuesday, 13 hours on Wednesday, 11 hours on
Thursday and 2 hours on Friday. What is the total number of hours that Gary worked during the
week?
13 In a batting innings, Phil hit 126
runs and Mario hit 19 runs. How
many more runs did Phil hit
compared to Mario?
Example 4a
Example 4b
Example 4c
Example 4d
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Number and Algebra 17
14 A farmer reduced his cattle numbers from 86 to 54. How many cows were taken away?
15 Bag A has 18 marbles and bag B has 7 fewer marbles than bag A. What is the total number of
marbles?
16 Matt has 36 cards and Andy has 35 more cards than Matt. If they combine their cards, how many
do they have in total?
17 Each side on a magic triangle adds up to the same number, as shown in this
example with a sum of 12 on each side.
4
3
5
12 12
12
1 6
2
a Place each of the digits from 1 to 6 in a magic triangle with three digits along each side so that
each side adds up to the given number.
i 9 ii 10
b Place each of the digits from 1 to 9 in a magic triangle with four digits
along each side so that each side adds up to the given number.
i 20 ii 23
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Chapter 1 Computation with 18 positive integers
18 a The mental strategy of partitioning is easy to apply for 23 + 54 but harder for 23 + 59.
Explain why.
b The mental strategy of partitioning is easy to apply for 158 − 46 but harder for 151 − 46.
Explain why.
19 Complete these number sentences if the letters a, b and c represent numbers.
a a + b = c so c − __ = a b a + c = b so b − a = __
20 This magic triangle uses the digits 1 to 6, and has each side adding to the same
total. This example shows a side total of 9.
a How many different side totals are possible using the same digits?
b Explain your method.
21 The sum of two numbers is 87 and their difference is 29. What are the two
numbers?
Enrichment: Magic squares
22 A magic square has every row, column and main diagonal adding to the
same number, called the magic sum. For example, this magic square
has a magic sum of 15.
Find the magic sums for these squares, then fi ll in the
missing numbers.
a
6
7 5
2
b
10
11 13
12
c
15 20
14
19
d
1 15 4
6 9
11
13 2 16
6 5
2 4 3
1
4 9 2 15
3 5 7 15
8 1 6 15
15 15 15 15 15
This magic square was known in ancient China as a ‘Lo Shu’ square and uses only the numbers 1 to 9. It is shown in
the middle of this ancient design as symbols on a turtle shell, surrounded by the animals which represent the traditional
Chinese names for the years.
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Number and Algebra 19
Algorithms for adding and subtracting
It is not always practical to mentally solve problems involving addition and subtraction. For complex
computations a procedure involving a number of steps can be used and this helps to give the answer. Such
a procedure is called an algorithm.
For the addition algorithm, if two digits add to more than 9, then the higher place value digit in the
sum can be carried to the next column.
For the subtraction algorithm, if two digits subtract to be less than 0 then a ‘1’ can be borrowed from
the next column.
let’s start: The missing digits
Discuss what digits should go in the empty boxes.
Give reasons for your answers.
1 4
9 5
1 2 5
+
5
1 4
9 4
−
1D
■■ An algorithm is a procedure involving a number of steps that eventually leads to the answer to
a problem.
■■ Addition algorithm
– Arrange the numbers vertically so that the digits with similar place
value are in the same column.
– Add digits in the same column, starting on the right.
– If the digits add to more than 9, carry the 10 to the next column.
■■ Subtraction algorithm
– Arrange the numbers vertically so that the digits with similar place
value are in the same column.
– Subtract digits in the same column top-down and starting on the right.
– If the digits subtract to less than 0, borrow a 1 from the column to
the left.
■■ Calculators may be used to check your answers.
1234
192
426
4 2 6
3 9 12
1 2 1 4
+ =
+ =
+ + =
9 2 7
15 8 7
1 1 0
− =
− =
− =
12159
1 82
77
Example 5 using the addition algorithm
Give the result for each of these sums.
a 26
+ 66
b 439
+ 172
Key ideas
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Chapter 1 Computation with 20 positive integers
SOlutiON ExplANAtiON
a 126
66
92
+
Add the digits vertically.
6 + 6 = 12, so carry the 1 to the tens column.
b 1413 9
1 7 2
6 1 1
+
9 + 2 = 11, so carry a 1 to the tens column.
1 + 3 + 7 = 11, so carry a 1 to the hundreds column.
Example 6 using the subtraction algorithm
Give the result for each of these differences.
a 74
− 15
b 3240
− 2721
SOlutiON ExplANAtiON
a 6 7 14
1 5
5 9
−
Borrow 1 from 7 to make 14 – 5 = 9.
Then subtract 1 from 6 (not 7).
b 2 3 123 4 10
2 7 2 1
5 1 9
−
Borrow 1 from 4 to make 10 – 1 = 9.
Subtract 2 from 3 (not 4).
Borrow 1 from 3 to make 12 – 7 = 5.
Note that 2 – 2 = 0 and you do not need to show a
0 before the 5.
1 Mentally fi nd the results to these simple sums.
a 8 + 9 b 87 + 14 c 138 + 6 d 99 + 11
e 998 + 7 f 19 + 124 g 102 + 99 h 52 + 1053
2 Mentally fi nd the results to these simple differences.
a 13 − 5 b 36 − 9 c 75 − 8 d 100 − 16
e 37 − 22 f 104 − 12 g 46 − 17 h 1001 − 22
Exercise 1D
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Number and Algebra 21
3 What is the missing digit in each of these computations?
a 2 7
3 1
5
+
b 3 6
1 5
5
+
c 1 2 3
9 1
2 4
+
d 4 6
5
1 1 1
+
e 2 4
1
1 2
−
f 6 7
4 8
9
−
g 1 6 2
1
8 1
−
h
1 4 2
6 2 3
8 0 9
−
4 Give the answer to each of these additions. Check your answers with a calculator.
a 36
+51
b 74
+25
c 17
+24
d 47
+39
e 129
+ 97
f 458
+287
g 1041
+ 882
h 3092
+1988
5 Show your working to find the result for each of these sums.
a 85 + 76 b 131 + 94 c 1732 + 497 d 988 + 987
6 Give the result for each of these sums.
a 17
26
+34
b 126
47
+ 19
c 152
247
+ 19
d 2197
1204
+ 807
e 946 + 241 + 27 + 9 f 1052 + 839 + 7 + 84
7 Find the answers to these subtractions. Check your answers with a calculator.
a 54
−23
b 85
−65
c 46
−27
d 94
−36
e 125
− 89
f 241
−129
g 358
−279
h 491
−419
8 Show your working to find the answer to each of these differences.
a 32 − 16 b 124 − 77 c 613 − 128 d 1004 − 838
9 Farmer Green owns 287 sheep, farmer Brown
owns 526 sheep and farmer Grey owns 1041
sheep. How many sheep are there in total?
Example 5
Example 6
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Chapter 1 Computation with 22 positive integers
10 A car’s odometer shows 12 138 kilometres at the start of a journey and 12 714 kilometres
at the end of the journey. What was the journey’s distance?
11 Two different schools have 871 and 950 students enrolled.
a How many students are there in total?
b Find the difference between the number of students in the schools.
12 Find the missing digits in these sums.
a 3
5 3
1
+
b 1 4
7
9 1
+
c
+ 4 7
9 1 4
13 Find the missing numbers in these differences.
a 6
2 8
4
−
b 2 5
8
8 1
−
c 3 2
9 2
1 6 5
−
14 a First work out the answer to these simple computations before doing part b.
i 28 + 18 − 17 ii 36 − 19 + 20
b For part i above, is it possible to work out 18 − 17 and then add this total to 28?
c For part ii above, is it possible to work out 19 + 20 and then subtract this total from 36?
d Can you suggest a good mental strategy for part ii above that gives the correct answer?
15 a What are the missing digits in this sum?
b Explain why there is more than one possible set of missing numbers in the
sum given opposite. Give some examples.
16 The sum of two numbers is 978 and their difference is 74. What are the two numbers?
17 Make up some of your own problems like Question 16 and test them on a friend.
Enrichment: More magic squares
18 Complete these magic squares.
a
62 67 60
65
b
101 114
106 109
110
113 103 102 116
2 3
4 2 1
+
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1D
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Number and Algebra 23
Multiplying small positive integers
The multiplication of two numbers represents a repeated addition.
For example, 4 × 2
could be thought of
as 4 groups of 2 or
2 + 2 + 2 + 2.
Similarly, 4 × 2
could be thought of
as 2 groups of 4 or
2 × 4 or 4 + 4.
The product 4 × 4 involves 4 groups of 4,
which means the dots form a square. 4 × 4 can
be read as ‘4 squared’ and written as 42. The
number 16 is a square number.
● ● ● ●
4 × 2
● ● ● ●
● ●
2 × 4
● ●
● ●
● ●
● ● ● ●
4 × 4
● ● ● ●
● ● ● ●
● ● ● ●
let’s start: Which is the correct mental strategy?
Three students explain their method for fi nding the answer to 124 × 8.
• Billy says that you can do 124 × 10 to get 1240, then subtract 2 to get 1238.
• Lea says that you halve 124 and 8 twice each to give 31 × 2 = 62.
• Surai says that you multiply 8 by 4 to give 32, 8 by 2 to give 16 and 8 by 1 to give 8. She says the
total is therefore 32 + 16 + 8 = 56.
Are any of the students correct and can you explain any errors in their thinking?
1E
■■ Finding the product of two numbers involves multiplication. We say ‘the product of 2 and 3 is 6’.
■■ a × b = b × a e.g. 2 × 3 = 3 × 2
– This is the commutative law for multiplication, meaning that the order does not matter.
■■ (a × b) × c = a × (b × c) e.g. (3 × 5) × 4 = 3 × (5 × 4)
– This is the associative law for multiplication, meaning it does not matter which pair is
multiplied fi rst.
■■ The multiplication algorithm for multiplying by a single digit involves:
– Multiplying the single digit by each digit in the other number,
starting from the right.
– Carry and add any digits with a higher place value to the total in
the next column.
■■ Mental strategies for multiplication include:
– Memorising the multiplication tables.
– Using the commutative law by changing the order. For
example, 43 × 2 might be thought of more easily as 2 groups
of 43 or 2 × 43.
– Using the commutative and associative law by altering the
order if more than one number is being multiplied.
■■ When a number is multiplied by itself it is said to be ‘squared’.
a × a = a2 e.g. 5 × 5 = 52 = 25
When the number 5 is squared, the result is 25. So the number 25 is a square number.
123
4
92
× 4 × 3 = 12
4 × 2 + 1 = 9
9 × 7 = 63 12 × 3 = 36
55 × 20 = 5 × 11 × 2 × 10
= 5 × 2 × 11 × 10
= 10 × 11 × 10
= 1100
Key ideas
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Chapter 1 Computation with 24 positive integers
Example 7 Using mental strategies for multiplication
Use a mental strategy to find the answer to each of these products.
a 7 × 6 b 3 × 13 c 4 × 29
d 5 × 24 e 7 × 14 f 122
Solution Explana tion
a 7 × 6 = 42 7 × 6 or 6 × 7 should be memorised
(from multiplication tables).
b 3 × 13 = 39 3 × 13 = (3 × 10) + (3 × 3) = 30 + 9 = 39
(The distributive law is being used.)
c 4 × 29 = 116 4 × 29 = (4 × 30) − (4 × 1) = 120 − 4 = 116
(The distributive law is being used.)
d 5 × 24 = 120 5 × 24 = 10 × 12 = 120
(The doubling and halving strategy is being used.)
e 7 × 14 = 98 7 × 14 = 7 × 7 × 2 = 49 × 2 = 98
(Factors of 14 are used.)
f 122 = 12 × 12 = 144 This is read as ‘12 squared’, which means that 12 is
multiplied by itself. The number 144 is a square number.
Example 8 Using the multiplication algorithm
Give the result for each of these products.
a 31 × 4 b 197 × 7
–– Using the distributive law by making a 10, 100 etc. and then adjusting by adding or
subtracting. The distributive law is a × (b + c) = (a × b) + (a × c) or
a × (b − c) = (a × b) − (b × c). This will be used more extensively in the algebra chapters.
6 × 21 = (6 × 20) + (6 × 1) 7 × 18 = (7 × 20) − (7 × 2)
= 120 + 6 = 140 − 14
= 126 = 126
–– Using the doubling and halving strategy by doubling one
number and halving the other.
–– Using factors to split a number.
5 × 7 × 4 = 10 × 7 × 2
= 70 × 2
= 140
11 × 12 = 11 × 6 × 2
= 66 × 2
= 132
Key ideas
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Number and Algebra 25
3 What is the missing digit in these products?
a 2 1
3
6
×
b 3 6
5
18
×
c 7 6
2
1 2
×
d 4 0 2
3
1 0 6
×
4 Using your knowledge of multiplication tables, give the answer to these products.
a 8 × 7 b 6 × 9 c 12 × 4 d 112
e 6 × 12 f 7 × 5 g 12 × 9 h 13 × 3
5 Find the results to these products mentally. Hint: Use the distributive law strategy – subtraction for
a to d and addition for e to h.
a 3 × 19 b 6 × 29 c 4 × 28 d 38 × 7
e 5 × 21 f 4 × 31 g 6 × 42 h 53 × 3
6 Find the answer to these products mentally. Hint: Use the double and halve strategy or split a
number using its factors.
a 4 × 24 b 3 × 18 c 6 × 16 d 24 × 3
7 Give the result of each of these products, using the multiplication algorithm. Check your
results with a calculator.
a 33
× 2
b 43
× 3
c 72
× 6
d 55
× 3
e 129
× 2
f 407
× 7
g 526
× 5
h 3509
× 9
Example 7a,f
Example 7b,c
Example 7d,e
Example 8
1 Write the next three numbers in these patterns.
a 4, 8, 12, 16, __ b 11, 22, 33, __ c 17, 34, 51, __
2 Are these statements true or false?
a 4 × 3 = 3 × 4 b 2 × 5 × 6 = 6 × 5 × 2 c 11 × 5 = 10 × 5
d 3 × 32 = 3 × 30 + 3 × 2 e 5 × 18 = 10 × 9 f 21 × 4 = 2 × 42
g 19 × 7 = 20 × 7 − 19 h 39 × 4 = 40 × 4 − 1 × 4 i 64 × 4 = 128 × 8
Exercise 1E
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SOlutiON ExplANAtiON
a 31
4
124
×
4 × 1 = 4
4 × 3 = 12
b 61497
7
1379
×
7 × 7 = 49 (carry the 4)
7 × 9 + 4 = 67 (carry the 6)
7 × 1 + 6 = 13
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Chapter 1 Computation with 26 positive integers
8 Find the answer to these products, showing your working.
a 47 × 5 b 1391 × 3 c 9 × 425 d 7 × 4170
9 A circular race track is 240 metres long and Rory runs seven laps.
How far does Rory run in total?
10 Eight tickets costing $33 each are purchased for a concert. What
is the total cost of the tickets?
11 Reggie and Angelo combine their packs of cards. Reggie has fi ve
sets of 13 cards and Angelo has three sets of 17 cards. How many
cards are there in total?
12 Sala purchases some goods for a party at an outlet store and has $100 to spend. She selects eight
bottles of drink for $2 each, 13 food packs at $6 each and 18 party hats at 50 cents each. Does she
have enough money to pay for all the items?
13 Find the missing digits in these products.
a 3 9
7
2 3
×
b 2 5
1 2 5
×
c 7 9
3 7
×
d 1 3 2
10 6
×
e 2
7
8 9
×
f
× 9
3 5 1
g 2 3
5
1 6 0
×
h 4
1 9 8
×
14 The commutative and associative laws for multiplication mean that numbers can be multiplied
in any order. So (a × b) × c = (b × a) × c = b × (a × c) = ___, where the brackets show which
numbers are multiplied fi rst. In how many ways can 2 × 3 × 5 be calculated?
15 The distributive law can help to work out products mentally.
e.g. 7 × 31 = (7 × 30) + (7 × 1) = 210 + 7 = 217
Ωριτε εαχη οφ τηε φολλοωινγ ασ σινγλε προδυχτσ. Δο νοτ φινδ τηε ανσωερ.
a 3 × 20 + 3 × 1 b 9 × 50 + 9 × 2 c 7 × 30 + 7 × 2
d 5 × 100 − 5 × 3 e a × 40 − a × 2 f a × 200 + a × 3
16 How many different ways can the two spaces be fi lled in this problem? Explain why.
Enrichment: Missing digits
17 Find all the missing digits in these products.
a 1
7
5 1
×
b 2 9
3
8
×
18 The product of two numbers is 132 and their sum is 28. What are the two numbers?
2 3
4
8 2
×
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1E
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Number and Algebra 27
Multiplying large positive integers
There are many situations that require the multiplication of large numbers; for example, the total revenue
from selling 40 000 tickets at $23 each, or the area of a rectangular park with length and width dimensions
of 65 metres by 122 metres. To complete such calculations by hand requires the use of a suitable
algorithm.
let’s start: Spot the errors
There are three types of errors in the working shown for
this problem. Find the errors and describe them.
1F
How much revenue came from selling tickets to this game?
271
13
613
271
1273
×
■■ When multiplying by 10, 100, 1000, 10 000 etc. each digit moves to the left by the number
of zeros; e.g. 45 × 1000 = 45 000.
■■ A strategy for multiplying by multiples of 10, 100 etc. is to fi rst multiply by the number
without the zeros then add the zeros to the answer later.
For example, 21 × 3000 = 21 × 3 × 1000 = 63 × 1000 = 63 000
■■ The algorithm for multiplying large numbers involves
dividing the problem into smaller products and then adding
the totals; e.g.
143
14
572
1430
2002
×
←
←
←
143 × 4
143 × 10
1430 + 572
Key ideas
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Chapter 1 Computation with 28 positive integers
SOlutiON ExplANAtiON
a 37 × 100 = 3700 Move the 3 and the 7 two places to the left and add
two zeros.
b 45 × 70 = 45 × 7 × 10
= 315 × 10
= 3150
First multiply by 7, then multiply by 10 later.
45
7
315
×
c 614
14
2456
6140
8596
×
First multiply 614 × 4.
Then multiply 614 × 10.
Add the totals to give the answer.
d First multiply 25 × 5.
Then multiply 25 × 20.
Add the totals to give the answer.
This can be written as 252 = 625.
Exercise 1F
1 What is the missing digit in these products?
a 72 × 10 = 7?0 b 13 × 100 = 130? c 49 × 100 = 49?0 d 924 × 10 = 92?0
2 What is the missing number in these products?
a 15 × __ = 1500 b 329 × __ = 3290 c 92 × __ = 920 000
3 State if the following calculations are correct. If they are incorrect, fi nd the correct answer.
a 26
4
84
×
b 39
14
156
39
195
×
c 92
24
368
1840
2208
×
d 102
24
408
240
648
×
Example 9 Multiplying large numbers
Give the result for each of these products.
a 37 × 100 b 45 × 70 c 614 × 14 d 252
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25
125
500
625
×
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4 Give the result of each of these products. C
a 4 × 100 b 29 × 10 c 183 × 10 d 46 × 100
e 50 × 1000 f 630 × 100 g 1441 × 10 h 2910 × 10 000
Example 9a
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Number and Algebra 29
5 Use the suggested strategy in Example 9b to find these products.
a 17 × 20 b 36 × 40 c 92 × 70 d 45 × 500
e 138 × 300 f 92 × 5000 g 317 × 200 h 1043 × 9000
6 Use the multiplication algorithm to find
these products.
a 37
× 11
b 72
× 19
c 126
× 15
d 428
× 22
e 46
× 46
f 416
× 98
g 380
× 49
h 1026
× 33
Example 9b
Example 9c,d
How could you calculate the number of pieces of fruit on
this stall without counting them all?
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7 Estimate the answers to these products, then use a calculator to check.
a 19 × 11 b 26 × 21
c 37 × 15 d 121 × 18
8 A pool area includes 68 square metres
of paving at $32 per square metre. What is the
total cost of paving?
9 Waldo buys 215 metres of pipe at $28
per metre. What is the total cost of piping?
10 How many seconds are there in one day?
11 Find the missing digits in these products.
a 2
1 7
1 1
2 0
1
×
b 1 3
1
2 9
1 3
5
×
c
× 3 7
3 4 3
4
d 2
2
1 2 6
5 2
6
×
12 There are 360 degrees in a full turn. How many degrees does the minute hand on a clock turn in one
week?
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Chapter 1 Computation with 30 positive integers
13 The product of two whole numbers is less than their sum. Neither number is zero or less.
What must be true about one of the numbers?
14 If both numbers in a multiplication computation have at least three digits, then the algorithm needs
to be expanded. Use the algorithm to fi nd these products.
a 294
× 136
b 1013
× 916
c 3947
× 1204
d 47126
× 3 107
15 Can you work out these problems using an effective mental strategy? Look to see if you can fi rst
simplify each question.
a 98 × 16 + 2 × 16
b 33 × 26 − 3 × 26
c 19 × 15 + 34 × 17 − 4 × 17 + 1 × 15
d 22 × 19 − 3 × 17 + 51 × 9 − 1 × 9 + 13 × 17 − 2 × 19
Enrichment: Multiplication puzzle
16 a What is the largest number you can make by choosing fi ve digits from the list
1, 2, 3, 4, 5, 6, 7, 8, 9 and placing them into the product shown at right?
b What is the smallest number you can make by choosing fi ve digits from the
list 1, 2, 3, 4, 5, 6, 7, 8, 9 and placing them into the product shown at right?
17 The product of two whole numbers is 14 391 and their difference is 6. What are the two numbers?
18 a 82 = 8 × 8 = 64
64 is a square number.
Find all the three-digit square numbers in which the fi rst digit is 1 or 2.
b What is the largest three-digit square number?
c What is the largest four-digit square number?
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Number and Algebra 31
Dividing positive integers and dealing with
remainders
Division involves fi nding the number of equal groups into which a particular number can be divided.
This can be achieved both with and without a remainder or ‘left over’. Dividing 20 apples amongst fi ve
people or $10 000 between three bank accounts are examples of when division can be used.
Multiplication and division are reverse operations, and this is shown in this simple example:
7 × 3 = 21 So, 21 ÷ 3 = 7 or 21 ÷ 7 = 3
let’s start: Arranging counters
A total of 24 counters sit on a table. Using whole numbers, in how many ways can the
counters be divided into equal-sized groups with no counters remaining?
• Is it also possible to divide the counters into equal-sized groups but with two
counters remaining?
• If fi ve counters are to remain, how many equal-sized groups can be formed
and why?
1G
• • • • •
• • • • •
• • • • •
• • • • •
• • • •
■■ The number of equal-sized groups formed from the division operation is called the quotient.
■■ The total being divided is called the dividend and the size of the equal groups is called the
divisor.
■■ Any amount remaining after division into
equal-sized groups is called the remainder.
– 79 ÷ 7 = 11 and 2 remainder means
79 = 11 × 7 + 2.
■■ a ÷ b ≠ b ÷ a (generally)
– The commutative law does not hold for division.
■■ (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) (generally)
– The associative law does not hold for division.
■■ The short division algorithm involves fi rst dividing
into the digit with the highest place value and then
carrying any remainder to the next digit, working
from left to right.
■■ Mental division can be done using different strategies.
– Knowing your multiplication tables off by heart.
– 63 ÷ 9 = ? is the same as asking 9 × ? = 63.
– Making a convenient multiple of the divisor and then adjusting by adding or subtracting.
Key ideas
8 ÷ 2 =/ 2 ÷ 8
(8 ÷ 4) ÷ 2 =/ 8 ÷ (4 ÷ 2)
79 ÷ 7 = 11 and 2 remainder = 11
2
7
total being divided
(dividend)
size of equal
groups (divisor)
quotient
4 ÷ 3 = 1 and
1 rem.
11 ÷ 3 = 3 and
2 rem.
23 ÷ 3 = 7 and
2 rem.
3)4 1 3
137
1123
413 ÷ 3 = 137 and 2 remainder
= 137 2
3
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Chapter 1 Computation with 32 positive integers
Example 10 Using mental strategies for division
Use a mental strategy to find the quotient.
a 84 ÷ 7 b 93 ÷ 3 c 128 ÷ 8
Solution Explana tion
a 84 ÷ 7 = 12 7 × ? = 84
(Use your knowledge from multiplication tables.)
b 93 ÷ 3 = 31 93 ÷ 3 = (90 ÷ 3) + (3 ÷ 3) = 30 + 1
(This uses the distributive law.)
c 128 ÷ 8 = 16 128 ÷ 8 = 64 ÷ 4 = 32 ÷ 2 = 16
(Halve both numbers repeatedly.)
or
128 ÷ 2 = 64
64 ÷ 2 = 32
32 ÷ 2 = 16
Example 11 Using the short division algorithm
Use the short division algorithm to find the quotient and remainder.
a 3)37 b 7)195
Solution Explana tion
a 3)37
13
37 ÷ 3 = 12 and 1 remainder.
= 12
1
3
3 ÷ 3 = 1 with no remainder.
7 ÷ 3 = 2 with 1 remainder.
b ) 7 19 5
2 7
5
195 ÷ 7 = 27 and 6 remainder.
= 27 6
7
7 does not divide into 1.
19 ÷ 7 = 2 with 5 remainder.
55 ÷ 7 = 7 with 6 remainder.
Below is an application of the distributive law.
84 ÷ 3 = (60 + 24) ÷ 3 84 ÷ 3 = (90 − 6) ÷ 3
= (60 ÷ 3) + (24 ÷ 3) = (90 ÷ 3) − (6 ÷ 3)
= 20 + 8 = 30 − 2
= 28 = 28
–– Halving both numbers. If both numbers in the division are even,
then halve both numbers.
70 14 35 7
5
÷ = ÷
=
Key ideas
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Number and Algebra 33
1 Write the number that is missing in these statements.
a 8 ÷ 2 = 4 is the same as 4 × ? = 8. b 36 ÷ 12 = 3 is the same as ? × 12 = 36.
c 42 ÷ ? = 6 is the same as 6 × 7 = 42. d 72 ÷ 6 = ? is the same as 12 × 6 = 72.
2 What is the remainder when:
a 2 is divided into 7? b 5 is divided into 37?
c 42 is divided by 8? d 50 is divided by 9?
3 Write the missing digit in each of these divisions.
a 3) 5 1
7 b 7) 8 4
2 c ) 5 1 2 5
2
d ) 9 1 3 5
1
4 Use your knowledge of multiplication tables to fi nd the quotient.
a 28 ÷ 7 b 36 ÷ 12 c 48 ÷ 8 d 45 ÷ 9
e 56 ÷ 8 f 63 ÷ 7 g 96 ÷ 12 h 121 ÷ 11
5 Find the answer to these using a mental strategy. Hint: Use the distributive law strategy.
a 63 ÷ 3 b 76 ÷ 4 c 57 ÷ 3 d 205 ÷ 5
e 203 ÷ 7 f 189 ÷ 9 g 906 ÷ 3 h 490 ÷ 5
6 Find the answers to these using a mental strategy. Hint: Use the halving strategy by halving
both numbers.
a 88 ÷ 4 b 124 ÷ 4 c 136 ÷ 8 d 112 ÷ 16
7 Write the answers to these divisions, which involve 0s and 1s.
a 26 ÷ 1 b 1094 ÷ 1 c 0 ÷ 7 d 0 ÷ 458
8 Use the short division algorithm to fi nd the quotient and remainder. Check your answers using
a calculator.
a 3)71 b 7)92 c 2)139 d 6)247
e 4)2173 f 3)61 001 g 5)4093 h 9)90 009
9 Use the short division algorithm and express the remainder as a fraction.
a 526 ÷ 4 b 1691 ÷ 7 c 2345 ÷ 6 d 92 337 ÷ 8
Example 10a
Example 10b
Example 10c
Example 11
Exercise 1G
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Chapter 1 Computation with 34 positive integers
10 If 117 food packs are divided equally among nine families, how many packs does each family
receive?
11 Spring Fresh Company sells mineral water in
packs of six bottles. How many packs are there in
a truck containing 744 bottles?
12 A bricklayer earns $1215 in a week.
a How much does he earn per day if he he
works Monday to Friday?
b How much does he earn per hour if he he
works 9 hours per day Monday to Friday?
13 A straight fence has two end posts as well as as
other posts that are divided evenly along the fence
4 metres apart. If the fence is to be 264 metres long,
how many posts are needed, including the end
posts?
14 Friendly Taxis can take up to four passengers
each. How many taxis are required to transport
59 people?
15 A truck can carry up to 7 tonnes of rock. What is the minimum number of trips needed to transport
130 tonnes of rock?
16 All the rows, columns and main diagonals in the magic square multiply to
give 216. Can you fi nd the missing numbers?
9 12
1
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Number and Algebra 35
17 Write down the missing numbers.
a 37 ÷ 3 = 12 and remainder means 37 = × 3 + 1.
b 96 ÷ 7 = and 5 remainder means 96 = 13 × + 5.
c 104 ÷ 20 = 5 and remainder means 104 = × 20 + 4.
18 Pies are purchased wholesale at
nine for $4. How much will it cost
to purchase 153 pies?
19 Give the results to these problems,
if a represents any number other
than 0.
a a ÷ 1
b 0 ÷ a
c a ÷ a
20 A number less than 30 leaves a
remainder of 3 when divided by 5
and a remainder of 2 when divided by 3.
What two numbers meet the given conditions?
21 As you know a ÷ b is not generally equal to b ÷ a. However, can you find a situation where
a ÷ b = b ÷ a?
22 The short division algorithm can also be used to divide by numbers with more than one digit.
e.g. 215 ÷ 12 = 17 and 11 remainder.
21 ÷ 12 = 1 and 9 remainder.
95 ÷ 12 = 7 and 11 remainder.
Use the short division algorithm and express the remainder as a fraction.
a 371 ÷ 11 b 926 ÷ 17 c 404 ÷ 13
d 1621 ÷ 15 e 2109 ÷ 23 f 6914 ÷ 56
Enrichment: Long, short division
23 Use the short division algorithm to find the quotient and remainder.
a 1247 ÷ 326 b 1094 ÷ 99 c 26 401 ÷ 1432
24 The magic product for this square is 6720. Find the missing numbers.
1 6 56
40 2 3
14
10
12)21 5
1 7
9
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Chapter 1 Computation with 36 positive integers
Estimating and rounding positive integers
Many theoretical and practical problems do not need precise or exact answers. In such situations
reasonable estimations can provide enough informaton to solve the problem.
The total revenue from the Australian Open tennis tournament depends on crowd numbers. Estimates
would be used before the tournament begins to predict these numbers. An estimate for the total revenue
might be $8 million.
let’s start: The tennis crowd
Here is a photo of a crowd at a tennis
match. Describe how you might estimate
the number of people in the photo. What
is your answer? How different is your
answer from those of others in your
class?
1H
How can you estimate the number of spectators?
■■ Estimates or approximations to the answers of problems can be found by rounding numbers
to the nearest 10, 100, 1000 etc.
■■ If the next digit is 0, 1, 2, 3 or 4, then round down.
■■ If the next digit is 5, 6, 7, 8 or 9, then round up.
■■ Leading digit approximation rounds the fi rst digit to the nearest 10 or 100 or 1000 etc.
e.g. For 932 use 900 For 968 use 1000
■■ The symbols ≈ or =.
. mean ‘is approximately equal to’.
Key ideas
Example 12 Rounding
Round these numbers as indicated.
a 86 (to the nearest 10) b 4142 (to the nearest 100)
SOlutiON ExplANAtiON
a 86 ≈ 90 or 86 = .
. 90 The digit after the 8 is greater than or equal to 5, so round up.
b 4142 ≈ 4100 or 4142 = .
. 4100 The digit after the 1 is less than or equal to 4, so round down.
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Number and Algebra 37
Example 13 using leading digit approximation
Estimate the answers to these problems by rounding each number to the leading digit.
a 42 × 7 b 95 × 326
SOlutiON ExplANAtiON
a 42 × 7 ≈ 40 × 7
= 280
The leading digit in 42 is the 4 in the ‘tens’ column.
b 95 × 326 = .
. 100 × 300
= 30 000
The nearest ‘ten’ to 95 is 100, and the leading digit in
326 is in the ‘hundreds’ column.
Example 14 Estimating with operations
Estimate the answers to these problems by rounding both numbers as indicated.
a 115 × 92 (to the nearest 100) b 2266 ÷ 9 (to the nearest 10)
SOlutiON ExplANAtiON
a 115 × 92 ≈ 100 × 100
∴ 115 × 92 ≈ 10 000
115 rounds to 100 and 92 rounds to 100.
b 2266 ÷ 9 = .
. 2270 ÷ 10
∴ 2266 ÷ 9 = .
. 227
2266 rounds to 2270 and 9 rounds to 10.
1 State whether these numbers have been rounded up or down.
a 59 ≈ 60 b 14 ≈ 10 c 137 ≈ 140
d 255 = .
. 260 e 924 = .
. 900 f 1413 = .
. 1000
2 For the given estimates, decide if the approximate answer is going to give a larger or smaller result
compared to the true answer.
a 58 + 97 ≈ 60 + 100 b 24 × 31 ≈ 20 × 30
c 130 − 79 = .
. 130 − 80 d 267 − 110 = .
. 270 − 110
3 Round these numbers as indicated.
a 59 (nearest 10) b 32 (nearest 10) c 124 (nearest 10)
d 185 (nearest 10) e 231 (nearest 100) f 894 (nearest 100)
g 96 (nearest 10) h 584 (nearest 100) i 1512 (nearest 1000)
Example 12
Exercise 1H
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Chapter 1 Computation with 38 positive integers
4 Round these numbers using leading digit approximation; i.e. round to the fi rst digit.
a 21 b 29 c 136 d 857
e 5600 f 92 104 g 9999 h 14
5 Estimate the answers to these problems by fi rst rounding both numbers as indicated.
a 72 + 59 (nearest 10) b 138 − 61 (nearest 10)
c 275 − 134 (nearest 10) d 841 + 99 (nearest 10)
e 203 − 104 (nearest 100) f 815 + 183 (nearest 100)
g 990 + 125 (nearest 100) h 96 + 2473 (nearest 100)
i 1555 − 555 (nearest 1000)
6 Use leading digit approximation to estimate the answer.
a 29 × 4 b 124 + 58 c 232 − 106 d 61 ÷ 5
e 394 ÷ 10 f 97 × 21 g 1390 + 3244 h 999 − 888
7 Estimate the answers to these problems by rounding both numbers as indicated.
a 29 × 41 (nearest 10) b 92 × 67 (nearest 10)
c 124 × 173 (nearest 100) d 2402 × 3817 (nearest 1000)
e 48 ÷ 11 (nearest 10) f 159 ÷ 12 (nearest 10)
g 104 ÷ 11 (nearest 10) h 2493 ÷ 103 (nearest 100)
8 Here is a photo of stars in the night sky. Use the grid to estimate the number of visible stars
in the photo.
Example 13
Example 14
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Number and Algebra 39
9 A digger can dig 29 scoops per hour and work 7 hours per day. Approximately how many
scoops can be dug over 10 days?
10 Most of the pens at a stockyard are full of sheep. There are 55 pens and one of the pens has 22 sheep.
Give an estimate for the total number of sheep at the stockyard.
11 A whole year group of 159 students is roughly divided into 19 groups. Estimate the number in each group.
12 It is sensible sometimes to round one number up if the other number is going to be rounded down.
Use leading digit approximation to estimate the answers to these problems.
a 11 × 19 b 129 × 954 c 25 × 36 d 1500 × 2500
13 The letters a and b represent numbers. Which of the words ‘smaller’ or ‘larger’ completes
these sentences?
a If a and b are both rounded up, then compared to the true answer the approximate answer to:
i a + b will be .
ii a × b will be .
b If only a is rounded up, but b is left as it is, then compared to the true answer the approximate answer to:
i a − b will be .
ii a ÷ b will be .
c If only b is rounded up, but a is left as it is, then compared to the true answer the approximate answer to:
i a − b will be .
ii a ÷ b will be .
d If only b is rounded down, but a is left as it is, then compared to the true answer the approximate
answer to:
i a − b will be .
ii a ÷ b will be .
Enrichment: Maximum error
14 When rounding numbers before a calculation is completed, it is most likely that there will be an
error. This error can be large or small, depending on the type of rounding involved.
e.g. When rounding to the nearest 10, 71 × 11 ≈ 70 × 10 = 700.
But 71 × 11 = 781, so the error is 81.
a Calculate the error if these numbers are rounded to the nearest 10 before the multiplication is
calculated.
i 23 × 17 ii 23 × 24 iii 65 × 54 iv 67 × 56
b Explain why the error in parts i and iii is much less than the error in parts ii and iv.
c Calculate the error if these numbers are rounded to the nearest 10 before the division is calculated.
i 261 ÷ 9 ii 323 ÷ 17 iii 99 ÷ 11 iv 396 ÷ 22
d Explain why the approximate answers in parts i and ii are less than the correct answer, and why
the approximate answers in parts iii and iv are more than the correct answer.
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Chapter 1 Computation with 40 positive integers
Order of operations with positive integers
When combining the operations of addition, subtraction,
multiplication and division, a particular order needs to
be followed. Multiplication and division sit higher in the
order than addition and subtraction, and this relates to
how we might logically interpret simple mathematical
problems put into words.
Consider these two statements.
• 2 groups of 3 chairs plus 5 chairs.
• 5 chairs plus 2 groups of 3 chairs.
In both cases, there are 2 × 3 + 5 = 11 chairs. This means that 2 × 3 + 5 = 5 + 2 × 3.
This also suggests that for 5 + 2 × 3, the multiplication should be done fi rst.
let’s start: Minimum brackets
• How might you use brackets to make this statement true?
2 + 3 × 5 − 3 ÷ 6 + 1 = 2
• What is the minimum number of pairs of brackets needed to make it true?
1I
■■ When working with more than one operation:
– Deal with brackets (also known as parentheses) fi rst.
– Do multiplication and division next, working from left to right.
– Do addition and subtraction last, working from left to right.
■■ Recall (a + b) + c = a + (b + c) but (a − b) − c ≠ a − (b − c)
(a × b) × c = a × (b × c) but (a ÷ b) ÷ c ≠ a ÷ (b ÷ c)
■■ Brackets can sit inside other brackets.
– Square brackets can also be used.
e.g. [2 × (3 + 4) − 1] × 3
– Always deal with the inner brackets fi rst.
■■ Note that some calculators apply the order of operations and some do not.
For example: 5 + 2 × 3 = 11 (not 21)
Try this on a variety of calculators and mobile phones.
Key ideas
4 × (2 + 3) − 12 ÷ 6
1st
5
2nd 3rd
20 2
last
18
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Number and Algebra 41
Example 15 using order of operations
Use order of operations to evaluate these expressions.
a 6 + 10 ÷ 2 b 18 − 2 × (4 + 6) ÷ 5
SOlutiON ExplANAtiON
a 6 + 10 ÷ 2 = 6 + 5
= 11
Do the division before the addition.
b 18 − 2 × (4 + 6) ÷ 5 = 18 − 2 × 10 ÷ 5
= 18 − 20 ÷ 5
= 18 − 4
= 14
Deal with brackets fi rst.
Do the multiplication and division next, working from
left to right.
Do the subtraction last.
Example 16 using order of operations in worded problems
Find the difference between 76 and 43, triple this result and, fi nally, subtract the quotient of 35 and 7.
SOlutiON ExplANAtiON
3 × (76 − 43) − 35 ÷ 7 = 3 × 33 − 5
= 99 − 5
= 94
First, write the problem using symbols and numbers.
Use brackets for the difference since this operation is
to be completed fi rst.
1 Which operation (addition, subtraction, multiplication or division) is done fi rst in these
computations?
a 2 + 5 − 3 b 5 ÷ 5 × 2 c 2 × 3 ÷ 6
d 5 × 2 + 3 e 7 ÷ 7 − 1 f (6 + 2) × 3
g (8 ÷ 4) − 1 h 4 + 7 × 2 i 8 − 10 ÷ 5
j 10 − 2 + 3 k 6 + 2 × 3 − 1 l 5 × (2 + 3 ÷ 3) − 1
2 Classify these statements as true or false.
a 5 × 2 + 1 = (5 × 2) + 1 b 10 × (3 + 4) = 10 × 3 + 4
c 21 − 7 ÷ 7 = (21 − 7) ÷ 7 d 9 − 3 × 2 = 9 − (3 × 2)
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Chapter 1 Computation with 42 positive integers
3 Use order of operations to fi nd the answers to these computations. Use a calculator to check
your answers and to see if it applies the order of operations.
a 2 + 3 × 7 b 5 + 8 × 2 c 10 − 20 ÷ 2
d 22 − 16 ÷ 4 e 6 × 3 + 2 × 7 f 1 × 8 − 2 × 3
g 18 ÷ 9 + 60 ÷ 3 h 2 + 3 × 7 − 1 i 40 − 25 ÷ 5 + 3
j 63 ÷ 3 × 7 + 2 × 3 k 78 − 14 × 4 + 6 l 300 − 100 × 4 ÷ 4
4 Use order of operations to fi nd the answer to these computations. Use a calculator to check your
answers.
a 2 × (3 + 2) b 18 ÷ (10 − 4) c (19 − 9) ÷ 5
d (100 + 5) ÷ 5 + 1 e 2 × (9 − 4) ÷ 5 f 50 ÷ (13 − 3) + 4
g 16 − 2 × (7 − 5) + 6 h (7 + 2) ÷ (53 − 50) i 14 − (7 ÷ 7 + 1) × 2
j (20 − 10) × (5 + 7) + 1 k 3 × (72 ÷ 12 + 1) − 1 l 48 ÷ (4 + 4) ÷ (3 × 2)
5 These computations involve brackets within brackets. Ensure you work with the inner brackets
fi rst.
a 2 × [(2 + 3) × 5 − 1] b [10 ÷ (2 + 3) + 1] × 6
c 26 ÷ [10 − (17 − 9)] d [6 − (5 − 3)] × 7
e 2 + [103 − (21 + 52)] − (9 + 11) × 6 ÷ 12
6 Find the answer to these worded problems by fi rst writing the sentence using numbers and
symbols.
a Triple the sum of 3 and 6.
b Double the quotient of 20 and 4.
c The quotient of 44 and 11 plus 4.
d 5 more than the product of 6 and 12.
e The quotient of 60 and 12 is subtracted from the product of 5 and 7.
f 15 less than the difference of 48 and 12.
g The product of 9 and 12 is subtracted from double the product of 10 and 15.
7 A delivery of 15 boxes of books arrives, each box
containing eight books. The bookstore owner removes
three books from each box. How many books still remain
in total?
Example 15a
Example 15b
Example 16
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Number and Algebra 43
8 In a class, eight students have three TV
sets at home, four have two TV sets,
13 have one TV set and two students
have no TV sets. How many TV sets
are there in total?
9 Insert brackets into these statements to make them true.
a 4 + 2 × 3 = 18 b 9 ÷ 12 − 9 = 3 c 2 × 3 + 4 − 5 = 9
d 3 + 2 × 7 − 3 = 20 e 10 − 7 ÷ 21 − 18 = 1 f 4 + 10 ÷ 21 ÷ 3 = 2
g 20 − 31 − 19 × 2 = 16 h 50 ÷ 2 × 5 − 4 = 1 i 25 − 19 × 3 + 7 ÷ 12 + 1 = 6
10 The amount of $100 is divided into two first prizes of equal value and three second prizes of equal
value. Each prize is a whole number of dollars and first prize is at least 4 times the value of the
second prize. If second prize is more than $6, find the amount of each prize.
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11 Decide if the brackets given in each statement are actually necessary. C
a 2 + (3 × 6) = 20 b (2 + 3) × 6 = 30 c (20 × 2) × 3 = 120
d 10 − (5 + 2) = 3 e 22 − (11 − 7) = 18 f 19 − (10 ÷ 2) = 14
g (40 ÷ 10) ÷ 4 = 1 h 100 ÷ (20 ÷ 5) = 25 i 2 × (3 + 2) ÷ 5 = 2
12 The letters a, b and c represent numbers. Decide if the brackets are necessary in these expressions.
a a + (b + c) b a − (b − c) c a × (b × c) d a ÷ (b ÷ c)
13 Write a simpler statement for these. Assume a ≠ 0 and b ≠ 0.
a a + b − a b (a − a) × b c a + b ÷ b d a × b ÷ a
Enrichment: Operation in rules
14 Using whole numbers and any of the four operations (+, –, ×, ÷), describe how you would obtain the
‘Finish’ number from the ‘Start’ number in each of these tables. Your rule must work for every pair of
numbers in its table.
a Start Finish
1 3
2 5
3 7
4 9
b Start Finish
1 0
2 3
3 6
4 9
c Start Finish
3 10
4 17
5 26
6 37
Make up your own table with a ‘secret’ rule and test it on a friend.
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Chapter 1 Computation with 44 positive integers
the abacus
The abacus is a counting device that has been used for thousands
of years. They were used extensively by merchants, traders, tax
collectors and clerks before modern-day numerals systems were
developed. Counting boards called Abax date back to 500 bc. These
were wood or stone tablets with grooves, which would hold beans
or pebbles.
The modern abacus is said to have originated in China in about
the 13th century and includes beads on wires held in a wooden
frame.
A modern abacus with thirteen wires
There are 5 beads on one side of a modern abacus
worth 1 each and 2 beads on the opposite side worth
5 each.
■■ Each wire represents a different unit, e.g. ones,
tens, hundreds etc.
■■ Beads are counted only when they are pushed
toward the centre.
ones
tens
hundreds
thousands
Here is a diagram showing the number 5716.
investigation
A German woodcut from 1508 showing an abacus in use by gentleman on right,
while a mathematician (at left) writes algorithms.
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Number and Algebra 45
1 What numbers are showing on the abacus diagrams below? Only the first six wires
are showing.
a b
c d
2 Draw abacus diagrams showing these numbers.
a 57 b 392 c 6804 d 290 316
3 Imagine adding two numbers using an abacus by sliding beads along their wires. Clearly explain
the steps taken to add these numbers.
a 11 + 7 b 2394 + 536
4 Imagine subtracting two numbers using an abacus by sliding beads along their wires. Clearly
explain the steps taken to subtract these numbers.
a 23 − 14 b 329 − 243
5 Multiplication is calculated as a repeated addition.
e.g. 3 × 21 = 21 + 21 + 21
Clearly explain the steps involved when using an abacus to multiply these numbers.
a 3 × 42 b 5 × 156
6 Division is calculated as a repeated subtraction.
e.g. 63 ÷ 21 = 3, since 63 − 21 − 21 − 21 = 0
Clearly explain the steps involved when using an abacus to divide these numbers.
a 28 ÷ 7 b 405 ÷ 135
7 See if you can find a real abacus or computer abacus with
which to work. Use the abacus to show how you can do the
problems in Questions 3 to 6 above.
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Chapter 1 Computation with 46 positive integers
puzzles and challenges
1 The extra dollar. The cost of dinner for two people is $45 and they both give the waiter $25 each.
Of the extra $5 the waiter is allowed to keep $3 as a tip and returns $1 to each person.
So the two people paid $24 each, making a total of $48, and the waiter has $3. The total is therefore
$48 + $3 = $51. Where did the extra $1 come from?
2 The sum along each line is 15. Place each of the digits 1, 2, 3, 4, 5, 6, 7, 8
and 9 to make this true.
3 The sum along each side of this triangle is 17. Place each of the
digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make this true.
4 Make the total of 100 out of all the numbers 2, 3, 4, 7 and 11, using each number only once. You
can use any of the operations (+, –, ×, ÷), as well as brackets.
5 Sudoku is a popular logic number puzzle made up of a 9 by 9 square, where each column and row
can use the digits 1, 2, 3, 4, 5, 6, 7, 8 and 9 only once. Also, each digit is to be used only once in
each 3 by 3 square. Solve these puzzles.
7 6 9 3
4 1 8 7
8 2 9 1
3 1
2 8 5 3
5 6 9 2
3 9 5
6 8 4
5 9 7
4 2 8 7
2 8 7 9 1
6 3 5
3 7 2 8
6 5 4 7 8 2
2 6
7 5
8 3 9 2 7
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Number and Algebra 47
Chapter summary
Order of Operations
Brackets first, then ×
and ÷, then + and –
from left to right.
2 + 3 × 4 ÷ (9 ÷ 3)
= 2 + 12 ÷ 3
= 2 + 4
= 6
Place value
Addition and
Subtraction
Ancient
Number Systems
Estimation
Roman
955 to the nearest 10 is 960
950 to the nearest 100 is 1000
371
_+_ _8_43_
1214
29
× 13 ____
87
__2_9_0
377
937
_–_ _6_4_3
294
68
3 205
with 1
remainder
Leading digit approximation
39 × 326 ≈ 40 × 300 = 12 000
Mental strategies
172 + 216 = 300 + 80 + 8
= 388
98 – 19 = 98 – 20 + 1
= 79
Mental strategies
7 × 31 = 7 × 30 + 7 × 1 = 217
5 × 14 = 10 × 7 = 70
64 ÷ 8 = 32 ÷ 4 = 16 ÷ 2 = 8
156 ÷ 4 = 160 ÷ 4 – 4 ÷ 4
= 40 – 1
= 39
Multiplying by 10, 100, ……
38 × 100 = 3800
38 × 700 = 38 × 7 × 100
= 26 600
The place value of 3 in
1327 is 300.
2 × 100 + 7 × 10 + 3 × 1
is the expanded form of 273.
Algorithms
Algorithms
Multiplication
and Division
Whole numbers
Egyptian
Babylonian
is 71
is 23
1 1
2
2
8
LXXVI is 76
XCIV is 94
is 143
is 21
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48 Chapter 1 Computation with positive integers
Multiple-choice questions
1 Which of the following is equal to 24?
A XXIII B XXIV C XXXLIV
D IVXX E IXXV
2 3 × 1000 + 9 × 10 + 2 × 1 is the expanded form of:
A 3920 B 392 C 3092
D 3902 E 329
3 Which of the following is not true?
A 2 + 3 = 3 + 2 B 2 × 3 = 3 × 2 C (2 × 3) × 4 = 2 × (3 × 4)
D 5 ÷ 2 ≠ 2 ÷ 5 E 7 − 2 = 2 − 7
4 The sum of 198 and 103 is:
A 301 B 304 C 299
D 199 E 95
5 The difference between 378 and 81 is:
A 459 B 297 C 303
D 317 E 299
6 The product of 7 and 21 is:
A 147 B 141 C 21
D 140 E 207
7 The missing digit in this division is:
A 6 B 1 C 9
D 8 E 7
8 The remainder when 317 is divided by 9 is:
A 7 B 5 C 2
D 1 E 0
9 458 rounded to the nearest 100 is:
A 400 B 500 C 460
D 450 E 1000
10 The answer to [2 + 3 × (7 – 4)] ÷ 11 is:
A 1 B 5 C 11
D 121 E 0
7) 2 6
1 1 8
1 5
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Number and Algebra 49
Short-answer questions
1 Write these numbers using the given number systems.
a Egyptian
i 3 ii 31 iii 326
b Babylonian
i 12 ii 60 iii 132
c Roman
i 14 ii 40 iii 146
2 Write down the place value of the digit 5 in these numbers.
a 357 b 5249 c 356 612
3 Use a mental strategy to find these sums and differences.
a 124 + 335 b 687 − 324 c 59 + 36 d 256 − 39
4 Use an algorithm and show your working for these sums and differences.
a 76
+52
b 1528
+ 796
c 329
−138
d 2109
−1814
5 Use a mental strategy to perform these computations.
a 5 × 19 b 22 × 6 c 5 × 44
d 123 ÷ 3 e 264 ÷ 8 f 96 ÷ 4
g 29 × 1000 h 36 × 300 i 14 678 ÷ 1
6 Use an algorithm and show your working for these computations.
a 157
× 9
b 27
×13
c 7)327 d 4)30162
7 Find the missing digits in these computations.
a 2 3
7 3
9 6 1
+
b 2
4 3
2 5 6
−
c 3
2
7 1
0 6 0
3 1
×
d
5) 1
1 3
4 1
8 Round these numbers as indicated.
a 72 (nearest 10) b 3268 (nearest 100) c 951 (nearest 100)
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50 Chapter 1 Computation with positive integers
9 Use leading digit approximation to estimate the answers to these computations.
a 289 + 532 b 22 × 19 c 452 × 11 d 99 ÷ 11
10 Use order of operations to find the answers to these computations.
a 3 × (2 + 6) b 6 − 8 ÷ 4 c 2 × 8 − 12 ÷ 6
d (5 + 2) × 3 − (8 − 7) e 0 × (9 88 234 ÷ 3) f 1 × (3 + 2 × 5)
Extended-response questions
1 A city tower construction uses 4520 tonnes of cement trucked from a factory that is 7 kilometres from
the construction site. Each cement mixer can carry 7 tonnes of cement, and the cement costs $85 per
truck load for the first 30 loads and $55 per load after that.
a How many loads of cement are needed? Add a full load for any remainder.
b Find the total distance travelled by the cement mixers to deliver all loads, assuming they need to
return to the factory after each load.
c Find the total cost of cement needed to make concrete for the tower construction.
d A different cement supplier offers a price of $65 per 8-tonne truck, no matter how many loads are
needed. Find the difference in the cost of cement for the tower by this supplier compared to the
original supplier.
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Number and Algebra 51
2 One night Ricky and her brother Micky decide to have some fun at their father’s sweet shop.
In the shop they find 7 tins of 135 jelly beans each, 9 packets of 121 choc buds, 12 jars of
70 smarties and 32 packets of 5 liquorice sticks.
a Find the total number of sweets that Ricky and Micky find that night.
b Find the difference between the number of choc buds and the number of smarties.
c Ricky and Micky decide to divide each type of sweet into groups of 7 and then eat any
remainder. Which type of sweet will they eat the most of and how many?
d After eating the remainders, they round the total of each sweet using leading digit
approximation. If they round down they put the spare sweets in their pockets. If they round
up they borrow any spare sweets from their pockets. Any leftover in their pockets they can
eat. Do Ricky and Micky get to eat any more sweets?
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Chapter
Angle
relationships
2W
hat you will learn
Points, lines, intervals and angles
Measuring and classifying angles REVISION
Adjacent angles and vertically opposite angles
Transversal lines and parallel lines
Solving geometry problems
Circles and constructions with ruler and compasses FRINGE
Constructions with dynamic geometry software EXTENSION
2A
2B
2C
2D
2E
2F
2G
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thales, pyramids and
the solar eclipse
From the Egyptian pyramids to modern architecture,
points, lines and angles are everywhere. Geometry
is a very visual element of mathematics where the
designs of buildings and the orbits of planets can
be studied using basic objects like points, lines and
circles.
Thales (624–546 BC) is known to be the founder
of Greek geometry. He was an astronomer and
philosopher, and records show he was the fi rst person
to use mathematical geometry to calculate the height
of an Egyptian pyramid using the Sun’s rays and to
accurately predict the timing of a solar eclipse.
NSW Syllabus
for the australian
Curriculum
Strand: Measurement and Geometry
Substrand: aNGlE RElatioNSHipS
Outcome
A student identifi es and uses angle
relationships, including those relating to
transversals on sets of parallel lines.
(MA4–18MG)
53
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Chapter 2 54 Angle relationships
1 Here are some objects labelled I, II, III, IV, V and VI.
i A
B C
ii
A
B iii P
iV V Vi
E
F
D
Which object is:
a a point b a line
c a circle d a segment AB (or interval AB )
e an angle ABC f an angle DEF ?
2 Estimate or guess the size of these angles. Remember there are 360° in a full circle.
a b c
d e f
3 What angle measurements are shown on these protractors?
a b c
90 80 70 60 50 40 30 20 10 0
180 170 160 150 140 130 120 110 100
90 100 110 120 130 140 150 160170 180
0 10 20 30 40 50 60 70 80
4 Decide if the following pairs of numbers add to 180°.
a 75°, 125° b 132°, 48° c 19°, 151°
5 Find the missing value in these simple statements.
a ? + 10° = 90° b ? + 30° = 180°
90 80 70 60 50 40 30 20 10 0
180 170 160 150 140 130 120 110 100
90 100 110 120 130 140 150 160 170 180
0 10 20 30 40 50 60 70 80
90 80 70 60 50 40 30 20 10 0
180 170 160 150 140 130 120 110 100
90 100 110 120 130 140 150 160 170 180
0 10 20 30 40 50 60 70 80
pre-test
c ? + 210° = 360° d ? + 20° + 85° = 360°
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Measurement and Geometry 55
points, lines, intervals and angles
The fundamental building blocks of geometry are
the point, line and plane. They are the basic objects
used to construct angles, triangles and other more
complex shapes and objects. Points and lines do not
actually occupy any area but can be represented on
a page using drawing equipment.
let’s start: Geometry around you
Take a look around the room you are in or consider any solid object near where you are seated (e.g. a
book). Discuss what parts of the room or object could be described using:
• single points
• straight lines
• fl at planes.
2A
The Sun emits light in rays.
■■ A point is usually labelled with a capital letter.
■■ A line passing through two points, A and B, can be called line AB
or line BA and extends indefi nitely in both directions.
■■ A plane is a fl at surface and extends indefi nitely.
■■ Points that all lie on a single line are collinear.
■■ If two lines meet, an intersection point is formed.
■■ More than two lines that meet at the same point are concurrent.
■■ A line segment (or interval) is part of a line with a fi xed length and
end points. If the end points are A and B, then it would be named
segment AB or segment BA (or interval AB or interval BA).
P
B
A
B
C
A
A
B
Key ideas
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Chapter 2 56 Angle relationships
■■ A ray AB is a part of a line with one end point A and passing through
point B. It extends indefinitely in one direction.
■■ When two rays (or lines) meet, an angle is formed at the intersection point
called the vertex. The two rays are called arms of the angle.
■■ An angle is named using three points, with the
B
C
A or
a°
vertex as the middle point. A common type of
notation is ∠ABC or ∠CBA.
■■ Lower-case letters are often used to represent the number of
degrees in an unknown angle.
■■ These two lines are parallel. This is written AB || DC.
■■ These two lines are perpendicular. This is written AB ⊥ CD.
■■ The markings on this diagram show that AB = CD, AD = BC,
∠BAD = ∠BCD and ∠ABC = ∠ADC.
B
A
arm
arm
vertex
B
A
C
D
C B
A D
A B
D C
Key ideas
Example 1 Naming objects
Name these objects.
a
A B
b
Q
R
P c
Q
R
P
Solution Explanation
a segment AB Segment BA, interval AB or interval BA are also acceptable.
b ∠PQR Point Q is the vertex and sits in between P and R. ∠RQP is also correct.
c reflex ∠PQR In diagrams b and c, there are two different angles called PQR: one is acute
and the other is reflex. The word ‘reflex’ is used in situations such as this.
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Measurement and Geometry 57
1 Draw a representation of the following objects.
a a point P b a line AN c an angle ∠ABC
d a ray ST e a plane f three collinear points A, B and C
2 Explain what it means to say:
a three points D, E and F are collinear b three lines are concurrent
3 Match the words line, segment or ray to the correct description.
a Starts from a point and extends indefi nitely in one direction.
b Extends indefi nitely in both directions, passing through two points.
c Starts and ends at two points.
4 Match the words point, line or plane with the
following descriptions.
a the edge of a sheet of paper
b a fl at wall
c the surface of a pool of water on a calm day
d where two walls and a fl oor meet in a room
e where two walls meet in a room
f one side of a cereal packet
g where two sides meet on a box
h where three sides meet on a box
5 Name the following objects.
a T b D
C
c
C
B
A
d e
Q
P
f S
T
Example 1
Exercise 2A
WORKING
MATHEMATICALLY
U F
R PS
C
This mosaic around a fountain in Morocco is made up entirely
of straight lines, even though it looks circular.
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 2 58 Angle relationships
6 Name the angle marked in these diagrams.
a A
B
C
D
O
b
A
B
C
D
c
A
D
B
E
C
d
A B
O
C D
E
e
B D
A
C
7 For each diagram, name the fi ve line segments and the four marked angles using the given labels.
a A
B
D
C
×
b R
Q
S
P
8 Name the set of three labelled points that are collinear in these diagrams.
a
D
A
B
C
b
A
B
C
D
9 State whether the following sets of lines are concurrent.
a b
10 In the following diagrams, name the equal sides and equal angles.
a B
C
A
D
b
C
D B
A
WORKING
MATHEMATICALLY
U F
R PS
C
2A
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Measurement and Geometry 59
11 Count the number of angles formed inside these shapes. Count all angles, including ones that
may be the same size and those angles that are divided by another segment.
a b
12 How many line segments are there on this line? Do
not count AB and BA as they represent the same
segment.
13 A line contains a certain number of labelled points.
For example, this line has three points.
a Complete this table by counting the total number of
segments for the given number of labelled points.
Number of points 1 2 3 4 5 6
Number of segments
b Explain any patterns you see in the table. Is there a quick way of finding the next number in
the table?
14 The lines joining each vertex (corner) of a triangle with the midpoint (middle point) of the
opposite side are drawn here.
a Draw any triangle and use a ruler to measure and mark the
midpoints of each side.
b Join each vertex with the midpoint of the opposite side.
c Are your segments from part b concurrent?
d Do you think your answer to part c will always be true for any triangle? Try one other
triangle of a different size to check.
15 a If points A, B and C are collinear and points A, B and D are collinear, does this mean that
points B, C and D are also collinear?
b If points A, B and C are collinear and points C, D and E are collinear, does this mean that
points B, C and D are also collinear?
Enrichment: The general rule
16 In Question 13 you may have determined a quick
method of finding the number of segments for
the given number of points. If n is the number of
points on the line, can you find a rule (in terms of
n) for the number of segments? Test your rule to
see if it works for at least three cases.
A B
C D E
A B
C
Number of segments
= (2n + 1) × n?
WORKING
MATHEMATICALLY
U F
R PS
C
WORK ING
MATHEMATICALLY
U F
R PS
C
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Chapter 2 60 Angle relationships
Measuring and classifying angles REVISION
Angles are usually described using the unit of measurement
called the degree, where 360 degrees (360°) describes one
full turn. The idea to divide a circle into 360° dates back to
the Babylonians, who used a sexagesimal number system
based on the number 60. Because both 60 and 360 are
numbers that have a large number of factors, many fractions
of these numbers are very easy to calculate.
let’s start: Estimating angles
How good are you at estimating the size of angles?
Estimate the size of these
angles and then check
with a protractor.
Alternatively,
construct an angle using
computer geometry.
Estimate and then check
your angle using the
angle-measuring tool.
2B
What angle is between each spoke on this Ferris
wheel?
■■ Angles are classifi ed according to their size.
angle type Size Examples
acute between 0° and 90°
right 90°
obtuse between 90° and 180°
straight 180°
refl ex between 180° and 360°
revolution 360°
Key ideas
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Measurement and Geometry 61
■■ A protractor can be used to measure angles to within an accuracy of about half a degree.
Some protractors have increasing scales marked both clockwise and anticlockwise from
zero. To use a protractor:
– Place the centre of the protractor on the vertex of the angle.
– Align the base line of the protractor along one arm of the angle.
– Measure the angle using the other arm and the scale on the protractor.
– A reflex angle can be measured by subtracting a measured angle from 360°.
Key ideas Example 2
Measuring with a
protractor
For the angles shown, state the type of angle and measure its size.
a
O B
A b
E
F
G c
O
E
D
Solution Explanation
a acute
∠AOB = 60°
B
A
O
b obtuse
∠EFG = 125°
E
F
G
c reflex
obtuse ∠DOE = 130°
reflex ∠DOE = 360° – 130°
= 230°
O
E
D
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Chapter 2 62 Angle relationships
Example 3 Drawing angles
Use a protractor to draw each of the following angles.
a ∠AOB = 65° b ∠WXY = 130° c ∠MNO = 260°
SolutioN ExplaNatioN
a
O
B
A Step 1: Draw a base line OB.
Step 2: Align the protractor along the base line with
the centre at point O.
Step 3: Measure 65° and mark a point, A.
Step 4: Draw the arm OA.
b
X
W
Y Step 1: Draw a base line XW.
Step 2: Align the protractor along the base line with
the centre at point X.
Step 3: Measure 130° and mark a point, Y.
Step 4: Draw the arm XY.
c
N
M
O Step 1: Draw an angle of 360° – 260° = 100°.
Step 2: Mark the refl ex angle on the opposite side to
the obtuse angle of 100°.
Alternatively, draw a 180° angle and measure an
80° angle to add to the 180° angle.
Exercise 2B REVISION
WORKING
MATHEMATICALLY
U F
R PS
1 Without using a protractor, draw an example of the following types of angles. C
a acute b right c obtuse
d straight e refl ex f revolution
2 How many right angles (i.e. angles of 90°) make up:
a a straight angle b 270° c a revolution?
3 What is the size of the angle measured with these protractors?
a b
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Measurement and Geometry 63
c d
4 For the angles shown, state the type of angle and measure its size.
a b c
d e f
g h i
5 a Write down the size of the angles shown on
this protractor.
i ∠AOB ii ∠BOC
iii ∠COD iv ∠DOE
v ∠EOF
b Find the sum of all the angles from part a. Name a
single angle in the diagram that equals this sum.
6 Use a protractor to draw each of the following angles.
a 40° b 75° c 90° d 135° e 175°
f 205° g 260° h 270° i 295° j 352°
Example 2
O
A
B
C
D
E
F
Example 3
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 2 64 Angle relationships
7 For each of the angles marked in the situations shown,
measure:
a the angle that this ramp makes with the ground
b the angle the Sun’s rays make with the ground
c the angle or pitch of this roof
d the angle between this laptop
screen and the keyboard
8 In the diagram shown at right, there are two marked angles,
∠AOB and ∠BOC. Measure ∠AOB, ∠BOC and ∠AOC.
Does ∠AOB + ∠BOC = ∠AOC ? Why or why not?
A
B
C
O
WORKING
MATHEMATICALLY
U F
R PS
C
2B
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Measurement and Geometry 65
9 A clock face is numbered 1 to 12. Find the angle the minute hand turns in:
a 30 minutes b 1 hour c 15 minutes d 45 minutes
e 5 minutes f 20 minutes g 55 minutes h 1 minute
i 9 minutes j 10.5 minutes k 42.5 minutes l 21.5 minutes
10 A clock face is numbered 1 to 12. Find the angle between the hour hand and the minute hand at:
a 6 p.m. b 3 p.m. c 4 p.m. d 11 a.m.
11 The arrow on this dial starts in an upright position. It then turns by a given
number of degrees clockwise or anticlockwise.
a Find the angle between the arrow in its final position with the arrow
in its original position, as shown in the diagram opposite, which
illustrates part i. Answer with an acute or obtuse angle.
i 290° clockwise ii 290° anticlockwise
iii 450° clockwise iv 450° anticlockwise
v 1000° clockwise vi 1000° anticlockwise
b Did it matter to the answer if the dial was turning clockwise or anticlockwise?
c Explain how you calculated your answer for turns larger than 360°.
12 An acute angle ∠AOB is equal to 60°. Why is it unnecessary to use a
protractor to work out the size of the reflex angle ∠AOB ?
Enrichment: Time challenge
13 Find the angle between the hour hand and the minute hand of a
clock at these times.
a 10:10 a.m. b 4:45 a.m.
c 11:10 p.m. d 2:25 a.m.
e 7:16 p.m. f 9:17 p.m.
?
290°
60°
O
?
A
B
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 2 66 Angle relationships
adjacent angles and vertically opposite angles
Not all angles in a diagram or construction need to be measured directly. Special relationships exist
between pairs of angles at a point and this allows some angles to be calculated exactly without
measurement, even if diagrams are not drawn to scale.
let’s start: Special pairs of angles
By making a drawing or using computer geometry, construct the diagrams below. Measure the two
marked angles. What do you notice about the two marked angles?
A
B
C
O
O
C
A
B
O
D
A B
C
2C
■■ A djacent angles are side by side and share a vertex and an arm. ∠AOB
and ∠BOC in this diagram at right are adjacent angles.
■■ Complementary adjacent ■ Supplementary adjacent ■ Angles in a revolution
angles sum to 90°. angles sum to 180°. sum to 360°.
b°
a°
a°
b°
a°
b°
a + b = 90 a + b = 180 a + b = 360
■■ Vertically opposite angles are formed when two lines intersect. The
opposite angles are equal. The name comes from the fact that the pair
of angles has a common vertex and they sit in opposite positions
across the vertex.
■■ Perpendicular lines meet at a right angle (90°).
– If the line segment AB is perpendicular to line segment CD,
we write AB ⊥ CD.
A
B
C
O
D
C
A
B
Key ideas
a° a°
b°
b°
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Measurement and Geometry 67
Example 4 Finding angles at a point
Without using a protractor, fi nd the size of each angle marked with the letter a.
a
a°
35°
b
a°
55°
c a°
120°
SolutioN ExplaNatioN
a a + 35 = 90
a = 55
Angles in a right angle add to 90.
90 – 35 = 55
b a + 55 = 180
a = 125
Angles on a straight line add to 180.
180 – 55 = 125
c a + 90 + 120 = 360
a + 210 = 360
a = 150
The sum of angles in a revolution is 360.
Simplify by adding 90 and 120.
a is the difference between 210 and 360.
Exercise 2C
WORKING
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1 a Give a value for a and b in this diagram. C
b Calculate a + b. Is your answer 90? If not, check your
measurements.
c Write the missing word: a° and b° are ____________ angles.
2 a Give a value for a and b in this diagram.
b Calculate a + b. Is your answer 180? If not,
check your measurements.
c Write the missing word: a° and b° are
____________ angles.
3 a Give a value for a, b, c and d in this
diagram.
b What do you notice about the sum of the
four angles?
c Write the missing words: b and d
are ______ ______ angles.
a°
b°
a°
b°
c°
d°
a°
b°
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Chapter 2 68 Angle relationships
4 a Name the angle that is complementary
to ∠AOB in this diagram.
A B
C
O
b Name the two angles that are
supplementary to ∠AOB in this
diagram.
A
B
C
D
O
c Name the angle that is vertically opposite to ∠AOB in this diagram.
5 Without using a protractor, fi nd the value of the letter a in each angle. (The diagrams shown
may not be drawn to scale.)
a
a°
30°
b
a°
75°
c
a°
45°
d
a°
110°
e
a°
50°
f
a°
115°
g
a°
37°
h a°
120°
i
a°
140°
6 For each of the given pairs of angles, write C if they are complementary, S if they are
supplementary or N if they are neither.
a 21°, 79° b 130°, 60° c 98°, 82° d 180°, 90°
e 17°, 73° f 31°, 59° g 68°, 22° h 93°, 87°
7 Write a statement like AB ⊥ CD for these pairs of perpendicular line segments.
a H
F
G
E
b V
U
S
T
c P
A
N
X
Example 4
WORKING
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C
A
B C
D
O
WORKING
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C
2C
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Measurement and Geometry 69
8 Without using a protractor, find the value of a in these diagrams.
a
30° a° 30°
b
a°
40°
65°
c
a°
100° 110° ×
d
a° × 45°
e
a°
135°
×
f
a°
40°
×
9 Do these diagrams have the correct information? Give reasons.
a
60°
× 25°
b
140°
40°
c
310°
50°
d
35°
35°
e
42° 138°
f
80° 250°
10 Find the value of a in these diagrams.
a a°
a° a°
b
(2a)°
a°
c
(3a)° (2a)°
a°
×
d
(a + 10)°
(a − 10)°
e
(a − 60)°
(a + 60)°
f
(2a)°
(3a)°
WORKING
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WORKING
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Chapter 2 70 Angle relationships
11 A pizza is divided between four people. Bella is to get twice as much
as Bobo, who gets twice as much as Rick, who gets twice as much as
Marie. Assuming the pizza is cut into triangular pieces, fi nd the angle at
the centre of the pizza for Marie’s piece.
12 Write down a rule connecting the letters in these diagrams; e.g. a + b = 180.
a
a°
b°
b
a°
b°
c°
c
b°
a°
13 What is the minimum number of angles needed in this diagram to
determine all other angles? Explain your answer.
Enrichment: pentagon turns
14 Consider walking around a path represented by this regular
pentagon. All sides have the same length and all internal angles are
equal. At each corner (vertex) you turn an angle of a, as marked.
a How many degrees would you turn in total after walking around
the entire shape. Assume that you face the same direction at the
end as you did at the start.
b Find the value of a.
c Find the value of b.
d Explore the outside and inside angles of other regular
polygons using the same idea. Complete this table to
summarise your results.
b°
b°
b°
b° b°
a°
a°
a°
a°
a°
Regular shape a b
triangle
square
pentagon
hexagon
heptagon
octagon
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2C
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Measurement and Geometry 71
transversal lines and parallel lines
When a line, called a transversal, cuts two or more
other lines a number of angles are formed. Pairs of
these angles are either corresponding, alternate or
cointerior angles, depending on their relative position.
If the transversal cuts two parallel lines then there is a
relationship between the sizes of these special pairs of
angles.
let’s start: What’s formed by a transversal?
Draw a pair of parallel lines using either:
• two sides of a ruler; or
• computer geometry (parallel line tool).
Then cross the two lines with a third line (transversal) at any angle.
Measure each of the eight angles formed and discuss what you fi nd. If
computer geometry is used, drag the transversal and see if your observations
apply to all the cases that you observe.
2D
Multiple angles are formed when this transversal road
intersects with the freeway.
■■ A transversal is a line intersecting two or more
other lines that are usually, but not necessarily,
parallel.
■■ A transversal crossing two lines will form special
pairs of angles. These are:
– corresponding (in corresponding positions)
– alternate (on opposite sides of the transversal
and inside the other two lines)
– cointerior (on the same side of the transversal
and inside the other two lines).
transversal
transversal
×
×
× ×
Key ideas
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Chapter 2 72 Angle relationships
■■ Parallel lines are marked with the same arrow set.
–– If AB is parallel to CD then we write AB || CD.
■■ If a transversal crosses two parallel lines, then:
–– corresponding angles are equal
–– alternate angles are equal
–– cointerior angles are supplementary (i.e. sum to 180°).
corresponding alternate cointerior
B
A
D
C
a° b°
a = b
a° b°
a = b
a°
b°
a = b
a°
b°
a = b a + b = 180
a°
b°
a + b = 180
a°b°
Key ideas
Example 5 Naming pairs of angles
Name the angle that is:
a corresponding to ∠ABF
b alternate to ∠ABF
c cointerior to ∠ABF
d vertically opposite to ∠ABF
Solution Explanation
a ∠HFG
b ∠EFB
c ∠HFB ×
d ∠CBD
A H
G
C
B F
D E
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Measurement and Geometry 73
Example 6 Finding angles in parallel lines
Find the value of a in these diagrams and give a reason for each answer.
a
a°
115°
b
a°
55°
c
a°
110°
Solution Explanation
a a = 115
alternate angles in parallel lines
Alternate angles in parallel lines are equal.
b a = 55
corresponding angles in parallel lines
Corresponding angles in parallel lines are equal.
c a = 180 – 110
= 70
cointerior angles in parallel lines
Cointerior angles in parallel lines sum to 180°.
Example 7 Proving lines are parallel
Giving reasons, state whether the two lines cut by the transversal are parallel.
a
75°
78°
b
122°
58°
Solution Explanation
a not parallel
Alternate angles are not equal.
Parallel lines have equal alternate angles.
b parallel
The cointerior angles sum to 180°.
122° + 58° = 180°
Cointerior angles inside parallel lines are supplementary
(i.e. sum to 180°).
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Chapter 2 74 Angle relationships
1 Use a protractor to measure each of the eight
angles in this diagram.
a How many different angle measurements did you fi nd?
b Do you think that the two lines cut by the transversal are
parallel?
2 Use a protractor to measure each of the eight angles in this diagram.
a How many different angle measurements did you fi nd?
b Do you think that the two lines cut by the transversal are parallel?
3 Choose the word equal or supplementary to complete these sentences.
If a transversal cuts two parallel lines, then:
a alternate angles are _____________. b cointerior angles are
_____________.
c corresponding angles are ________. d vertically opposite angles are ______.
4 Name the angle that is:
a corresponding to ∠ABE
b alternate to ∠ABE
c cointerior to ∠ABE
d vertically opposite to ∠ABE
5 Name the angle that is:
a corresponding to ∠EBH
b alternate to ∠EBH
c cointerior to ∠EBH
d vertically opposite to ∠EBH
6 Name an angle that is:
a corresponding to ∠ACD
b vertically opposite to ∠ACD
Example 5
G
A
D
H
F
C
E
B
G
F
A
C D
E
H
B
H
E B A
C
D
F
G
Exercise 2D
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Measurement and Geometry 75
7 Find the value of a in these diagrams, giving a reason.
a
130° a°
b
a°
70°
c
a°
110°
d
a°
120°
e a°
130°
f
a°
67°
g
a°
115°
h
a°
62°
i
a°
100°
j
a°
117°
k
a°
64°
l
a°
116°
8 Find the value of each unknown pronumeral in the following diagrams.
a
c°
b°
a°
70°
b
c°b°
a° 120°
c
b° c°
d°
a°
82°
d
b° c°
a°
e
c° b°
a° 85°
f
b°
a°
119°
9 Giving reasons, state whether the two lines cut by the transversal are parallel.
a 59°
58°
b
81°
81°
c
112°
68°
d
132° 132°
e
79° 78°
f
60°
100°
Example 6
Example 7
WORK ING
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Chapter 2 76 Angle relationships
10 Find the value of a in these diagrams.
a
35°
a°
b
a° 41°
c
70°
a°
d
150°
a°
e
60°
a°
f
141°
a°
11 Find the value of a in these diagrams.
a
115°
a°
b
80° a° c
62°
a°
d
67°
a°
e
42°
a°
f
57°
a°
g
80°
a° h
130°
a°
i
121°
a°
12 A transversal cuts a set of three parallel lines.
a How many angles are formed?
b How many angles of different sizes are formed if the transversal is not perpendicular to the
three lines?
13 Two roads merge into a freeway at the same angle, as shown. Find the
value of a between the parallel roads and the freeway.
60°
freeway
a°
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2D
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Measurement and Geometry 77
14 This diagram includes two triangles with two sides that are parallel.
a Give a reason why:
i a = 20 ii b = 45
b Now find the values of a and b in the diagrams below.
i
50°
b° 25°
a°
ii
41°
35°
a°
b°
iii
15 This shape is a parallelogram with two pairs of parallel sides.
a Use the 60° angle to find the value of a and b.
b Find the value of c.
c What do you notice about the angles inside a parallelogram?
16 Explain why these diagrams do not contain a pair of parallel lines.
a
150°
40°
b
130°
300°
c
140°
130°
Enrichment: Adding parallel lines
17 Consider this triangle and parallel lines.
a Giving a reason for your answer, name an angle equal to:
i ∠ABD ii ∠CBE
b What do you know about the three angles ∠ABD, ∠DBE
and ∠CBE?
c What do these results tell you about the three inside angles of
the triangle BDE ? Is this true for any triangle? Try a new diagram to check.
18 Use the ideas explored in Question 17 to show that the angles inside a
quadrilateral (i.e. a four-sided shape) must sum to 360°. Use this diagram
to help.
20°
45°
a°
b°
25°
35°
a°
b°
60°
a°
b°
c°
A B C
D E
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Chapter 2 78 Angle relationships
Solving geometry problems
Parallel lines are at the foundation of construction in all
its forms. Imagine the sorts of problems engineers and
builders would face if drawings and constructions could not
accurately use and apply parallel lines. Angles formed by
intersecting beams would be diffi cult to calculate and could
not be transferred to other parts of the building.
let’s start: Not so obvious
Some geometrical problems require a combination of two or more
ideas before a solution can be found. This diagram includes the
unknown angle a°.
• Discuss if it is possible to fi nd the value of a.
• Describe the steps you would take to fi nd the value of a. Discuss your
reasons for each step.
2E
Parallel support beams in the foyer of Parliament
House in Canberra
65°
a°
■■ Some geometrical problems involve more than one step.
Step 1: ∠ABC = 75° (corresponding angles on parallel lines)
Step 2: a = 360 − 75 (angles in a revolution sum to 360°)
= 285
Key ideas
75°
A E
C D
B
a°
Example 8 Finding angles with two steps
Find the value of a in these diagrams.
a
170° A
E
F
C
B
D
a°
b
60° 70°
A
C
B
D
a°
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Measurement and Geometry 79
1 In these diagrams, fi rst fi nd the value of a then fi nd the value of b.
a
65°
a° b°
b
a° 74°
b°
c
125° a°
b°
2 Name the angle in these diagrams (e.g. ∠ABC ) that you would need to fi nd fi rst before fi nding the
value of a. Then fi nd the value of a.
a
60°
E
F
D
C
A
B
a°
b
70°
E
G
F
D
C
A
B
a°
c
70°
60° D
C
A
B
a°
SolutioN ExplaNatioN
a ∠BDE = 360° – 90° – 170°
= 100°
a = 100
Angles in a revolution add to 360°.
∠ ABC corresponds with ∠BDE, and BC and DE are
parallel.
b ∠ ABC = 180° – 70°
= 110°
a = 110 – 60
= 50
∠ ABC and ∠BCD are cointerior angles, with AB and
DC parallel.
∠ ABC = 110° and a + 60° = 110°
Exercise 2E
WORKING
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Chapter 2 80 Angle relationships
3 Find the value of a in these diagrams.
a 300°
D
E
C
a°
A
B
b
150°
D
E F
C
A
B
a°
×
c
65°
D
E
F
C
A
B
a°
×
d
57°
D
E
F
G
H
C
A
B
a°
×
e
130°
D
E
F
H
a°
I
G
C
A
B
×
f
38°
62°
D
C
A
B
a°
g
45°
75°
D
C
A
B a°
h
85°
30°
D
C
A
B
a°
i
45°
C D E
A
a° B
j
a°
40°
80°
E D C
A B
k
a°
35°
D
C
A E
B
l
a°
25°
D
C
E
A B
Example 8
WORKING
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2E
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Measurement and Geometry 81
4 Find the size of ∠ABC in these diagrams.
a
60°
70°
C
A
B
b
130°
110°
A
B
C
c
130°
75° B
A C
d
25°
50°
A
C
B
e
30°
60°
A
B
C
f
40°
35°
B
A
C
5 Find the value of x in each of these diagrams.
a
110°
60°
x°
b
130°
280°
x°
c
140°
100°
x°
6 What is the minimum number of angles you need to know to find all the angles marked in
these diagrams?
a
b°
c°
d°
e°
f °
g°
h°
a°
b
b°
c°
d°
e°
f °
a°
c
c°
b°
e°
d°
a°
7 In these diagrams, the letter x represents a number and 2x means 2 × x. Find the value of x.
a
60°
(2x)°
b
120°
(2x)°
c
(3x)° 60°
WORKING
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WORKING
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Chapter 2 82 Angle relationships
d
50°
(x + 20)°
e
70° (x − 10)°
f
60°
(4x)° 80°
8 Find the value of a in these diagrams.
a
(2a)°
a°
b
(3a)° a°
c
(5a)° 150°
Enrichment: adding parallel lines
9 Find the value of a in these diagrams. You may wish to add one or more parallel lines to each diagram.
a
60°
50°
a°
b
50°
120°
a°
c
a°
50°
80°
d
260°
30°
a°
e
300°
280°
20°
a°
f
140° 70°
a°
WORKING
MATHEMATICALLY
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2E
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Measurement and Geometry 83
Circles and constructions with ruler and compasses
One of the most important characteristics of a circle is that
the distance from the centre to the circle, called the radius,
is always the same. This fact is critical in the construction of
geometrical diagrams and other objects that contain circular
parts like gears and wheels.
let’s start: Features of a circle
Here is a circle with some common features.
Which of the features (radius, diameter, chord or arc)
would change in length if:
• point A is moved around the circle?
• point B is moved away from O so that the size of the
circle changes?
If possible, try constructing this diagram using computer software.
Measure lengths and drag the points to explore other possibilities.
2F
radius
chord
diameter
O
arc
B
A
Gears in a car’s gearbox must be circular.
■■ Common circle features include:
– centre (point at an equal distance from all points on the
circle)
– radius (line interval joining the centre to a point on the
circle. Plural: radii)
– chord (line interval joining two points on the circle)
– diameter (longest chord passing through the centre)
– arc (part of a circle. It is possible for a circle to have either a
minor or major arc.)
– sector (region bounded by two radii and an arc).
■■ A pair of compasses (sometimes called a
compass) and a ruler can be used to construct
geometrical fi gures precisely.
■■ The word bisect means to cut in half.
■■ A tangent to a circle is a line that touches the circle at a point
and is at 90° (perpendicular) to the radius.
radius
chord
diameter
major arc
minor
arc
sector
tangent
Key ideas
F R I N G E
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Chapter 2 84 Angle relationships
Example 9 Constructing an angle bisector
Use a pair of compasses and a ruler to bisect an angle ∠AOB by following steps a to e.
a Draw and label an angle ∠AOB.
b Construct an arc with centre O so that it cuts OA at point C and OB at point D.
c With the same radius construct an arc with centre C and another with centre D. Ensure these arcs
intersect at a point E.
d Mark in the ray OE.
e Measure ∠AOE and ∠DOE. What do you notice?
SolutioN ExplaNatioN
a, b A
C
O D
B
First, draw an angle ∠AOB. The size of the angle is not
important.
Construct an arc using O as the centre to produce points
C and D.
c, d A
C
E
O D
B
Construct E so that the intersecting arcs have the same
radius.
Ray OE completes the construction.
e ∠AOE = ∠BOE The angles are equal, so ray OE bisects ∠AOB.
Exercise 2F FRINGE
WORKING
MATHEMATICALLY
U F
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1 Use a pair of compasses and a ruler to draw a circle with a radius of about 3 cm. Then mark C
and label these features.
a centre O b two points, A and B, at any place on the circle
c radius OA d chord AB e minor arc AB
2 Name the features marked on the circle shown opposite.
3 Use a ruler to draw a segment AB of length 6 cm and then complete
the following.
a Construct a circle with radius 3 cm with centre A. (Use a ruler to
help set the pair of compasses.)
b Construct a circle with radius 3 cm with centre B.
c Do your two circles miss, touch or overlap? Is this what
you expected?
d
b
f
c
a
e
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Measurement and Geometry 85
4 Use a ruler to draw a line segment, AB, of about 5 cm in length.
a Using a pair of compasses, construct arcs with radius 4 cm,
as shown, using:
i centre A ii centre B
b Mark point C as shown and use a ruler to draw the segments:
i AC ii BC
c Measure the angles ∠BAC and ∠ABC. What do you notice?
5 Follow steps a to e to construct a perpendicular line.
a Draw a line segment, AB, of about 5 cm in length.
b Construct overlapping circles of different sizes
using the two centres A and B.
c Mark the intersecting points of the circles and label
these points C and D.
d Draw the line CD and mark the intersection of line
CD and segment AB with the point E.
e Measure ∠AEC with a protractor. What do you
notice?
6 Follow steps a to d to construct a 60° angle.
a Draw a line segment, AB, of about 5 cm in length.
b Construct an arc with centre A and intersecting the segment AB
at C.
c With the same radius construct an arc with centre C and
intersecting the first arc at D.
d Draw the ray AD and measure ∠BAD. What do you notice?
7 Follow steps a to e to construct an angle bisector.
a Draw any angle and label ∠AOB.
b Construct an arc with centre O so that it cuts OA and OB at
points C and D.
c With the same radius, construct an arc with centre C and
another with centre D. Ensure these arcs intersect at a point, E.
d Mark in the ray OE.
e Measure ∠AOE and ∠BOE. What do you notice?
8 Consider the construction of the perpendicular line. (See diagram in Question 5.)
a Explain how to alter the construction so that the point E is the exact midpoint of the
segment AB.
b If point E is at the centre of segment AB, then the line CD will be called the perpendicular
bisector of segment AB. Complete the full construction to produce a perpendicular bisector.
A
C
4 cm 4 cm
B
Example 9
A B
D
E
C
D
60°
C A B
A
C
E
O D
B
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Chapter 2 86 Angle relationships
9 Using the results from Questions 6 and 7, explain how you could construct the angles below.
Try each construction and then check each angle with a protractor.
a 30° b 15°
10 Show how you could construct these angles. After each construction, measure the angle using a
protractor. (You may wish to use the results from Questions 5 and 7 for help.)
a 45° b 22.5°
11 Consider the construction of a perpendicular line. (See the diagram in Question 5.) Do you
think it is possible to construct a perpendicular line using circles with radii of any size? Explain.
12 The diagram in Question 7 shows an acute angle, ∠AOB.
a Do you think it is possible to bisect an obtuse angle? If so, show how.
b Do you think it is possible to bisect a refl ex angle? If so, show how.
Enrichment: No measurement allowed
13 Using only a pair of compasses and a ruler’s edge, see if you can construct these objects. No
measurement is allowed.
a Perpendicular line at the end point of a segment. Start with segment AB.
A B
b Two parallel lines. Start with line AB.
A
B
c A square. Start with segment AB.
A B
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2F
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Measurement and Geometry 87
Constructions with dynamic geometry software EXTENSION
Dynamic geometry software is an ideal tool for constructing geometrical fi gures. Constructing with
dynamic geometry is like constructing with a ruler and a pair of compasses, but there is the added
freedom to drag objects and explore different variations of the same construction. With dynamic geometry
the focus is on ‘construction’ as opposed to ‘drawing’. Although this is more of a challenge initially, the
results are more precise and allow for greater exploration.
let’s start: The disappearing line
Use computer geometry to construct this fi gure.
Add the line CD and check that it makes a right angle.
Drag the points A and B or increase the size of the circles.
Can you drag point A or B to make the line CD disappear? Why
would this happen?
2G
C
D
A B
■■ Using dynamic geometry is like using a
pair of compasses and a ruler.
■■ Objects can be dragged to explore
different cases.
■■ Upon dragging, the geometrical
construction should retain
the desired properties.
■■ The same segment can be used
to ensure two circles have
exactly the same radius.
Key ideas
drag
Example 10 Constructing a 60° angle
Construct an angle of 60° using dynamic geometry software. Then drag one of the starting points to
check the construction.
SolutioN ExplaNatioN
C
A B
Step 1: Construct and label a segment AB.
Step 2: Construct two circles with radius AB and
centres A and B.
Step 3: Mark the intersection C and draw the ray AC.
Step 4: Measure ∠BAC to check.
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Chapter 2 88 Angle relationships
1 a Use dynamic geometry software to construct an angle ∠ABC. Any size
will do.
b Mark and measure the angle using geometry software. Drag the point
A around B to enlarge the angle. See whether you can form all these
types of angles.
i acute ii right iii straight
iv refl ex v revolution
2 Look at the 60° angle construction in Example 10.
a Why do the two circles have exactly the same radius?
b What other common geometrical object could be easily constructed simply by adding one
more segment?
3 Construct each of the following using dynamic geometry software. If necessary, refer back to
Section 2.6 and Exercise 2F to assist you. Check each construction by dragging one of the
starting points. All desired properties should be retained.
a perpendicular line b perpendicular bisector c 60° angle d angle bisector
4 a Use the ‘parallel line’ tool to construct a pair of parallel lines and a transversal.
b Measure the eight angles formed.
c Drag the transversal to change the size of the angles. Check that:
i alternate angles are equal
ii corresponding angles are equal
iii cointerior angles are always supplementary
5 Use dynamic geometry software to construct these angles. You may wish to use the ‘angle bisector’
shortcut tool.
a 30° b 15° c 45°
6 Use dynamic geometry software to construct a six-pointed fl ower. Then drag one of the starting
points to increase or decrease its size.
A
B C
58.2°
Example 10
Exercise 2G EXTENSION
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Measurement and Geometry 89
7 a When using geometry software it may be necessary to use a full circle instead of an arc.
Explain why.
b When constructing a perpendicular bisector, the starting segment AB is used as the radius
of the circles. This is instead of using two circles with different radii. Explain why.
8 Explain why geometrical construction is a precise process, whereas drawing using measurement
is not.
Enrichment: Intricate designs
9 Sketch your own intricate design or use the internet to find a design that uses circles and lines.
Use dynamic geometry to see if it is possible to precisely construct the design. Use colour to
enhance your design.
WORK ING
MATHEMATICALLY
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Chapter 2 90 Angle relationships
the perfect billiard ball path
When a billiard ball bounces off a straight
wall (with no side spin) we can assume
that the angle at which it hits the wall
(incoming angle) is the same as the angle
at which it leaves the wall (outgoing
angle). This is similar to how light refl ects
off a mirror.
Single bounce
Use a ruler and protractor to draw a diagram for each
part and then answer the questions.
a Find the outgoing angle if:
i the incoming angle is 30°
ii the centre angle is 104°
b What geometrical reason did you use to calculate the answer to part a ii above?
Two bounces
Two bounces of a billiard ball on a rectangular table are
shown here.
a Find the values of angles a, b, c, d and e, in that order.
Give a reason for each.
b What can be said about the incoming angle on the fi rst bounce and the outgoing angle on the
second bounce? Give reasons for your answer.
c Accurately draw the path of two bounces using:
i an initial incoming bounce of 20° ii an initial incoming bounce of 55°
More than two bounces
a Draw paths of billiard balls for more than two bounces starting at the
midpoint of one side of a rectangular shape, using the starting incoming
angles below.
i 45° ii 30°
b Repeat part a but use different starting positions. Show accurate diagrams, using the same starting
incoming angle but different starting positions.
c Summarise your fi ndings of this investigation in a report that clearly explains what you have found.
Show clear diagrams for each part of your report.
wall
incoming
angle centre
angle
outgoing
angle
a°
b° c°
d°
e°
30°
investigation
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Measurement and Geometry 91
1 What is the angle between the hour hand and minute hand of a clock at 9:35 a.m.?
2 Two circles are the same size. The shaded circle rolls around the other circle. How many degrees
will it turn before returning to its starting position?
3 Move three matchsticks to turn the fi sh to face the opposite direction.
4 Move three matchsticks to get three squares of the same size.
5 How many angles of different sizes can you form from joining dots in this 2 by 3 grid? One
possible angle is shown for you. Do not count the 180° angle or refl ex angles outside the grid.
6 What is the angle between the hour hand and minute hand of a clock at 2:37 p.m.?
puzzles and challenges
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Chapter 2 92 Angle relationships
Chapter summary
Angles
acute 0° to 90°
right 90°
obtuse 90° to 180°
straight 180°
reflex 180° to 360°
revolution 360°
Angles at a point
Complementary
a + b = 90
Supplementary
c + d = 180
Vertically opposite
a = c
Revolution
a + b + 90 + c + d = 360
a° b°
c° d°
Parallel lines
a = b (corresponding)
a = d (alternate)
a + c = 180 (cointerior)
If a = 120, b = 120,
d = 120 and c = 60.
a°c° d°
transversal
b°
Circle features
radius
diameter
tangent
minor
arc
chord
sector
centre
Geometrical objects
ABC
vertex B
ray BD
segment AB
collinear points B, C, D
line BE
A
B
C
E
D
Constructions
angle
bisector
triangle
perpendicular
line
Measuring angles
3 cm
5 cm
4 cm
Introduction
to Geometry
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Measurement and Geometry 93
Multiple-choice questions
1 Three points are collinear if:
A they are at right angles B they form a 60° angle
C they all lie in a straight line D they are all at the same point
E they form an arc on a circle
2 The angle shown here can be named:
A ∠QRP B ∠PQR C ∠QPR
D ∠QRR E ∠PQP
3 Complementary angles:
A sum to 180° B sum to 270° C sum to 360°
D sum to 90° E sum to 45°
4 A reflex angle is:
A 90° B 180° C between 180° and 360°
D between 0° and 90° E between 90° and 180°
5 What is the reading on this protractor?
A 15° B 30°
C 105° D 165°
E 195°
6 The angle a minute hand on a clock turns in 20 minutes is:
A 72° B 36° C 18° D 144° E 120°
7 If a transversal cuts two parallel lines, then:
A cointerior angles are equal
B alternate angles are supplementary
C corresponding angles are equal
D vertically opposite angles are supplementary
E supplementary angles add to 90°
8 An angle bisector:
A cuts an angle in half B cuts a segment in half
C cuts a line in half D makes a 90° angle
E makes a 180° angle
9 The value of a in this diagram is:
A 115 B 75 C 60 D 55 E 65
Q R
P
90 80 70 60 50 40 30 20 10 0
180 170 160 150 140 130 120 110 100
90 100 110 120 130 140 150 160 170 180
0 10 20 30 40 50 60 70 80
a°
115°
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Chapter 2 94 Angle relationships
10 In this diagram, if line CD is to cut segment AB in half, then:
A segment AB has to be 5 cm
B the radii of the arcs must be the same
C the radii of the arcs must not be the same
D line CD should be 10 cm
E AB should be a line not a segment
Short-answer questions
1 Name each of these objects.
a
C
D b
O
B
A c
P
d e
C
A
f
T
S
2 For the angles shown, state the type of angle and measure its size using a protractor.
a b
c
C
A B
D
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Measurement and Geometry 95
3 Find the angle between the hour and minute hands on a clock at the following times.
a 6 a.m. b 9 p.m. c 3 p.m. d 5 a.m.
4 Without using a protractor, find the value of a in these diagrams.
a
70°
a°
b
130°
a°
c
145°
a°
d
41°
a°
e
a° 75°
f
a°
52°
g
(2a)°
a°
h
(2a)°
a°
i
(a + 30)°
a°
5 Using the letters a, b, c or d given in the diagram, write down
a pair of angles that are:
a vertically opposite
b cointerior
c alternate
d corresponding
e supplementary but not cointerior
6 For each of the following, state whether the two lines cut by the transversal are parallel.
Give reasons for each answer.
a
65°
65°
b
92°
89°
c
130°
60°
b°
a°
d°
c°
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Chapter 2 96 Angle relationships
7 Find the value of a in these diagrams.
a
80°
a°
b
85° a°
c
59°
70°
a°
d
150°
a°
e
140°
a°
f
70°
a° 32°
8 Find the size of ∠ABC in these diagrams.
a
70°
75°
B
A
C
b
80° 60°
B
A
C
c
70°
30°
B
A
C
9 Use these diagrams to help draw your own construction. You will
need a pair of compasses and a ruler.
a angle bisector (Start with any angle size.)
b perpendicular line (Start with a segment of about 6 cm in length.)
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Measurement and Geometry 97
Extended-response questions
1 A factory roof is made up of three sloping sections. The sloping sections are all parallel and the
upright supports are at 90° to the horizontal, as shown. Each roof section makes a 32° angle (or
pitch) with the horizontal.
B C
E
factory
F G
A D
a State the size of each of these angles.
i ∠EAB ii ∠GCD iii ∠ABF iv ∠EBF
b Complete these sentences.
i ∠BAE is to ∠CBF.
ii ∠FBC is to ∠GCB.
iii ∠BCG is to ∠GCD.
c Solar panels are to be placed on the sloping roofs and it is decided that the angle to the
horizontal is to be reduced by 11°. Find the size of these new angles.
i ∠FBC ii ∠FBA iii ∠FCG
2 A circular birthday cake is cut into pieces of equal size, cutting from the centre outwards. Each cut
has an angle of a° at the centre.
Tanya’s family takes four pieces. George’s family takes three pieces.
Sienna’s family takes two pieces. Anita’s family takes two pieces.
Marcus takes one piece.
a How many pieces were taken all together?
b If there is no cake left after all the pieces are taken, find the value of a.
c Find the value of a if:
i half of the cake still remains
ii one-quarter of the cake still remains
iii one-third of the cake still remains
iv one-fifth of the cake still remains
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Chapter
Computation
with positive and
negative integers 3
What you will learn
Working with negative integers
Adding or subtracting a positive integer
Adding or subtracting a negative integer
Multiplying or dividing by an integer
Order of operations with positive and negative integers
The Cartesian plane
3A
3B
3C
3D
3E
3F
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Number and Algebra The coldest place on Earth
The coldest place on Earth is Antarctica, where
maximum daily temperatures below 0° Celsius are
commonplace. The coldest temperature ever recorded
on Earth was about –89°C in 1983 at the Russian
Vostok Station in Antarctica. Temperatures on the
coast of Antarctica can reach 15°C in summer but
average temperatures are generally below 0°C. A
layer of ice surrounds and covers almost all of the
continent. The surrounding ice sits both above and
below sea level, making it very diffi cult for ships to
reach their destinations. Like the temperature scale,
heights above and below sea level can be recorded
using both positive and negative numbers.
NSW Syllabus
for the Australian
Curriculum
Strand: Number and Algebra
Substrand: COMPUTATION WITH INTEGERS
Outcome
A student compares, orders and calculates
with integers, applying a range of
strategies to aid computation.
(MA4–4NA)
99
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Chapter 3 Computation with positive and 100 negative integers
1 Insert the symbols < (is less than) or > (is greater than) to make each statement true.
a 5 7 b 0 10 c 9 11 d 3 0
2 Read the temperature on these thermometers.
a
-10
-5
0 5
10
15
20
°C b
-5
0
5
°C c
-2
0 2 4 6 8
10
°C
3 Evaluate these products.
a 2 × 15 b 11 × 7 c 3 × 13 d 28 × 4
4 Evaluate these quotients.
a 35 ÷ 7 b 121 ÷ 11 c 84 ÷ 12 d 340 ÷ 20
5 Use order of operations to evaluate the following.
a 2 + 5 × 4 b 10 ÷ 2 - 3 c (11 + 15) × 2
d 24 ÷ (8 - 2) e (6 - 3) × (1 + 9) f 8 × (4 - 2) + 10 ÷ 5
6 Decide if the answers to these expressions are positive (i.e. greater than zero) or negative
(i.e. less than zero).
a 5 - 4 b 4 - 5 c 10 × 2 - 21 d 30 - 5 × 4
7 Write down the coordinates (x, y) of A, B and C
for this Cartesian plane.
8 Plot these points on the given Cartesian plane.
a A (2, 3)
b B (4, 1)
c C (5, 4)
d D (0, 2)
2
3
1
0 1 2 3
y
x
A
B
C
2
3
1
0 1 2 3 4 5
4
5
y
x
Pre-test
e E (3, 0)
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Number and Algebra 101
Working with negative integers
The numbers 1, 2, 3, … are considered to be positive
because they are greater than zero (0). Negative numbers
extend the number system to include numbers less than
zero. All the whole numbers less than zero, zero itself and
the whole numbers greater than zero are called integers.
The use of negative numbers dates back to 100 bc
when the Chinese used black rods for positive numbers
and red rods for negative numbers in their rod number
system. These coloured rods were used for commercial
and tax calculations. Later, a great Indian mathematician
named Brahmagupta (598-670) set out the rules for the
use of negative numbers, using the word fortune for
positive and debt for negative. Negative numbers were
used to represent loss in a fi nancial situation.
An English mathematician named John Wallis
(1616-1703) invented the number line and the idea that
numbers have a direction. This helped defi ne our number
system as an infi nite set of numbers extending in both
the positive and negative directions. Today negative numbers are used in all sorts of mathematical
calculations and are considered to be an essential element of our number system.
Let’s start: Simple applications of negative numbers
• Try to name as many situations as possible in which negative numbers are used.
• Give examples of the numbers in each case.
3A
John Wallis invented the number line.
■ Negative numbers are numbers less than zero.
■ Integers are whole numbers that can be negative, zero or positive.
… -4, -3, -2, -1, 0, 1, 2, 3, 4, …
■ The number -4 is read as ‘negative 4’.
■ The number 4 is sometimes written as +4 and
is sometimes read as ‘positive 4’.
■ Every number has direction and magnitude.
■ A number line shows:
– positive numbers to the right of zero
– negative numbers to the left of zero.
■ A thermometer shows:
– positive temperatures above zero
– negative temperatures below zero.
■ Each number other than zero has an opposite.
3 and -3 are examples of opposite numbers. They are equal in magnitude
but opposite in sign.
-4 -3 -2 -1 0 1 2 3 4
negative positive
-4
direction magnitude
or sign
-5
0
5
°C
Key ideas
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Chapter 3 Computation with positive and 102 negative integers
1 What are the missing numbers on these number lines?
a
−3 −1 0 1 3
b
−2 −1 1 3
c
−10 −9 −8 −6
d
−4 −2 −1
2 -5 is the opposite number of 5, and 5 is the opposite number of -5. Write down the opposite
to these numbers.
a 2 b 6 c -3 d -7
e -15 f 21 g 132 h -1071
3 Fill in the blanks using the words greater and less.
a 5 is ___________ than 0 b -3 is ___________ than 0
c 0 is ___________ than -6 d 0 is ___________ than 1
Exercise 3A
WORKING
MATHEMATICALLY
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Example 1 Drawing a number line
Draw a number line, showing all integers from -4 to 2.
SOLUTION EXPLANATION
−4 −3 −2 −1 0 1 2
Use equally spaced markings and put -4 on the left and
2 on the right.
Example 2 Less than or greater than
Insert the symbol < (is less than) or > (is greater than) into these statements to make them true.
a -2 3 b -1 -6
SOLUTION EXPLANATION
a -2 < 3 -2 is to the left of 3 on a number line.
−2 −1 0 1 2 3
b -1 > -6 -1 is to the right of -6 on a number line.
−6 −5 −4 −3 −2 −1 0
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Number and Algebra 103
4 Draw a number line for each description, showing all the given integers.
a from -2 to 2 b from -5 to 1 c from -10 to -6 d from -32 to -25
5 List all the integers that fit the given description.
a from -2 up to 4 b from -7 up to 0
c greater than -3 and less than 2 d greater than -5 and less than 1
e less than 4 and greater than -4 f less than -3 and greater than -10
6 Insert the symbol < (is less than) or > (is greater than) into these statements to make them true.
a 7 9 b 3 2 c 0 -2 d -4 0
e -1 -5 f -7 -6 g -11 -2 h -9 -13
i -3 3 j 3 -3 k -130 1 l -2 -147
7 Give the temperature for these thermometers.
a
-5
0
5
10
°C b
-10
0
10
20
°C c
-20
-10
0
10
°C d
-40
-20
0
20
40
°C
Example 1
Example 2
WORKING
MATHEMATICALLY
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WORKING
MATHEMATICALLY
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8 Arrange these numbers in ascending order. C
a -3, -6, 0, 2, -10, 4, -1 b -304, 126, -142, -2, 1, 71, 0
9 Write the next three numbers in these simple patterns.
a 3, 2, 1, ___, ___, ___ b -8, -6, -4, ___, ___, ___
c 10, 5, 0, ___, ___, ___ d -38, -40, -42, ___, ___, ___
e -91, -87, -83, ___, ___, ___ f 199, 99, -1, ___, ___, ___
10 These lists of numbers show deposits (positive numbers) and withdrawals (negative numbers) for
a month of bank transactions. Find the balance at the end of the month.
a Starting balance $200
-$10
-$130
$25
-$100
$20
_____
Final balance _____
b Starting balance $0
$50
-$60
-$100
$200
-$100
_____
Final balance _____
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Chapter 3 Computation with positive and 104 negative integers
11 If the height above sea level for a plane is a positive number, then the height for a
submarine could be written as a negative number. What is the height relative to sea level
for a submarine at these depths?
a 50 metres b 212.5 metres c 0 metres
12 The difference between two numbers could be thought of as the distance between the numbers on a
number line. For example, the difference between -2 and 1 is 3.
-3 -2 -1 0 1 2 3
3
Find the difference between these pairs of numbers.
a -1 and 1 b -2 and 2 c -3 and 1 d -4 and 3
e -3 and 0 f -4 and -1 g -10 and -4 h -30 and 14
WORKING
MATHEMATICALLY
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Enrichment: The fi nal position
13 For these sets of numbers, a positive number means to move right and a negative number means to
move left. Start at zero each time and fi nd the fi nal position.
-3 -2 -1 0 1 2 3
negative positive
a -1, 4, -5 b 3, -5, -1, 4
c -5, -1, 3, 1, -2, -1, 4 d -10, 20, -7, -14, 8, -4
e -250, 300, -49, -7, 36, -81 f -7001, 6214, -132, 1493, -217
3A
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Number and Algebra 105
■ If a positive number is added to a number, you move right on a number line.
2 + 3 = 5 Start at 2 and move right by 3.
-5 + 2 = -3 Start at -5 and move right by 2.
■ If a positive number is subtracted from a number, you move left on a number line.
2 - 3 = -1 Start at 2 and move left by 3.
-4 - 2 = -6 Start at -4 and move left by 2.
1 2 3 4 5 6
+3
+2
−6 −5 −4 −3 −2
0 1 2
−3
−2 −1 3
−2
−7 −6 −5 −4 −3
Key ideas
Adding or subtracting a positive integer
Adding and subtracting a positive integer can give both positive and
negative answers. For example, when the temperature inside a newly
installed fridge is 20°C but, after being switched on, the temperature
then falls by 25°C, the result is -5°C; i.e. 20 - 25 = -5. If a temperature
of -10°C rises by 5°C, the result is -5°C; i.e. -10 + 5 = -5.
Let’s start: Positive and negative possibilities
Decide if it is possible to fi nd an example of the following.
If so, give a specifi c example.
• A positive number added to a positive number gives
a positive number.
• A positive number added to a positive number gives
a negative number.
• A positive number added to a negative number gives
a positive number.
• A positive number added to a negative number gives
a negative number.
• A positive number subtracted from a positive number gives a positive number.
• A positive number subtracted from a positive number gives a negative number.
• A positive number subtracted from a negative number gives a positive number.
• A positive number subtracted from a negative number gives a negative number.
3B
When food goes into a fridge its
temperature begins to change.
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Chapter 3 Computation with positive and 106 negative integers
Example 3 Adding and subtracting positive integers
Calculate the answer to these additions and subtractions.
a -2 + 3 b -8 + 1 c 5 - 7 d -3 - 3
SOLUTION EXPLANATION
a -2 + 3 = 1
0 1
+3
−3 −2 −1 2
b -8 + 1 = -7
+1
−9 −8 −7 −6 −5
c 5 - 7 = -2
−3 0 1
−7
−2 −1 2 3 4 5 6
d -3 - 3 = -6
−3
−7 −6 −5 −4 −3 −2 −1
1 In which direction (i.e. right or left) on a number line do you move for the following
calculations?
a 2 is added to -5 b 6 is added to -4
c 4 is subtracted from 2 d 3 is subtracted from -4
2 Match up the problems a to d with the number lines A to D.
a 5 - 6 = -1 A
b -2 + 4 = 2 B
c -1 - 3 = -4 C
d -6 + 3 = -3 D
−3 −2 −1 0 1 2 3
−5 −4 −3 −2 −1 0
−7 −6 −5 −4 −3 −2
−2 −1 0 1 2 3 4 5 6
Exercise 3B
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Number and Algebra 107
3 Calculate the answer to these additions. Check your answers using a calculator.
a -1 + 2 b -1 + 4 c -3 + 5 d -10 + 11
e -4 + 3 f -5 + 2 g -11 + 9 h -20 + 18
i -4 + 0 j -8 + 0 k -30 + 29 l -39 + 41
m -130 + 132 n -181 + 172 o -57 + 63 p -99 + 68
4 Calculate the answer to these subtractions. Check your answers using a calculator.
a 4 − 6 b 7 − 8 c 3 − 11 d 1 − 20
e -3 − 1 f -5 − 5 g -2 − 13 h -7 − 0
i -37 − 4 j 39 − 51 k 62 − 84 l -21 − 26
m -100 − 200 n 100 − 200 o 328 − 421 p -496 − 138
5 Find the missing number.
a 2 + = 7 b -2 + = 7 c -2 + = 3 d -4 + = -2
e 5 − = 0 f 3 − = -4 g -9 − = -12 h -20 − = -30
i -6 + = -1 j -8 − = -24 k + 1 = -3 l + 7 = 2
m − 4 = -10 n − 7 = -20 o + 6 = -24 p − 100 = -213
6 Evaluate the following. Remember to work from left to right.
a 3 − 4 + 6 b 2 − 7 − 4 c -1 − 4 + 6 d -5 − 7 − 1
e -3 + 2 − 7 + 9 f -6 + 1 − 20 + 3 g 0 − 9 + 7 − 30 h -15 − 20 + 32 − 1
Example 3a,b
Example 3c,d
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7 Determine how much debt remains in these financial situations.
a owes $300 and pays back $155 b owes $20 and borrows another $35
c owes $21 500 and pays back $16 250
8 a The reading on a thermometer measuring temperature rises 18°C from -15°C. What is
the final temperature?
b The reading on a thermometer measuring temperature falls 7°C from 4°C. What is the
final temperature?
c The reading on a thermometer measuring temperature falls 32°C from -14°C. What is the
final temperature?
9 For an experiment, a chemical solution starts at a temperature of 25°C, falls to -3°C, rises to 15°C
and then falls again to -8°C. What is the total change
in temperature? Add all the changes together for
each rise and fall.
10 An ocean sensor is raised and lowered to different depths
in the sea. Note that -100 metres means 100 metres
below sea level.
a If the sensor is initially at -100 metres and then
raised to -41 metres, how far does the sensor rise?
b If the sensor is initially at -37 metres and then lowered
to -93 metres, how far is the sensor lowered?
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Chapter 3 Computation with positive and 108 negative integers
11 Give an example that suits the description.
a A positive number subtract a positive number equals a negative number.
b A negative number subtract a positive number equals a negative number.
c A negative number add a positive number equals a positive number.
d A negative number add a positive number equals a negative number.
12 a a is a positive integer, b is a positive integer and a > b. For each of the following, decide if the
result will be positive, negative or zero.
i a + b ii a - b iii b - a iv a - a
b a is a negative integer and b is a positive integer. Decide if each of the following is always true.
i a + b is positive ii a - b is negative
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Enrichment: + or – combinations
13 Insert + or - signs into these statements to make them true.
a 3 4 5 = 4 b 1 7 9 4 = -5
c -4 2 1 3 4 = 0 d -20 10 7 36 1 18 = -4
e -a b a b = 0 f -a a 3a b b = a - 2b
Month
Mean Temperature °C
Daily
Minimum
Daily
Maximum
Jan −12.9 −5.8
Feb −11.6 −3.3
Mar −7.5 0.7
Apr −1.8 5.9
May 3.8 12.3
Jun 8.4 16.2
Jul 10.7 18.6
Aug 9.7 17.1
Sep 5.2 12.9
Oct −1.8 4.7
Nov −9.4 −2.4
Dec −12.3 −5.1
3B
Positive and negative numbers are used in everyday life.
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Number and Algebra 109
Adding or subtracting a negative integer
By observing patterns in number calculations, we can see
the effect of adding and subtracting negative integers.
Addition Subtraction
2 + 3 = 5 -1 2 - 3 = -1 +1
2 + 2 = 4 -1 2 - 2 = 0 +1
2 + 1 = 3 -1 2 - 1 = 1 +1
2 + 0 = 2 -1 2 - 0 = 2 +1
2 + (-1) = 1 -1 2 - (-1) = 3 +1
2 + (-2) = 0 -1 2 - (-2) = 4 +1
2 + (-3) = -1 2 - (-3) = 5
So adding -3 is equivalent to subtracting 3, and
subtracting -3 is equivalent to adding 3.
Let’s start: Dealing with debt
Let -$10 represent $10 of debt. Can you write a statement (e.g. 5 + (-10) = -5) to represent the following
fi nancial situations?
• $10 of debt is added to a balance of $5.
• $10 of debt is added to a balance of -$5.
• $10 of debt is removed from a balance of -$15.
3C
If I have $100 of debt and subtract $50 of debt, how
do I calculate my fi nal balance?
■ Adding a negative number is equivalent to subtracting its opposite.
a + (-b) = a - b
2 + (-3) = 2 - 3 = -1
-4 + (-2) = -4 - 2 = -6
■ Subtracting a negative number is equivalent to adding its opposite.
a - (-b) = a + b
5 - (-2) = 5 + 2 = 7
-2 - (-3) = -2 + 3 = 1
■ On a number line, the effect of adding or subtracting a negative number is to reverse the
direction of the operation.
0 1 2
−3
−2 −1 3
−2
−7 −6 −5 −4 −3
+2
4 5 6 7 8
0 1
+3
−3 −2 −1 2
Key ideas
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Chapter 3 Computation with positive and 110 negative integers
1 Write down the missing numbers in these sentences. The fi rst one has been done for you.
a 2 + 5 means that 5 is added to 2.
b -3 + 6 means that is added to .
c 1 + (-3) means that is added to .
d -7 + (-11) means that is added to .
e 5 - 3 means that is subtracted from .
f -2 - 6 means that is subtracted from .
g 7 - (-3) means that is subtracted from .
h -7 - (-11) means that is subtracted from .
2 Complete these sentences.
a Adding -4 is equivalent to subtracting .
b Adding -6 is equivalent to _____________ 6.
c Adding 5 is equivalent to subtracting .
d Adding -11 is equivalent to _____________ 11.
e Subtracting -2 is equivalent to adding .
f Subtracting -7 is equivalent to _____________ 7.
3 State whether each of the following is true or false.
a 2 + (-3) = 5 b 10 + (-1) = 9 c -5 + (-3) = -8 d -6 + (-2) = -4
e 5 - (-1) = 4 f 3 - (-9) = 12 g 2 - (-3) = 1 h -11 - (-12) = -1
Exercise 3C
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Example 4 Adding and subtracting negative integers
Calculate the answer to these additions and subtractions.
a 7 + (-2) b -2 + (-3) c 1 - (-3) d -6 - (-2)
SOLUTION EXPLANATION
a 7 + (-2) = 7 - 2
= 5
Adding -2 is equivalent to subtracting 2.
b -2 + (-3) = -2 - 3
= -5
Adding -3 is equivalent to subtracting 3.
c 1 - (-3) = 1 + 3
= 4
Subtracting -3 is equivalent to adding 3.
d -6 - (-2) = -6 + 2
= -4
Subtracting -2 is equivalent to adding 2.
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Number and Algebra 111
4 Calculate the answer to each of these additions. Check your answer using a calculator.
a 3 + (-2) b 8 + (-3) c 12 + (-6) d 9 + (-7)
e 1 + (-4) f 6 + (-11) g 20 + (-22) h 0 + (-4)
i -2 + (-1) j -7 + (-15) k -5 + (-30) l -28 + (-52)
m -7 + (-3) n -20 + (-9) o -31 + (-19) p -103 + (-9)
5 Calculate the answer to each of these subtractions. Check your answer using a calculator.
a 2 − (-3) b 5 − (-6) c 20 − (-30) d 29 − (-61)
e -5 − (-1) f -7 − (-4) g -11 − (-6) h -41 − (-7)
i -4 − (-6) j -9 − (-10) k -20 − (-20) l -96 − (-104)
m 5 − (-23) n 28 − (-6) o -31 − (-19) p -104 − (-28)
6 Find the missing number.
a 2 + = -1 b 3 + = -7 c -2 + = -6 d + (-3) = 1
e + (-10) = -11 f + (-4) = 0 g 5 − = 6 h 2 − = 7
i -1 − = 3 j − (-3) = 7 k − (-10) = 12 l − (-4) = -20
m 5 − = 11 n − (-2) = -3 o -2 − = -4 p + (-5) = -1
7 Calculate the answer, working from left to right.
a 3 + (-2) + (-1) b 2 + (-1) + (-6) c 3 − (-1) − (-4)
d 10 − (-6) + (-4) e -7 − (-1) + (-3) f -20 − (-10) − (-15)
g -9 − (-19) + (-16) h -15 − (-20) + (-96) i -13 − (-19) + (-21)
j -2 − (-3) − (-5) k -18 − (-16) − (-19) l 5 + (-20) − (-26)
Example 4a,b
Example 4c,d
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8 A diver is at a height of -90 metres from the surface of the sea. During a diving exercise, the
diver rises 50 metres, falls 138 metres and then rises once again by 35 metres. What is the
diver’s final height from sea level?
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Chapter 3 Computation with positive and 112 negative integers
9 A small business has a bank balance of -$50 000. An amount of $20 000 of extra debt is
added to the balance and, later, $35 000 is paid back. What is the fi nal balance?
10 $100 of debt is added to an existing balance of $50 of debt. Later, $120 of debt is removed from
the balance. What is the fi nal balance?
11 Here is a profi t graph showing the profi t for each
month of the fi rst half of the year for a bakery shop.
a What is the profi t for:
i February?
ii April?
b What is the overall profi t for the 6 months?
12 Complete these magic squares, using addition. The sum of each row, column and diagonal should
be the same.
a
-2 5
1
4
b
-6
-3 -17
-7
4
6
2
-4
-6
-2
-8
-10
0 J F M J
8
10
Profit ($1000s)
Month
A M
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3C
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Number and Algebra 113
13 Write these sentences as mathematical statements, e.g. 2 + (-3).
a The sum of 3 and 4. b The sum of -2 and -9.
c The difference between 5 and -2. d The difference between -2 and 1.
e The sum of a and the opposite of b.
f The difference between a and the opposite of b.
14 Simplify these numbers. Hint: In part a, -(-4) is the same as 0 − (-4).
a -(-4) b -(-(-1)) c -(-(-(-(-3))))
15 a If a is a positive number and b is a negative number, decide if each of the following statements
is always true or false.
i a + b is negative ii a − b is positive
b If a is a negative number and b is a negative number, decide if each of the following statements
is always true or false.
i a + b is negative ii a − b is positive
c If a and b are both negative numbers and b < a, is a − b always positive? Give reasons.
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Enrichment: Have some fun!
16 Write down the value of these expressions.
a 1 + 2 + 3 + 4 + 5 + … + 99 + 100
b 1 - 2 + 3 - 4 + 5 - … + 99 - 100
c 1 + 2 - 3 - 4 + 5 + 6 - 7 - 8 + 9 + 10 - 11 - 12 … - 99 - 100
17 Use a spreadsheet to check your answers to Question 16.
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Chapter 3 Computation with positive and 114 negative integers
Multiplying or dividing by an integer
The rules for multiplication and division of integers can be developed by considering repeated addition.
For example: 4 groups of -3 is -3 + (-3) + (-3) + (-3) = -12.
So, 4 × (-3) = -12.
Also, -3 × 4 = -12 since a × b = b × a.
We also know that if 5 × 7 = 35, then 35 ÷ 7 = 5.
Similarly, if 4 × (-3) = -12 then -12 ÷ (-3) = 4. This is saying there are 4 groups of -3 in -12, which we
know from the repeated addition above.
Also, -12 ÷ 4 = -3.
These examples give rise to the rules governing the multiplication and division of negative numbers.
Let’s start: Patterns in tables
Complete this table of values for multiplication by noticing the patterns. What does the table of values tell
you about the rules for multiplying negative integers?
× -3 -2 -1 0 1 2 3
-3 0
-2 0
-1 0
0 0 0 0 0 0 0 0
1 0 1
2 0 2 4
3 0
3D
■ The product or quotient of two numbers of the same sign (i.e. positive or negative) is a
positive number.
So a × b = ab and -a × (-b) = ab
e.g. 3 × 4 = 12 or -3 × (-4) = 12
and a ÷ b = a
b
and -a ÷ (-b) = a
b
e.g. 12 ÷ 4 = 3 or -12 ÷ (-4) = 3
■ The product or quotient of two numbers of the opposite sign (i.e. positive and negative) is a
negative number.
So -a × b = -ab and a × (-b) = -ab
e.g. -3 × 4 = -12 or 3 × (-4) = -12
and -a ÷ b = − a
b
and a ÷ (-b) = − a
b
e.g. -12 ÷ 3 = -4 or 12 ÷ (-3) = -4
Key ideas
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Number and Algebra 115
1 Complete these product tables.
a × -2 -1 0 1 2
-2 0
-1 0
0 0 0 0 0 0
1 0 1 2
2 0
b × -4 -2 0 2 4
-4 16
-2
0 0
2
4 8
2 Write down the missing number.
a 2 × (-3) = -6, so -6 ÷ (-3) = b 2 × (-3) = -6, so -6 ÷ 2 =
c -16 ÷ 4 = -4, so × 4 = -16 d 16 ÷ (-4) = -4, so × (-4) = 16
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Example 5 Multiplying and dividing integers
Calculate these products and quotients.
a 5 × (-6) b -3 × (-7) c -36 ÷ (-4) d -18 ÷ 9
SOLUTION EXPLANATION
a 5 × (-6) = -30 The two numbers are of opposite sign, so the answer
is negative.
b -3 × (-7) = 21 The two numbers are of the same sign, so the answer
is positive.
c -36 ÷ (-4) = 9 Both numbers are negative, so the answer is positive.
d -18 ÷ 9 = -2 The two numbers are of opposite sign, so the answer
is negative.
Example 6 Working with mixed operations
Work from left to right to fi nd the answer to -7 × 4 ÷ (-2).
SOLUTION EXPLANATION
-7 × 4 ÷ (-2) = -28 ÷ (-2)
= 14
First, calculate -7 × 4.
Then calculate -28 ÷ (-2).
Exercise 3D
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Chapter 3 Computation with positive and 116 negative integers
3 Complete each sentence by inserting the missing word positive or negative.
a The product (×) of two positive numbers is ________________.
b The product (×) of two negative numbers is ________________.
c The product (×) of two numbers with opposite signs is ________________.
d The quotient (÷) of two positive numbers is ________________.
e The quotient (÷) of two negative numbers is ________________.
f The quotient (÷) of two numbers with opposite signs is ________________.
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4 Calculate the answer to these products.
a 3 × (-5) b 1 × (-10) c -3 × 2 d -9 × 6
e -8 × (-4) f -2 × (-14) g -12 × (-12) h -11 × 9
i -13 × 3 j 7 × (-12) k -19 × (-2) l -36 × 3
m -6 × (-11) n 5 × (-9) o -21 × (-3) p -36 × (-2)
5 Calculate the answer to these quotients.
a 14 ÷ (-7) b 36 ÷ (-3) c -40 ÷ 20 d -100 ÷ 25
e -9 ÷ (-3) f -19 ÷ (-19) g -25 ÷ 5 h 38 ÷ (-2)
i 84 ÷ (-12) j -108 ÷ 9 k -136 ÷ 2 l -1000 ÷ (-125)
m -132 ÷ (-11) n -39 ÷ (-3) o 78 ÷ (-6) p -156 ÷ (-12)
6 Work from left to right to fi nd the answer. Check your answer using a calculator.
a 2 × (-3) × (-4) b -1 × 5 × (-3) c -10 ÷ 5 × 2
d -15 ÷ (-3) × 1 e -2 × 7 ÷ (-14) f 100 ÷ (-20) × 2
g 48 ÷ (-2) × (-3) h -36 × 2 ÷ (-4) i -125 ÷ 25 ÷ (-5)
j -8 ÷ (-8) ÷ (-1) k 46 ÷ (-2) × (-3) × (-1) l -108 ÷ (-12) ÷ (-3)
7 Write down the missing number in these calculations.
a 5 × = -35 b × (-2) = -8 c 16 ÷ = -4
d -32 ÷ = -4 e ÷ (-3) = -9 f ÷ 7 = -20
g -5000 × = -10 000 h -87 × = 261 i 243 ÷ = -81
j 50 ÷ = -50 k -92 × = 184 l -800 ÷ = -20
8 Remember that
9
3
means 9 ÷ 3. Use this knowledge to simplify each of the following.
a −12
4
b
21
-7 c
-
-
40
5 d
-124
-4
e −
−
15
5
f -
-100
20 g
-900
30
h
20 000
-200
9 Given that 32 = 3 × 3 = 9 and (-3)2 = -3 × (-3) = 9, simplify each of the following.
a (-2)2 b (-1)2 c (-9)2 d (-10)2
e (-6)2 f (-8)2 g (-3)2 h (-1.5)2
Example 5a,b
Example 5c,d
Example 6
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3D
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Number and Algebra 117
10 List the different pairs of integers that multiply to give these numbers.
a 6 b 16 c -5 d -24
11 Insert a multiplication or division sign between the numbers to make a true statement.
a 2 -3 -6 = 1 b -25 -5 3 = 15
c -36 2 -3 = 216 d -19 -19 15 = 15
12 a There are two distinct pairs of numbers whose product is -8 and difference is 6. What are the
two numbers?
b The quotient of two numbers is -11 and their difference is 36. What are the two numbers?
There are two distinct pairs to find.
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13 Given that 24 means 2 × 2 × 2 × 2 and (-2)4 = -2 × -2 × -2 × -2
a Calculate:
i (-2)3 ii (-2)6 iii (-3)3 iv (-3)4
b Which questions from part a give positive answers and why?
c Which questions from part a give negative answers and why?
14 a × b is equivalent to ab, and 2 × (-3) is equivalent to -(2 × 3). Use this information to simplify
these expressions.
a a × (-b) b -a × b c -a × (-b)
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Enrichment: Multiplication and division with negative integers
15 (-1) + (-2) + (-3) + (-4) = -10 and (-1) × (-2) × (-3) × (-4) = 24. Therefore, it is possible to use the
numbers -1, -2, -3 and -4 to achieve a ‘result’ of -10 and 24. What other ‘results’ can you find
using those four numbers and any mathematical operations?
For example: What is (-1) × (-2) + (-3) × (-4)? Can you find expressions for every integer from
-20 to 20?
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Chapter 3 Computation with positive and 118 negative integers
Order of operations with positive and
negative integers
We have learnt from our study of positive whole numbers that there is a particular order to follow when
dealing with mixed operations and brackets. This order also applies when dealing with negative numbers.
For example: -2 + 3 × (-4) is different from (-2 + 3) × (-4).
Let’s start: Brackets or not?
During a classroom debate about the statement 3 × (-4) - 8 ÷ (-2) = -8:
• Lil says that the statement needs to have brackets to make it true.
• Max says that even with brackets it is impossible to make it true.
• Riley says that it is correct as it is and there is no need for brackets.
Who is correct and why?
3E
■ When working with more than one operation and with positive
and/or negative numbers:
– Deal with brackets fi rst.
– Do multiplication and division next, working from left to right.
– Do addition and subtraction last, working from left to right.
-2 × 3 - (10 + (-2)) ÷ 4
2nd 1st
-6 8
3rd
2
last
-8
Key ideas
Example 7 Using order of operations
Use order of operations to evaluate the following.
a 5 + 2 × (-3) b -6 × 2 - 10 ÷ (-5)
SOLUTION EXPLANATION
a 5 + 2 × (-3) = 5 + (-6)
= -1
Do the multiplication before the addition.
b -6 × 2 - 10 ÷ (-5) = -12 - (-2)
= -12 + 2
= -10
Do the multiplication and division fi rst.
When subtracting -2, add its opposite.
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Number and Algebra 119
1 Which operation (i.e. addition, subtraction, multiplication or division) is done fi rst in each
of the following problems?
a -2 ÷ 2 + 1 b 8 × (-6) - 4 c -3 + 2 × (-6)
d 7 - (-8) ÷ 4 e (-2 + 3) ÷ 5 f -6 ÷ (4 - (-2))
g -4 × 3 ÷ (-6) h (2 + 3 × (-2)) + 1 i -11 ÷ (7 - 2 × (-2))
2 Classify each of the following statements as true or false.
a -4 + 2 × 3 = -4 + (2 × 3) b -4 + 2 × 3 = (-4 + 2) × 3
c 8 × (2 - (-2)) = 8 × 4 d 8 × (2 - (-2)) = 8 × 0
e -40 - 20 ÷ (-5) = (-40 - 20) ÷ (-5) f -40 - 20 ÷ (-5) = -40 - (20 ÷ (-5))
Exercise 3E
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3 Use order of operations to evaluate the following. Check your answer using a calculator.
a 2 + 3 × (-3) b 9 + 10 ÷ (-5) c 20 + (-4) ÷ 4
d 18 + (-9) × 1 e 10 - 2 × (-3) f 10 - 1 × (-4)
g -8 - (-7) × 2 h -2 × 4 + 8 × (-3) i -3 × (-1) + 4 × (-2)
j 12 ÷ (-6) + 4 ÷ (-2) k -30 ÷ 5 - 6 × 2 l -2 × 3 - 4 ÷ (-2)
m 8 × (-2) - (-3) × 2 n -1 × 0 - (-4) × 1 o 0 × (-3) - (-4) × 0 + 0
Example 7
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Example 8 Order of operations with brackets
Use order of operations to evaluate the following.
a (-2 - 1) × 8 b 5 ÷ (-10 + 5) + 5 c − − +
6
10 2
3
SOLUTION EXPLANATION
a (-2 - 1) × 8 = -3 × 8
= -24
Deal with brackets fi rst.
b 5 ÷ (-10 + 5) + 5 = 5 ÷ (-5) + 5
= -1 + 5
= 4
Deal with brackets fi rst. Then do the division before
the subtraction.
c − − + = − −
= − −
= −
6
10 2
3
6
12
3
6 4
10
Simplify the fraction fi rst.
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Chapter 3 Computation with positive and 120 negative integers
4 Use order of operations to evaluate the following. Check your answer using a calculator.
a (3 + 2) × (-2) b 8 4
2
−
−
c -3 × (-2 + 4)
d -1 × (7 - 8) e 10
4 − (−1)
f (2 + (-3)) × (-9)
g 24 12
16 4
−
+ (− )
h (3 - 7) ÷ (-1 + 0) i -2 × (8 - 4) + (-6)
j -2 - 3 × (-1 + 7) k 0 + (-2) ÷ (1 - 2) l 1
2 3
3 2
− × −
− − −
( )
( )
m (-3 + (-5)) × (-2 - (-1)) n −
− +
× 3
1 4
6 o -5 - (8 + (-2)) + 9 ÷ (-9)
5 A shop owner had bought socks at $5 a pair but, during an economic downturn, sold them
for $3 a pair. In a particular week, 124 pairs are sold and there are other costs of $280.
What is the shop owner’s overall loss for the week?
6 A debt of $550 is doubled and then $350 of debt is removed each month for 3 months. What is the
fi nal balance?
7 Insert brackets to make each statement true.
a -2 + 3 × 8 = 8 b -10 ÷ 4 + 1 = -2 c -1 + 7 × 2 - 15 = -3
d -5 - 1 ÷ (-6) = 1 e 3 - 8 ÷ 5 + 1 = 0 f 50 × 7 - 8 × (-1) = 50
g -2 × 3 - (-7) - 1 = -21 h -3 + 9 ÷ (-7) + 5 = -3 i 32 - (-8) ÷ (-3) + 7 = 10
8 By inserting only one pair of brackets, how many different answers are possible for this calculation?
Also include the answers for which brackets are not used.
-2 + 8 × (-4) - (-3)
Example 8
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
3E
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Number and Algebra 121
9 If brackets are removed from these problems, does the answer change?
a (2 × 3) − (-4) b (8 ÷ (-2)) − 1 c (-2 + 3) × 4
d 9 ÷ (-4 + 1) e (9 − (-3) × 2) + 1 f (-1 + 8 ÷ (-2)) × 2
10 State if each of the following is generally true or false.
a (-3 + 1) + (-7) = -3 + (1 + (-7)) b (-3 + 1) − (-7) = -3 + (1 − (-7))
c (a + b) + c = a + (b + c) d (a + b) − c = a + (b − c)
e (a − b) + c = a − (b + c) f (a − b) − c = a − (b − c)
11 a Given that 53 = 5 × 5 × 5, is the answer to each of the following positive or negative?
i -6 × (-4) × (-8) × (-108) × (-96) ii -100 ÷ (-2) ÷ 2 ÷ (-5)
iii (-3)3 iv -1 × (-2)3
v − × − × × × −
−
6 3 4 7 3
2 2
( ) ( )
( )
vi
( ) ( )
( ) ( )
− × −
− × −
1 1
1 1
2
3
b Explain the strategy you used to answer the questions in part a.
Enrichment: Powers and negative numbers
12 First, note that:
• 24 = 2 × 2 × 2 × 2 = 16
• (-2)4 = -2 × (-2) × (-2) × (-2) = 16
• -24 = -(2 × 2 × 2 × 2) = -16
When evaluating expressions with powers, the power is dealt with first in the order of operations.
For example: ((-2)3 − 1) ÷ (-3) = (-8 − 1) ÷ (-3) = -9 ÷ (-3) = 3
Evaluate each of the following.
a 22 b (-2)2 c -22
d (-2)5 e -25 f (32 − 1) × 4
g ((-3)3 − 1) ÷ (-14) h 30 ÷ (1 − 42) i -10 000 ÷ (-10)4
13 Kevin wants to raise -3 to the power of 4. He
types -34 into a calculator and gets -81. Explain
what Kevin has done wrong.
WORK ING
MATHEMATICALLY
U F
R PS
C
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Chapter 3 Computation with positive and 122 negative integers
The Cartesian plane
During the 17th century, two wellknown
mathematicians,
René Descartes and Pierre de Fermat, independently developed
the idea of a number plane. The precise positions of points
are illustrated using coordinates, and these points can be
plotted using the axes as measuring guides. This invention
revolutionised the study of mathematics and provided a
vital link between geometry and algebra. The number
plane, or coordinate plane, is also called the Cartesian plane
(named after Descartes). It uses two axes at right angles that
extend in both the positive and negative directions.
Let’s start: North, south, east and west
The units for this grid are in metres.
René starts at position O and moves:
• 3 m east
• 2 m south
• 4 m west
• 5 m north.
Pierre starts at position O and moves:
• 1 m west
• 3 m south
• 4 m east
• 5 m north.
Using the number plane, how would you describe René and Pierre’s fi nal positions?
3F
Mathematician and philosopher René Descartes
2
3
1
-2
-3
-3 -2 -1-1 1 2 3 O
N
S
W E
■ The Cartesian plane or number plane uses two axes
(xaxis
and yaxis)
at right angles. Each axis uses a
scale that includes both positive and negative
numbers.
■ A point plotted on the plane has an x and
y coordinate, which is written as (x, y). The
x coordinate is written before the y coordinate,
as in the alphabet.
■ The point (0, 0) is called the origin or O.
2
3
1
-2
-3
-1
(-1, -2)
(-3, 3)
(4, -3)
(2, 3)
-4 -3 -2 -1
-4
O 1 2 3 4
4
y
x
Key ideas
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Number and Algebra 123
Example 9 Finding coordinates
For the Cartesian plane shown, write down the coordinates
of the points labelled A, B, C and D.
SOLUTION EXPLANATION
A = (1, 1)
B = (3, -2)
C = (-2, -4)
D = (-3, 3)
For each point, write the x coordinate fi rst (from the
horizontal axis) followed by the y coordinate (from the
vertical axis).
2
3
1
-2
-3
-1
D
A
B
C
-4 -3 -2 -1
-4
O 1 2 3 4
4
y
x
1 Match the points A, B, C, D, E, F, G and H with the given coordinates.
a (-1, 3) b (2, -3)
c (2, 1) d (-2, -2)
e (3, 3) f (-3, 1)
g (1, -2) h (-1, -1)
2 Count the number of points, shown as dots, on this plane that have:
a both x and y coordinates as positive numbers
b an x coordinate as a positive number
c a y coordinate as a positive number
d an x coordinate as a negative number
e a y coordinate as a negative number
f both x and y coordinates as negative numbers
g neither x nor y as positive or negative numbers
2
3
1
-2
-3
-1
-3 -2 -1 O 1 2 3
y
x
A
F
B
G
C
D E
H
2
3
1
-2
-1
-1 O 1 2 3
y
x
Exercise 3F
WORKING
MATHEMATICALLY
U F
R PS
C
Example 9 Finding coordinates
For the Cartesian plane shown, write down the coordinates
of the points labelled A, B, C and D.
SOLUTION EXPLANATION
A = (1, 1)
B = (3, -2)
C = (-2, -4)
D = (-3, 3)
For each point, write the x coordinate fi rst (from the
horizontal axis) followed by the y coordinate (from the
vertical axis).
2
3
1
-2
-3
-1
D
A
B
C
-4 -3 -2 -1
-4
O 1 2 3 4
4
y
x
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Chapter 3 Computation with positive and 124 negative integers
3 For the Cartesian plane given, write down the
coordinates of the points labelled A, B, C, D, E, F,
G and H.
4 a Draw a set of axes, using 1 cm spacings. Use -4 to 4 on both axes.
b Now plot these points.
i (-3, 2) ii (1, 4) iii (2, -1)
iv (-2, -4) v (2, 2) vi (-1, 4)
vii (-3, -1) viii (1, -2)
5 For the number plane given, write down the
coordinates of the points labelled A, B, C, D, E, F,
G and H.
6 Seven points have the following x and y coordinates.
x -3 -2 -1 0 1 2 3
y -2 -1 0 1 2 3 4
a Plot the seven points on a Cartesian plane. Use -3 to 3 on the xaxis
and -2 to 4 on the yaxis.
b What do you notice about these seven points on the Cartesian plane?
7 Seven points have the following x and y coordinates.
x -3 -2 -1 0 1 2 3
y 5 3 1 -1 -3 -5 -7
a Plot the seven points on a number plane. Use -3 to 3 on the xaxis
and -7 to 5 on the yaxis.
b What do you notice about these seven points on the number plane?
Example 9
2
3
1
-2
-3
-1
H
A
B
C
-4 -3 -2 -1
-4
O 1 2 3 4
4
y
x
E
D
F
G
2
3
1
-2
-3
-1
B
-4 -3 -2 -1
-4
O 1 2 3 4
4
y
x
D
E
H
G
C F A
WORKING
MATHEMATICALLY
U F
R PS
C
3F
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Number and Algebra 125
8 When plotted on the Cartesian plane, what shape does each set of points form?
a A (-2, 0), B (0, 3), C (2, 0) b A (-3, -1), B (-3, 2), C (1, 2), D (1, -1)
c A (-4, -2), B (3, -2), C (1, 2), D (-1, 2) d A (-3, 1), B (-1, 3), C (4, 1), D (-1, -1)
9 Using the origin as one corner, the point A(3, 2) as the opposite corner
and the axes as two of the sides, a rectangle can be positioned on a set
of axes, as shown opposite. Its area is 6 square units. Find the area
of the rectangle if the point A is:
a (2, 2) b (-3, 2)
c (-1, -4) d (3, -5)
10 Karen’s bushwalk starts at a point (2, 2) on a grid
map. Each square on the map represents 1 kilometre.
If Karen walks to the points (2, -7), then (-4, -7),
then (-4, 0) and then (2, 0), how far has she walked
in total?
11 The points A(-2, 0), B (-1, ?) and C (0, 4) all lie on
a straight line. Find the y coordinate of point B.
12 The points A(-4, 8), B (-1, ?) and C (2, -2) all lie
on a straight line. Find the y coordinate of point B.
2
3
1
1 2 3
y
x
A
O
WORKING
MATHEMATICALLY
U F
R PS
C
13 Consider the points A(-2, 2), B (0, 2) and C (3, -2).
a Which point is closest to (0, 0)?
b Which point is farthest from (0, 0)?
c List the given points in order from closest to farthest from the origin, O.
14 A point (a, b) sits on the number plane in one of the four regions
1, 2, 3 or 4, as shown. These regions are called quadrants.
a Name the quadrant or quadrants that include the points where:
i a > 0 ii a > 0 and b < 0
iii b < 0 iv a < 0 and b < 0
b Shade the region that includes all points for which b > a.
y
x
2 1
3 4
O
WORKING
MATHEMATICALLY
U F
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C
Enrichment: Complete the shape
15 Consider the points A(0, 0) and B(3, 1).
a ABCD is a square. Write down the coordinates of C and D if C is in the first quadrant.
b ABE is an isosceles right-angled triangle. There are four possible locations for point E if AB is not
the hypotenuse. List them all.
c G is the point (1, 3) and ABGH is a parallelogram. Write down the coordinates of H.
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Chapter 3 Computation with positive and 126 negative integers
Account balance with spreadsheets
If you have money saved in a bank account, your account balance should be positive. If you take out or
spend too much money, your account balance may become negative.
a Set up a spreadsheet to record and calculate a bank balance. Enter the given information describing
one week of deposits and withdrawals, as shown.
b i For the given spreadsheet, what is the balance at the end of May 1st?
ii On which day does the balance become negative?
c Enter this formula into cell E5: = E4+C5-D5
Fill down to reveal the balance after each day.
d Enter another week of deposits and withdrawals so that the balance shows both positive
and negative amounts.
e Now alter your opening balance. What opening balance is needed so that the balance never becomes
negative? Is there more than one value? What is the least amount?
f Investigate how positive and negative numbers are used on credit card accounts. Give a
brief explanation.
Investigation
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Number and Algebra 127
1 Complete these magic squares. All rows, columns and diagonals sum to the same number.
a
-5
0
-6 -1
b
-9 5 -6
-4 -1
1
3 -8 6
c
-1
-3
-7 -5 -4 -10
-2 -13 1
2 Find the next three numbers in these patterns.
a 3, -9, 27, ___, ___, ___
b -32, 16, -8, ___, ___, ___
c 0, -1, -3, -6, ___, ___, ___
d -1, -1, -2, -3, -5, ___, ___, ___
3 Evaluate the following.
a -100 + (-98) + (-96) + … + 98 + 100
b (50 - 53) + (49 - 52) + (48 - 51) + … + (0 - 3)
4 Insert brackets and symbols (+, -, ×, ÷) into these number sentences to make them true.
a -3 4 -2 = -6
b -2 5 -1 11 = 21
c 1 30 -6 -2 = -3
5 a The difference between two numbers is 14 and their sum is 8. What are the two numbers?
b The difference between two numbers is 31 and their sum is 11. What are the two numbers?
6 Place the integers -3, -2, -1, 0, 1 and 2 into the triangle so that the sum of every side is:
a -3
b 0
c -2
Puzzles and challenges
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128
Chapter 3 Computation with positive and negative integers
Chapter summary
Integers
…, −3, −2, −1,
0, 1, 2, 3, …
−3 + 5 = 2
−4 + 3 = −1
5 − 7 = −2
−1 – 10 = −11 2 + (−3) = 2 − 3 = −1
−5 + (−4) = −5 − 4 = −9
4 − (−3) = 4 + 3 = 7
−10 − (−6) = −10 + 6 = −4
2 × 3 = 6
2 × (−3) = −6
−2 × 3 = −6
−2 × (−3) = 6
Cartesian plane
Number line
First brackets, then × or ÷
then + or –, from left to right.
Adding and subtracting
negative integers
Adding and subtracting
positive integers
Multiplication
10 ÷ 5 = 2
10 ÷ (−5) = −2
−10 ÷ 5 = −2
−10 ÷ (−5) = 2
Division
3 × (5 – (−2)) + 8 ÷ (−4)
= 3 × 7 + (−2)
= 21 + (−2)
= 19
2
3
1
−2
−3
−3 −2 −1−1 1 2 3 O
y
x
(−2, −2) (1, −2)
(0, 2)
(−3, 0)
(3, 1)
−3 −2 0 1 2 3
−2 < 3 1 > −1
−1
negative positive
Order of operations
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Number and Algebra 129
Multiple-choice questions
1 When the numbers -4, 0, -1, 7 and -6 are arranged from lowest to highest, the correct sequence is:
A 0, -1, -4, -6, 7 B 0, -4, -6, -1, 7 C -6, -4, -1, 0, 7
D -1, -4, -6, 0, 7 E -6, -1, 0, -4, 7
2 The difference between -19 and 8 is:
A 152 B -11 C -27 D 11 E 27
3 The missing number in 2 − = 3 is:
A 1 B -1 C 5 D -5 E 2
4 5 − (-2) + (-7) is equal to:
A -4 B 10 C 7 D 0 E 14
5 The temperature inside a mountain cabin is initially -5°C. After burning a fire for 2 hours the
temperature rises to 17°C. What is the rise in temperature?
A -12°C B 12°C C 22°C D -85°C E -22°C
6 The product and quotient of two negative numbers is:
A positive B negative C zero D added E different
7 -2 × (-5) ÷ (-10) is equal to:
A -5 B 10 C -20 D 1 E -1
8 Which operation (i.e. addition, subtraction, multiplication or division) is completed second in the
calculation of (-2 + 5) × (-2) + 1?
A addition B subtraction C multiplication
D division E brackets
9 (-2) × 5 - (-2) is equal to:
A -12 B -8 C 8 D 12 E 9
10 The points A(-2, 3), B(-3, -1), C(1, -1) and D(0, 3) are joined on a number plane. What shape do
they make?
A triangle B square C trapezium
D kite E parallelogram
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Chapter 3 Computation with positive and 130 negative integers
Short-answer questions
1 Insert the symbol < (is less than) or > (is greater than) into each statement to make it true.
a 0 7 b -1 4 c 3 -7 d -11 -6
2 Evaluate:
a 2 − 7 b -4 + 2 c 0 − 15 d -36 + 37
e 5 + (-7) f -1 + (-4) g 10 − (-2) h -21 − (-3)
i 1 − 5 + (-2) j -3 + 7 − (-1) k 0 + (-1) − 10 l -2 − (-3) − (-4)
3 Find the missing number for each of the following.
a -2 + = -3 b -1 + = -10 c 5 − = 6 d -2 − = -4
e -1 − = 20 f -15 − = -13 g 7 + = -80 h -15 + = 15
4 Evaluate:
a 5 × (-2) b -3 × 7 c -2 × (-15) d 10 ÷ (-2)
e -36 ÷ 12 f -100 ÷ (-25) g -3 × 2 ÷ (-6) h -38 ÷ (-19) × (-4)
5 Find the missing number.
a 4 × = -8 b ÷ -5 = 10 c ÷ 9 = -4 d -1 × = 1
6 Use order of operations to find the answers to these expressions.
a -2 + 5 × (-7) b -1 − 18 ÷ (-2) c -15 ÷ (1 + 4)
d 5 − 4 × (-3) ÷ (-3) e (-2 − 5) × (8 ÷ (-1)) f -7 × ((-4) − 7) + 3
7 Evaluate:
a − − −
+
3
3 3
3 3
b (1 - 2) × 3 - 4
8 For the Cartesian plane shown, write down the coordinates
of the points labelled A, B, C, D, E and F.
2
3
1
-2
-3
-1
E
-4 -3 -2 -1
-4
O 1 2 3 4
4
y
x
B
F
D
C
A
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Number and Algebra 131
Extended-response questions
1 A scientist, who is camped on the ice in Greenland, records the following details in her notepad
regarding the temperature over five days. Note that ‘min’ stands for minimum and ‘max’ stands for
maximum.
• Monday: min = -18°C, max = -2°C.
• Decreased 29°C from Monday’s max to give Tuesday’s min.
• Wednesday’s min was -23°C.
• Max was only -8°C on Thursday.
• Friday’s min is 19°C colder than Thursday’s max.
a What is the overall temperature increase on Monday?
b What is Tuesday’s minimum temperature?
c What is the difference between the minimum temperatures for Tuesday and Wednesday?
d What is the overall temperature drop from Thursday’s maximum to Friday’s minimum?
e By how much will the temperature need to rise on Friday if its maximum is 0°C?
2 When joined, these points form a picture on the number plane. What is the picture?
A(0, 5), B (1, 3), C (1, 1), D (2, 0), E (1, 0), F (1, -2), G (3, -5),
H (-3, -5), I (-1, -2), J (-1, 0), K (-2, 0), L (-1, 1), M (-1, 3), N (0, 5)
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Chapter
Understanding
fractions,
decimals and
percentages
4
What you will learn
Factors and multiples
Highest common factor and lowest common multiple
What are fractions?
Equivalent fractions and simplifi ed fractions
Mixed numerals and improper fractions
Ordering positive and negative fractions
Place value in decimals and ordering decimals
Rounding decimals
Decimal and fraction conversions
Connecting percentages with fractions and decimals
Decimal and percentage conversions
Fraction and percentage conversions
Percentage of a quantity
Using fractions and percentages to compare two
quantities
4A
4B
4C
4D
4E
4F
4G
4H
4I
4J
4K
4L
4M
4N
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ancient Egyptian fractions
The ancient Egyptians used fractions over 4000 years
ago. The Egyptian sky god Horus was a falcon-headed
man whose eyes were believed to have magical healing
powers. Egyptian jewellery, ornaments or clothing
decorated with the Eye of Horus design were regarded
as good luck charms for health and magical protection
from evil.
The six parts in the Eye of Horus design represent
the six ways that information enters the brain. These six
different parts or symbols represented the six fractions
used by ancient Egyptian mathematics. For example,
instead of writing 1
2
, Egyptians would write , and
instead of writing 1
8
they would write .
Eye of Horus fraction symbols are found in ancient
Egyptian medical prescriptions for mixing ‘magical’
medicine. Amazingly, modern doctors still use the eye
of Horus ( ) symbolism when they write (Rx)
at the start of a prescription.
nSW Syllabus
for the australian
Curriculum
Strand: number and algebra
Substrand: FRaCtionS, DECiMalS anD
pERCEntaGES
Outcome
A student operates with fractions,
decimals and percentages.
(MA4–5NA)
133
1
8
thought (eyebrow
closest to brain)
1
16
hearing (pointing
to ear)
1
2
smell (pointing to
nose)
1
4
sight (pupil of the
eye)
1
64
touch (leg touching
the ground)
1
32
taste (curled top of
wheat plant)
A proportion or fraction can be written using
a combination of these symbols. For example:
3
4 = and
3
16 = .
Which symbols would represent
7
8
? Can
1
3
be
written using the Eye of Horus symbols?
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Chapter 4 Understanding fractions, decimals 134 and percentages
1 In which diagram is one-third shaded?
a B C D
2 Which of the following is not equivalent to one whole?
a 2
2
B 6
6
C 1
4
D 12
12
3 Which of the following is not equivalent to one-half?
a 2
4
B 3
9
C 5
10
D 10
20
4 Find:
a 1 - 1
4
b 1 – 1
2
c 1 – 1
3
d 1 –
1
5
5 Find:
a 3 – 1
4
b 2 – 1
2
c 10 – 1
2
d 6 –
3
4
6 Tom eats half a block of chocolate on Monday and half of the remaining block on Tuesday.
How much chocolate is left for Wednesday?
7 Find the next three terms in these number sequences.
a 0
1
2
1 1
1
2
, , , ,
__
,
__
,
__
b
1
3
2
3
3
3
, , ,
__
,
__
,
__
c
1
4
2
4
3
4
4
4
, , , ,
__
,
__
,
__
d 1
6
1
3
1
2
, , ,
__
,
__
,
__
8 Copy and complete.
a
1
2
1
2
1
2
1
2
+ + =
−−
× b
3
4
3
4
3
4
3
4
3
4
+ + + =
−−
×
c
5
6
5
6
× =
__
d
6
8
6
8
÷ = __
9 Find:
a
1
2
of $15 b
1
4
of $160 c
3
4
of $1 d
1
3
of $6
1
2
16
4
1
4
3
4
1
pre-test
10 State whether each of the following is true or false.
a of 16 = 16 ÷ 2 b = of 16
c of 100 = 75 d one-tenth =
100
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number and algebra 135
Factors and multiples
Number patterns are fascinating. Factors and
multiples are key building blocks for a broad
fi eld known as Number Theory. Many famous
mathematicians have studied number patterns in an
attempt to better understand our world and to assist
with new scientifi c discoveries. Around 600 bc, the
Greeks built on the early work of the Egyptians and
Babylonians. Thales of Miletus, the ‘father of Greek
mathematics’, is credited for signifi cant advances
in Number Theory. One of his students, Pythagoras
of Samos, went on to become one of the most wellknown
mathematicians to have lived. Pythagoras
was primarily a religious leader, but he believed that
the understanding of the world could be enhanced through the understanding of number. We start
this chapter by explaining the concepts of factors and multiples.
One dozen doughnuts are generally packed into bags with 3 rows of 4 doughnuts each.
Since 3 × 4 = 12, we can say that 3 and 4 are factors of 12.
Purchasing ‘multiple’ packs of one dozen doughnuts could result in buying 24, 36, 48 or 60
doughnuts, depending on the number of packs. These numbers are known as multiples of 12.
let’s start: The most factors, the most multiples
Which number that is less than 100 has the most factors?
Which number that is less than 100 has the most multiples less than 100?
4A
How many factors are there in a set of 12?
■■ Factors of a particular number are numbers that divide evenly into that number.
– Therefore, written in ascending order, the factors of 20 are 1, 2, 4, 5, 10, 20.
– Every whole number is a factor of itself and also 1 is a factor of every whole number.
■■ Multiples of a particular number are numbers created by multiplying the particular number
by any positive integer.
– For example: The multiples of 20 are 20, 40, 60, 80, 100, 120, …
Multiples of 20 are also 480, 2000, 68 600. There is an infi nite number of multiples!
■■ Given the statements above, it follows that factors are less than or equal to the particular
number being considered and multiples are greater than or equal to the number being
considered.
Key ideas
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Chapter 4 Understanding fractions, decimals 136 and percentages
Example 1 Finding factors
Find the complete set of factors for each of these numbers.
a 15 b 40
Solution Explanation
a Factors of 15 are 1, 3, 5, 15. 1 × 15 = 15, 3 × 5 = 15
b Factors of 40 are:
1, 2, 4, 5, 8, 10, 20, 40.
1 × 40 = 40, 2 × 20 = 40
4 × 10 = 40, 5 × 8 = 40
The last number you need to check is 7.
Example 2 Listing multiples
Write down the first six multiples for each of these numbers.
a 11 b 35
Solution Explanation
a 11, 22, 33, 44, 55, 66 The first multiple is always the given number.
Add on the given number to find the next multiple.
Repeat this process to get more multiples.
b 35, 70, 105, 140, 175, 210 Start at 35, the given number, and repeatedly add 35 to
continue producing multiples.
Example 3 Finding factor pairs
Express 195 as a product of two factors, both of which are greater than 10.
Solution Explanation
195 = 13 × 15 Systematically divide 195 by numbers greater than 10 in
an attempt to find a large factor.
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number and algebra 137
1 For each of the following numbers, state whether they are factors (F), multiples (M) or neither
(N) of the number 60.
a 120 b 14 c 15 d 40
e 6 f 5 g 240 h 2
i 22 j 600 k 70 l 1
2 For each of the following numbers, state whether they are factors (F), multiples (M) or neither
(N) of the number 26.
a 2 b 54 c 52 d 4
e 210 f 27 g 3 h 182
i 1 j 26 000 k 13 l 39
3 List the complete set of factors for each of the following numbers.
a 10 b 24 c 17
d 36 e 60 f 42
g 80 h 12 i 28
4 Write down the fi rst six multiples for each of the following numbers.
a 5 b 8 c 12
d 7 e 20 f 75
g 15 h 100 i 37
5 Fill in the gaps to complete the set of factors for each of the following numbers.
a 18 1, 2, , 6, 9,
b 25 1, , 25
c 72 , 2, 3, , , 8, , , 18, , 36, 72
d 120 1, 2, , , , 6, , 10, , , 20, , 30, , 60,
6 Which number is the incorrect multiple for each of the following sequences?
a 3, 6, 9, 12, 15, 18, 22, 24, 27, 30
b 43, 86, 129, 162, 215, 258, 301, 344
c 11, 21, 33, 44, 55, 66, 77, 88, 99, 110
d 17, 34, 51, 68, 85, 102, 117, 136, 153, 170
7 Consider the set of whole numbers from 1 to 25 inclusive.
a Which number has the most factors?
b Which number has the fewest factors?
c Which numbers have an odd number of factors?
Example 1
Example 2
Exercise 4A
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138 Chapter 4 Understanding fractions, decimals and percentages
8 Express each of these numbers as a product of two factors, both of which are greater than 10.
a 192 b 315 c 180
d 121 e 336 f 494
9 Zane and Matt are both keen runners. Zane takes 4 minutes to jog around a running track and
Matt takes 5 minutes. They start at the same time and keep running until they both cross the
fi nish line at the same time.
a How long do they run for? b How many laps did Zane run?
c How many laps did Matt run?
10 Anson is preparing for his 12th birthday party.
He has invited 12 friends and is making each of
them a ‘lolly bag’ to take home after the party. To
be fair, he wants to make sure that each friend has
the same number of lollies. Anson has a total of
300 lollies to share among the lolly bags.
a How many lollies does Anson put in each of
his friends’ lolly bags?
b How many lollies does Anson have left over
to eat himself?
Anson then decides that he wants a lolly bag for himself also.
c How many lollies will now go into each of the 13 lolly bags?
After much pleading from his siblings, Anson prepares lolly bags for them also. His sister
Monique notices that the total number of lolly bags is now a factor of the total number of lollies.
d What are the different possible number of sibling(s) that Anson could have?
e How many siblings do you expect Anson has?
11 Are the following statements true or false?
a A multiple of a particular number is always smaller than that number.
b 2 is a factor of every even number.
c 3 is a factor of every odd number.
d A factor is always greater than or equal to the given number.
e When considering a particular number, that number is both a factor and a multiple of itself.
12 60 is a number with many factors. It has a total of 12 factors and, interestingly, it has each of the
numbers 1, 2, 3, 4, 5, 6 as a factor.
a What would be the smallest number that could boast having 1, 2, 3, 4, 5, 6, 7 and 8 as
factors?
b What would be the smallest number that could boast having 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10
as factors?
c Express, in written form, how you can determine the smallest number that contains a
particular set of factors.
Example 3
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Number and Algebra 139
13 a What numbers can claim the number 100 to be a multiple?
b What are the factors of 100?
14 All AM radio stations have frequencies
that are multiples of 9. For example, a
particular radio station has a frequency
of 774 kilohertz (or kHz). Find three
other AM radio stations and show their
frequencies are, indeed, multiples of 9.
15 Two numbers are chatting with one another when one number asks the other, “Are you a
multiple of mine?” The reply comes back, “Well, I have always considered you to be one of
my factors.” Explain why this response is enough to help the first number answer her question.
Which number is the larger number?
Enrichment: Designing some help from the computer
16 a Design a spreadsheet that will enable a user to enter any number between 1 and 100 and it
will automatically list the first 30 multiples of that number.
b Design a spreadsheet that will enable a user to enter any particular number between 1 and
100 and it will automatically list the number’s factors.
c Improve your factor program so that it finds the sum of the factors and also states the total
number of factors for the particular number.
d Use your spreadsheet program to help you find a pair of amicable numbers. A pair of
numbers is said to be amicable if the sum of the factors for each number, excluding the
number itself, is equal to the other number. Each number that makes up the first such pair of
amicable numbers falls between 200 and 300.
An example of a non-amicable pair of numbers:
12 – factor sum = 1 + 2 + 3 + 4 + 6 = 16
16 – factor sum = 1 + 2 + 4 + 8 = 15
The factor sum for 16 would need to be 12 for the pair to be amicable numbers.
Helpful Excel formulas
INT (number) – Rounds a number down to the nearest integer (whole number).
MOD (number, divisor) – Returns the remainder after a number is divided by its divisor.
IF (logical test, value if true, value if false) – Checks whether a condition is met and returns
one value if true and another value if false.
COUNTIF (range, criteria) – Counts the number of cells within a range that meet the given
condition.
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Chapter 4 Understanding fractions, decimals 140 and percentages
Highest common factor and lowest common multiple
In the previous exercise, factors and muliples of a number were explained. Remember that factors are less
than or equal to a given number and that multiples are greater than or equal to a given number.
given number
e.g. 12
factors
≤ 12
e.g. 1, 2, 3, 4, 6, 12
multiples
≥ 12
e.g. 12, 24, 36, 48, . . .
There are many applications in Mathematics for which the highest common factor (HCF) of two or
more numbers must be determined. In particular, the skill of fi nding the HCF is required for the future
topic of factorisation, which is an important aspect of Algebra.
Similarly, there are many occasions for which the lowest common multiple (LCM) of two or more
numbers must be determined. Adding and subtracting fractions with different denominators requires the
skill of fi nding the LCM.
let’s start: You provide the starting numbers!
For each of the following answers, you must determine possible starting numbers. On all occasions, the
numbers involved are less than 100.
1 The HCF of two numbers is 12. Suggest two possible starting numbers.
2 The HCF of three numbers is 11. Suggest three possible starting numbers.
3 The LCM of two numbers is 30. Suggest two possible starting numbers.
4 The LCM of three numbers is 75. Suggest three possible starting numbers.
5 The HCF of four numbers is 1. Suggest four possible numbers.
6 The LCM of four numbers is 24. Suggest four possible numbers.
4B
■■ HCF stands for highest common factor.
■■ As the name suggests, it refers to the highest (i.e. largest) factor that is common to the numbers
provided in the question.
– For example: Find the HCF of 24 and 40.
Factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.
Factors of 40 are 1, 2, 4, 5, 8, 10, 20 and 40.
Therefore, common factors of 24 and 40 are 1, 2, 4 and 8.
Therefore, the highest common factor of 24 and 40 is 8.
Key ideas
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Number and Algebra 141
■■ LCM stands for lowest common multiple.
■■ As the name suggests, it refers to the lowest (i.e. smallest) multiple that is common to the
numbers provided in the question.
– For example: Find the LCM of 20 and 12.
Multiples of 20 are 20, 40, 60, 80, 100, 120, 140, …
Multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, …
Therefore, common multiples of 20 and 12 are 60, 120, 180, …
Therefore, the lowest common multiple of 20 and 12 is 60.
■■ The LCM of two numbers can always be found by multiplying the two numbers together
and dividing by their HCF.
– For example: Find the LCM of 20 and 12.
The HCF of 20 and 12 is 4.
Therefore, the LCM of 20 and 12 is 20 × 12 ÷ 4 = 60.
Key ideas
Example 4 Finding the highest common factor (HCF)
Find the highest common factor (HCF) of 36 and 48.
Solution Explanation
Factors of 36 are:
1, 2, 3, 4, 6, 9, 12, 18 and 36.
Factors of 48 are:
1, 2, 3, 4, 6, 8, 12, 16, 24 and 48.
1 × 36 = 36, 2 × 18 = 36, 3 × 12 = 36,
4 × 9 = 36, 6 × 6 = 36
1 × 48 = 48, 2 × 24 = 48, 3 × 16 = 48,
4 × 12 = 48, 6 × 8 = 48
The HCF of 36 and 48 is 12. Common factors are 1, 2, 3, 4, 6 and 12, of
which 12 is the highest.
Example 5 Finding the lowest common multiple (LCM)
Find the lowest common multiple (LCM) of the following pairs of numbers.
a 5 and 11 b 6 and 10
Solution Explanation
a The LCM of 5 and 11 is 55. Note that the HCF of 5 and 11 is 1.
5 × 11 = 55
b The LCM of 6 and 10 is 30. Note that the HCF of 6 and 10 is 2.
The LCM of 6 and 10 is 6 × 10 ÷ 2 = 30.
Multiples of 6 are 6, 12, 18, 24, 30, 36, …
Multiples of 10 are 10, 20, 30, 40, …
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Chapter 4 Understanding fractions, decimals 142 and percentages
1 The factors of 12 are 1, 2, 3, 4, 6 and 12, and the factors of 16 are 1, 2, 4, 8 and 16.
a What are the common factors of 12 and 16?
b What is the HCF of 12 and 16?
2 Fill in the missing numbers to fi nd out the HCF of 18 and 30.
Factors of 18 are 1, , 3, , and 18.
Factors of are 1, , , 5, , 10, and 30.
Therefore, the HCF of 18 and 30 is .
3 The fi rst 10 multiples of 8 are 8, 16, 24, 32, 40, 48, 56, 64, 72 and 80.
The fi rst 10 multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54 and 60.
a What are two common multiples of 8 and 6?
b What is the LCM of 8 and 6?
4 Fill in the missing numbers to fi nd out the LCM of 9 and 15.
Multiples of 9 are 9, 18, , 36, , , , , 81 and .
Multiples of 15 are , 30, , 60, 75, , and 120.
Therefore, the LCM of 9 and 15 is .
5 Find the HCF of the following pairs of numbers.
a 4 and 5 b 8 and 13 c 2 and 12 d 3 and 15
e 16 and 20 f 15 and 60 g 50 and 150 h 48 and 72
i 80 and 120 j 75 and 125 k 42 and 63 l 28 and 42
6 Find the HCF of the following groups of numbers.
a 20, 40, 50 b 6, 15, 42 c 50, 100, 81
d 18, 13, 21 e 24, 72, 16 f 120, 84, 144
7 Find the LCM of the following pairs of numbers.
a 4 and 9 b 3 and 7 c 12 and 5 d 10 and 11
e 4 and 6 f 5 and 10 g 12 and 18 h 6 and 9
i 20 and 30 j 12 and 16 k 44 and 12 l 21 and 35
8 Find the LCM of the following groups of numbers.
a 2, 3, 5 b 3, 4, 7 c 2, 3, 4
d 3, 5, 9 e 4, 5, 8, 10 f 6, 12, 18, 3
9 Find the HCF of the following pairs of numbers and then use this information to help calculate
the LCM of the same pair of numbers.
a 15 and 20 b 12 and 24 c 14 and 21 d 45 and 27
Example 4
Example 5
Exercise 4B
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Number and Algebra 143
10 Find the LCM of 13 and 24.
11 Find the HCF of 45 and 72.
12 Find the LCM and HCF of 260 and 390.
13 Andrew runs laps of ‘the circuit’ in 4 minutes. Bryan runs laps of the same circuit in 3 minutes. Chris
can run laps of the same circuit in 6 minutes. They all start together on the starting line and run a
‘race’ that goes for 36 minutes.
a What is the first time, after the start, that they will all cross over the starting line together?
b How many laps will each boy complete in the race?
c How many times does Bryan overtake Andrew during this race?
d Prepare a 1-minute class presentation involving three PowerPoint slides that describes how you
solved this problem and explains your answer.
14 Given that the HCF of a pair of different numbers is 8, find the two numbers:
a if both numbers are less than 20
b when one number is in the 20s and the other in the 30s
15 Given that the LCM of a pair of numbers is 20, find the seven possible pairs of numbers.
Enrichment: LCM of large groups of numbers
16 a Find the LCM of these single-digit numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9.
b Find the LCM of these first 10 natural numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
c Compare your answers to parts a and b. What do you notice? Explain.
d Find the LCM of the first 11 natural numbers.
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Chapter 4 Understanding fractions, decimals 144 and percentages
What are fractions?
The word fraction comes from the Latin word
‘frangere’, which means ‘to break into pieces’.
Although the following sentences are
not directly related to the mathematical use
of fractions, they all contain words that are
related to the orginal Latin term ‘frangere’
and they help us gain an understanding of
exactly what a fraction is.
The fragile vase smashed into a hundred
pieces when it landed on the ground.
After the window was broken, several
fragments were found on the fl oor.
She fractured her leg in two places.
The computer was running slowly and
needed to be defragmented.
The elderly gentleman was becoming very frail in his old age.
Can you think of any other related sentences?
Brainstorm specifi c common uses of fractions in everyday life. The list could include cooking,
shopping, sporting, building examples and more.
let’s start: What strength do you like your cordial?
• Imagine preparing several jugs of different strength cordial. Samples could include 1
4
strength
cordial, 1
5
strength cordial, 1
6
strength cordial, 1
8
strength cordial.
• In each case, describe how much water and how much cordial is needed to make a 1 litre mixture.
Note: 1 litre (L) = 1000 millilitres (mL).
• On the label of a cordial container, it suggests ‘To make up by glass or jug: add fi ve parts water to one
part cordial, according to taste.’
• What fraction of the beverage is cordial?
4C
Fragments of a broken object are all fractions of the whole.
■■ A fraction is made up of a numerator (up) and a denominator (down).
For example: 3
5
numerator
denominator
– The denominator tells you how many parts the whole is divided up into.
– The numerator tells you how many of the divided parts you have selected.
– The horizontal line separating the numerator and the denominator is called the vinculum.
■■ A proper fraction or common fraction is less than a whole, and therefore the numerator must be
smaller than the denominator.
For example:
2
7
is a proper fraction.
Key ideas
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Number and Algebra 145
■■ An improper fraction is greater than a whole, and therefore the numerator must be larger
than the denominator.
For example: 5
3
is an improper fraction, which is equal to 1
2
3
.
■■ We can represent positive and negative fractions on a number line.
This number line shows the whole numbers from -2 to 2. Each unit has then been divided
equally into four segments, therefore creating ‘quarters’.
■■ Whole numbers can be represented as fractions.
On the number line above we see that 1 is the same as 4
4
and 2 is the same as 8
4
.
■■ We can represent fractions using area. If a shape is divided into regions of equal areas, then
shading a certain number of these regions will create a fraction of the whole shape.
For example: Fraction shaded = 3
4
0 1 1 2
2
0
2
4
1
4
4
4
3
4
6
4
5
4
8
4
7
4
10
4
9
4
11
2 21
2
-2 -1 1
2
7
4
8
4
5
4
6
4
3
4
4
4
1
4
2
4
-11
2
-
- - - - - - - -
Key ideas
Example 6 Understanding the numerator and the denominator
a Into how many pieces has the whole pizza been divided?
b How many pieces have been selected (i.e. shaded)?
c In simplest form, when representing the shaded fraction of the pizza:
i What must the denominator equal?
ii What must the numerator equal?
iii Write the amount of pizza selected (shaded) as a fraction.
Solution Explanation
a 8 Pizza cut into 8 equal pieces.
b 3 3 of the 8 pieces are shaded in blue.
c i 8 Denominator shows the number of parts the whole
has been divided into.
ii 3 Numerator tells how many of the divided parts you
have selected.
iii
3
8
Shaded fraction is the numerator over the denominator;
i.e. 3 out of 8 divided pieces.
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Chapter 4 Understanding fractions, decimals 146 and percentages
Example 7 Representing fractions on a number line
Represent the fractions
3
5
and
9
5
on a number line.
Solution Explanation
0 3 1 2
5
9
5
Draw a number line starting at 0 and mark on it the
whole numbers 0, 1 and 2.
Divide each whole unit into fi ve segments of equal
length. Each of these segments has a length of one-fi fth.
Example 8 Shading areas
Represent the fraction
3
4
in three different ways, using a square divided into four equal regions.
Solution Explanation
Ensure division of square creates four equal areas.
Shade in three of the four regions.
Exercise 4C
1 a State the denominator of this proper fraction: 2
9
.
b State the numerator of this improper fraction: 7
5
.
2 Group the following list of fractions into proper fractions, improper fractions and whole numbers.
a 7
6
b 2
7
c 50
7
d 3
3
e
3
4
f 5
11
g 1
99
h 9
4
i
11
8
j 10
10
k 5
1
l 121
5
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Number and Algebra 147
3 Answer the following questions for each of the pizzas (A to D) drawn below.
a Into how many pieces has the whole pizza been divided?
b How many pieces have been selected (shaded)?
c In representing the shaded fraction of the pizza:
i What must the denominator equal?
ii What must the numerator equal?
iii Write the amount of pizza selected (shaded) as a fraction.
A   B C D
4 Find the whole numbers amongst the following list of fractions. Hint: There are five whole
numbers to find.
a
15
4
b 14
8
c 12
5
d 30
15
e
17
3
f 30
12
g 12
12
h 33
10
i
53
3
j 9
3
k 50
20
l 28
7
m
96
8
n 24
5
o 62
4
p 1031
2
5 Represent the following fractions on a number line.
a
3
7
and
6
7
b 2
3
and
5
3
c 1
6
and
5
6
d
2
4
and
11
4
e 11
5
and - 8
5
f - 5
4
, - 9
4
and - 3
2
6 Represent each of these fractions in three different ways, using a rectangle divided into
equal regions.
a 1
4
b 3
8
c 2
6
7 Write the next three fractions for each of the following fraction sequences.
a
3
5
4
5
5
5
6
5
, , , ,
__
,
__
,
__
b 5
8
6
8
7
8
8
8
, , , ,
__
,
__
,
__
c
1
3
2
3
3
3
4
3
, , , ,
__
,
__
,
__
d 11
7
10
7
9
7
8
7
, , , ,
__
,
__
,
__
e 9
2
,
7
2
,
5
2
,
3
2
,
__
,
__
,
__
f 18
4
13
4
8
4
3
4
, , , ,
__
,
__
,
__
Example 6
Example 7
Example 8
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Chapter 4 Understanding fractions, decimals 148 and percentages
8 What fractions correspond to each of the different shapes positioned on these number lines?
a
0 1 2 3 4 5 6 7
b
0 1 2
c
0 1 2 3 4
d
-4 -3 -2 -1 0
9 What operation (i.e. +, –, × or ÷) does the vinculum relate to?
10 For each of the following, state what fraction of the diagram is shaded.
a b c
d e f
11 For each of the following, write the fraction that is
describing part of the total.
a After one day of a 43-kilometre hike, they had
completed 12 kilometres.
b From 15 starters, 13 went on and fi nished the race.
c Rainfall for 11 months of the year was below average.
d One egg is broken in a carton that contains a dozen eggs.
e Two players in the soccer team scored a goal.
f The lunch stop was 144 kilometres into the
475-kilometre trip.
g Seven members in the class of 20 have visited
Australia Zoo.
h One of the car tyres is worn and needs replacing.
i It rained three days this week.
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Number and Algebra 149
12 Explain the logic behind the terms ‘proper fraction’ and ‘improper fraction’.
13 Which diagram has one-quarter shaded?
a b c d
Enrichment: Adjusting concentration
14 a A 250-millilitre glass of cordial is made by mixing four parts water to one part cordial.
i What fraction of the glass is cordial?
ii What amount of cordial is required?
b Fairuz drinks 50 millilitres of the glass and thinks it’s too strong. So he fills the glass back
up with 50 millilitres of pure water.
i How much cordial is in the glass now?
ii What fraction of the glass is cordial?
c Fairuz drinks 50 millilitres of the drink but he still thinks it is too strong. So, once again, he
fills the glass back up with 50 millilitres of pure water.
i How much cordial is in the glass now?
ii What fraction of the glass is cordial?
d Lynn prefers her cordial much stronger compared with Fairuz. When she is given a glass of
the cordial that is mixed at four parts to one, she drinks 50 millilitres and decides it is too
weak. So she fills the glass back up with 50 millilitres of straight cordial.
i How much cordial is in Lynn’s glass after doing this once?
ii What fraction of the glass is cordial?
e Like Fairuz, Lynn needs to repeat the process to make her cordial even stronger. So, once
again, she drinks 50 millilitres and then tops the glass back up with 50 millilitres of straight
cordial.
i How much cordial is in Lynn’s glass now?
ii What fraction of the glass is cordial?
f If Fairuz continues diluting his cordial concentration in
this manner and Lynn continues strengthening her cordial
concentration in this manner, will either of them ever
reach pure water or pure cordial? Discuss.
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Chapter 4 Understanding fractions, decimals 150 and percentages
Equivalent fractions and simplifi ed fractions
Often fractions may look very different when in
fact they have the equivalent value.
For example, in an AFL football match,
‘half-time’ is the same as ‘the end of the second
quarter’. We can say that 1
2
and 2
4
are equivalent
fractions. In both situations, the equivalent
fraction of the game has been completed.
Consider a group of friends eating pizzas
during a sleepover. The pizzas are homemade
and each person cuts up their pizza as they like.
Trevor cuts his pizza into only two pieces,
Jackie cuts hers into four pieces, Tahlia cuts
hers into six pieces and Jared cuts his into eight pieces.
The shaded pieces are the amount that they have eaten
before it is time to start the second movie.
By looking at the pizzas, it is clear to see that Trevor,
Jackie, Tahlia and Jared have all eaten the same amount of
pizza. We can therefore conclude that
1
2
,
2
4
,
3
6
and
4
8
are
equivalent fractions.
This means that
1
2
=
2
4
=
3
6
=
4
8
.
let’s start: Fraction clumps
• Prepare a class set of fraction cards. (Two example sets
are provided below.)
• Hand out one fraction card to each student.
• Students then arrange themselves into groups of equivalent fractions.
• Set an appropriate time goal by which this task must be completed.
• Repeat the process with a second set of equivalent fraction cards.
Sample sets of fraction cards
Class set 1
1
2
, 3
12
, 3
24
, 10
80
, 1
3
, 8
40
, 1
5
, 3
6
, 1
8
, 5
40
, 3
9
, 1
4
, 1000
4000
, 100
200
, 10
50
, 2
16
, 10
30
, 13
39
, 5
10
, 7
14
, 2
6
, 7
28
, 2
10
, 4
20
, 2
8
Class set 2
2
3
, 6
14
, 3
18
, 4
10
, 2
12
, 24
64
, 11
66
, 4
6
, 3
7
, 30
70
, 12
32
, 3
8
, 10
15
, 5
30
, 1
6
, 2000
5000
, 21
49
, 300
800
, 6
9
, 9
21
, 2
5
, 14
35
, 20
30
, 6
16
, 22
55
4D
There are four quarters played out in a game of AFL football.
Trevor Jackie
Tahlia Jared
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Number and Algebra 151
■■ Equivalent fractions are fractions that mark the same place on a number line.
For example: 1
2
and 2
4
are equivalent fractions.
■■ Equivalent fractions are produced by multiplying the numerator and denominator by the
same number. This number can be any whole number greater than 1.
■■ Equivalent fractions can also be produced by dividing the numerator and denominator by the
same number.
■■ Simplifying fractions involves writing a fraction in its ‘simplest form’ or ‘easiest form’ or
‘most convenient form’. To do this, the numerator and the denominator must be divided by
their highest common factor (HCF).
■■ It is a mathematical convention to write all answers involving fractions in their simplest
form. Key ideas
Example 9 Producing equivalent fractions
Write four equivalent fractions for
2
3
4
6
6
9
8
12
10
15
.= = = =
Solution Explanation
2
3
4
6
6
9
8
12
10
15
= = = =
Many other fractions are also possible.
Other common choices include:
20
30
,
200
300
,
2000
3000
,
40
60
2
3
4
6
6
9
8
12
10 … etc.
15 =
×2
×2
×3
×3
×4
×4
×5
×5
= = =
Example 10 Checking for equivalence
By writing either = or ≠ between the fractions, state whether the following pairs of fractions are
equivalent or not equivalent.
a 1
3
3
7
b 4
5
20
25
Solution Explanation
a 1
3
≠ 3
7
Convert to a common denominator.
1
3
= 1 7
3 7
×
×
= 7
21
and 3
7
= 3 3
7 3
×
×
= 9
21
, 7
21
9
21

b 4
5
= 20
25
4
5
=
4 5
5 5
×
×
=
20
25
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Chapter 4 Understanding fractions, decimals 152 and percentages
Example 11 Converting to simplest form
Write these fractions in simplest form.
a 12
20
b
7
42
Solution Explanation
a
12
20
=
×
×
3 4
5 4
=
3
5
The HCF of 12 and 20 is 4.
Both the numerator and the denominator are divided by
the HCF of 4.
b
7
42
=
×
×
7 1
7 6
=
1
6
The HCF of 7 and 42 is 7.
The 7 is ‘cancelled’ from the numerator and the
denominator.
Exercise 4D
1 Which of the following fractions are equivalent to 1
2
?
3
5
,
3
6
,
3
10
,
2
4
,
11
22
,
7
15
,
8
12
,
2
1
,
5
10
,
6
10
2 Which of the following fractions are equivalent to
8
20
?
4
10
,
1
5
,
6
20
,
8
10
,
16
40
,
2
5
,
4
12
,
12
40
,
80
200
,
1
4
3 Fill in the missing numbers to complete the following strings of equivalent fractions.
a = = = = = 1
3 6
4
30 60
100
b 2
8 4 12
6
80
10 = = = = =
4 In each of these lists of equivalent fractions, circle the fraction that is in its simplest form.
a 3
15
,
10
50
, 2
10
,
1
5
b 100
600
,
3
18
,
1
6
,
7
42
c 4
6
,
2
3
,
16
24
,
20
30
d 9
12
,
15
20
,
6
8
,
3
4
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Number and Algebra 153
5 Fill in the gaps to reduce these fractions to their simplest form.
a
10
30
i HCF = ii
10
30
=
1
3
×
×
. Therefore, simplest form is
3
.
b
4
18
i HCF = ii
4
18
= 2
9
×
×
. Therefore, simplest form is
9
.
c
4
28
i HCF = ii
4
28
=
1
7
×
×
. Therefore, simplest form is
1
.
d
9
15
i HCF = ii
9
15
= 3
5
×
×
. Therefore, simplest form is .
6 Write four equivalent fractions for each of the fractions listed.
a 1
2
b 1
4
c 2
5
d 3
5
e 2
9
f 3
7
g 5
12
h 3
11
7 Find the unknown value to make the equation true.
a
3
4 12
= ?
b 5
8 80
= ?
c 6
11
18 =
?
d 2
7
16 =
?
e
3 15
? 40
= f ?
1
14
7
= g ?
10
24
20
= h 13
14 42
= ?
i
2
7
10 =
?
j 19
20
190 =
?
k 11
21
55 =
?
l 11 44
? 8
=
8 By writing either = or  between the fractions, state whether the following pairs of fractions are
equivalent or not equivalent.
a
1
2
5
8
b 4
8
2
4
c 3
7
30
60
d
5
9
15
18
e 11
15
33
45
f 1
2
402
804
g
12
36
1
3
h 18
24
21
28
i 6
18
11
33
9 Write the following fractions in simplest form.
a
15
20
b 12
18
c 10
30
d 8
22
e
14
35
f 2
22
g 8
56
h 9
27
i
35
45
j 36
96
k 120
144
l 700
140
Example 9
Example 10
Example 11
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Chapter 4 Understanding fractions, decimals 154 and percentages
10 These lists of fractions are meant to contain only fractions in their simplest form; however,
there is one mistake in each list. Find the fraction that is not in simplest form and rewrite it
in its simplest form.
a 1
3
3
8
5
9
7
14
, , , b 2
5
12
16
15
19
13
37
, , ,
c 12
19
4
42
5
24
6
61
, , , d 7
63
9
62
11
81
13
72
, , ,
11 A family block of chocolate consists of 12 rows of 6 individual squares. Tania eats 16
individual squares. What fraction of the block, in simplest terms, has Tania eaten?
12 Four people win a competition that allows them to receive 1
2
a tank of free petrol.
Find how many litres of petrol the drivers of these cars receive.
a Ford Territory with a 70-litre tank
b Nissan Patrol with a 90-litre tank
c Holden Commodore with a 60-litre tank
d Mazda 323 with a 48-litre tank
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Number and Algebra 155
13 Justin, Joanna and Jack are sharing a large pizza for dinner. The pizza has been cut into
12 equal pieces. Justin would like
1
3
of the pizza, Joanna would like 1
4
of the pizza and Jack
will eat whatever is remaining. By considering equivalent fractions, determine how many slices
each person gets served.
14 J. K. Rowling’s first book, Harry Potter and the Philosopher’s Stone, is 225 pages long. Sam
plans to read the book in three days, reading the same number of pages each day.
a How many pages should Sam read each day?
b The fraction 75
225
of the book is equivalent to what fraction in simplest form?
By the end of the second day, Sam is on track and has read 2
3
of the book.
c How many pages of the book is 2
3
equivalent to?
15 A fraction when simplified is written as
3
5
. What could the fraction have been before it was
simplified? Explain why the number of answers is infinite.
Enrichment: Equivalent bars of music
16 Each piece of music has a time signature. A common time signature is called
4
4
time, and is
actually referred to as Common time!
Common time, or 4
4
time, means that there are four ‘quarter notes’ (or crotchets) in each bar.
Listed below are the five most commonly used musical notes.
– whole note (fills the whole bar) – semibreve
– half note (fills half the bar) – minim
– quarter note (four of these to a bar) – crotchet
– eighth note (eight to a bar) – quaver
– sixteenth note (sixteen to a bar) – semi-quaver
a Write six different ‘bars’ of music in 4
4
time.
Carry out some research on other types of musical time signatures.
b Do you know what the time signature 12
8
means?
c Write three different bars of music for a 12
8
time signature.
d What are the musical symbols for different length rests?
e How does a dot (or dots) written after a note affect the length of the note?
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Chapter 4 Understanding fractions, decimals 156 and percentages
Mixed numerals and improper fractions
As we have seen in this chapter, a fraction is a
common way of representing part of a whole
number. For example, a particular car trip may
require
2
3
of a tank of petrol.
On many occasions, you may need whole
numbers plus a part of a whole number. For
example, a long interstate car trip may require
2
1
4
tanks of petrol. When you have a combination
of a whole number and a fraction this number
is known as a mixed numeral.
let’s start: Pizza frenzy
With a partner, attempt to solve the following
pizza problem. There is more than one answer.
At Pete’s pizza shop, small pizzas are cut into four equal slices, medium pizzas are cut into six equal
slices and large pizzas are cut into eight equal slices.
For a class party, the teacher orders 13 pizzas, which the students eat with ease. After the last slice is
eaten, a total of 82 slices of pizza have been eaten by the students. How many pizzas of each size did the
teacher order?
4E
A long car trip may require a full tank of petrol and another
fraction of a tank as well.
■■ A number is said to be a mixed numeral when it is a mix of a whole number plus a proper fraction.
is a mixed numeral
whole
number
proper
fraction
2
3
5
■■ Improper fractions (fractions greater than a whole, where the numerator is greater than the
denominator) can be converted to mixed numerals or whole numbers.
15
4
3
3
4
16
4
= = 4
improper
fraction
improper
fraction
mixed
numeral
whole
number
Key ideas
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Number and Algebra 157
■■ Mixed numerals can be converted to improper fractions.
■■ In general, improper fractions should be written as mixed numerals, with the fraction part
written in simplest form.
■■ A number line helps show the different types of fractions.
1 2
proper
fractions
mixed
numerals
whole
numbers
improper fractions
0 3
44
14
12
34
1 14
2 12
2 14
2 34
1 12
1 34
54
64
74
84
94
10
4
11
4
12
4
Key ideas
Example 12 Converting mixed numerals to improper fractions
Convert 3
1
5
to an improper fraction.
Solution Explanation
3
1
5
1 1 1
1
5
5
5
5
5
5
5
1
5
16
5
= + + +
= + + +
=
or
3
1
5
15
5
1
5
16
5
= +
=
3
1
5
3
1
5
= wholes + of a whole
+
+
=
=
+
+
+
+
Short-cut method:
Multiply the whole number part by the denominator and
then add the numerator.
3 × 5 + 1 = 16
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Chapter 4 Understanding fractions, decimals 158 and percentages
Example 13 Converting improper fractions to mixed numerals
Convert 11
4
to a mixed numeral.
Solution Explanation
Method 1
11
4
8 3
4
8
4
3
4
2
3
4
2
3
4
= + = + = + =
Method 2
Divided 11 by 4.
4)
rem. 3
11
2
=
2
3
4
11
4
= 11 quarters
+ + + + +
+
+ +
+ + + +
=
=
= 2
3
4
Example 14 Writing mixed numerals in simplest form
Convert 20
6
to a mixed numeral in simplest form.
Solution Explanation
= = ×
×
= 20
6
3
2
6
3
1 2
3 2
3
1
3
or
Method 1: Convert to a mixed numeral and then
simplify the fraction part.
= ×
×
= = 20
6
10 2
3 2
10
3
3
1
3
Method 2: Simplify the improper fraction first and then
convert to a mixed numeral.
Each pane of glass is
1
12
of all the glass.
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number and algebra 159
1 Between which two whole numbers do the following mixed numerals lie?
a 2
1
2
b 11
1
7
c 36
8
9
2 Work out the total number of pieces in each of these situations.
a four pizzas cut into six pieces each
b 10 Lego trucks, where each truck is made from 36 Lego pieces
c fi ve jigsaw puzzles with 12 pieces in each puzzle
d three cakes cut into eight pieces each
3 The mixed numeral 2 3
4
can be represented in ‘window shapes’ as
2 + +
3
4
=
Represent the following mixed numerals using ‘window shapes’.
a 1 1
4
b 1 3
4
c 3 2
4
d 5 2
4
4 A ‘window shape’ consists of four panes of glass.
How many panes of glass are there in the following
number of ‘window shapes’?
a 2 b 3 c 7 d 11
e 4
1
4
f 1
3
4
g 2
2
4
h 5
4
4
5 What mixed numerals correspond to the letters written on each number line?
a
8 9 11 12
A B
7 10
b
1 2 4 5
C D E
0 3
c
24
F G H I
22 23 25 26
d
−1
M L K J
−3 −2 0
6 a Write down how your calculator expresses a mixed numeral.
b Write down how your calculator converts between improper fractions and mixed numerals.
7 Convert these mixed numerals to improper fractions.
a 2
1
5
b 1
3
5
c 3
1
3
d 5
2
3
e 4
1
7
f 3
3
7
g 2
1
2
h 6
1
2
i 4
2
5
j 11
1
2
k 8
2
5
l 10
3
10
m 6
1
9
n 2
7
9
o 5
2
8
p 2
5
8
q 1
11
12
r 3
5
11
s 4
5
12
t 9
7
12
u 5
15
20
v 8
3
100
w 64
3
10
x 20
4
5
y 4
3
25
Example 12
Exercise 4E
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Chapter 4 Understanding fractions, decimals 160 and percentages
8 Convert these improper fractions to mixed numerals.
a
7
5
b 4
3
c 5
3
d 7
4
e
11
3
f 21
5
g 16
7
h 10
4
i
12
7
j 19
6
k 20
3
l 41
4
m
35
8
n 26
5
o 48
7
p 41
3
q
37
12
r 81
11
s 93
10
t 78
7
u
231
100
v 333
10
w 135
11
x 149
12
9 Convert these improper fractions to mixed numerals in their simplest form.
a
10
4
b 28
10
c 16
12
d 8
6
e
18
16
f 30
9
g 40
15
h 60
25
10 Draw a number line from 0 to 5 and mark on it the following fractions.
a
2
3
, 2,
5
3
, 3
1
3
b 3
4
, 12
4
, 2
1
4
, 3
1
2
c 4
5
, 14
5
, 3
1
5
, 10
5
, 19
5
11 Fill in the gaps for the following number patterns.
a 1
1
3
, 1
2
3
, 2,
___
, 2
2
3
, 3, 3
1
3
,
___
,
___
, 4
1
3
, 4
2
3
, 5
b
3
7
,
5
7
, 1, 1
2
7
,
___
, 1
6
7
,
___
, 2
3
7
, 2
5
7
,
___
, 3
2
7
,
___
,
___
c
3
5
, 1
1
5
, 1
4
5
,
___
, 3, 3
3
5
,
___
,
___
, 5
2
5
,
___
, 6
3
5
,
___
12 Four friends order three large pizzas for their
dinner. Each pizza is cut into eight equal slices.
Simone has three slices, Izabella has four slices,
Mark has fi ve slices and Alex has three slices.
a How many pizza slices do they eat in total?
b How much pizza do they eat in total? Give
your answer as a mixed numeral.
c How many pizza slices are left uneaten?
d How much pizza is left uneaten? Give your
answer as a mixed numeral.
Example 13
Example 14
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Number and Algebra 161
13 a Patricia has three sandwiches that are cut into quarters and she eats all but one-quarter.
How many quarters does she eat?
b Phillip has five sandwiches that are cut into halves and he eats all but one-half. How many
halves does he eat?
c Crystal has x sandwiches that are cut into quarters and she eats them all but one-quarter.
How many quarters does she eat?
d Byron has y sandwiches that are cut into thirds and he eats all but one-third. How many
thirds does he eat?
e Felicity has m sandwiches that are cut into n pieces and she eats them all. How many pieces
does she eat?
Enrichment: Mixed numeral swap meet
14 a Using the digits 1, 2 and 3 only once, three different mixed numerals can be written.
i Write down the three possible mixed numerals.
ii Find the difference between the smallest and highest mixed numerals.
b Repeat part a using the digits 2, 3 and 4.
c Repeat part a using the digits 3, 4 and 5.
d Predict the difference between the largest and smallest mixed numeral when using only the
digits 4, 5 and 6. Use subtraction to see if your prediction is correct.
e Write down a rule for the difference between the largest and smallest mixed numerals when
using any three consecutive integers.
f Extend your investigation to allow mixed numerals where the fraction part is an improper
fraction.
g Extend your investigation to produce mixed numerals from four consecutive digits.
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Chapter 4 Understanding fractions, decimals 162 and percentages
ordering positive and negative fractions
You already know how to order a set of whole numbers.
For example: 3, 7, 15, 6, 2, 10 are a set of six different whole numbers that you could place in ascending
or descending order.
In ascending order, the correct order is: 2, 3, 6, 7, 10, 15.
In descending order, the correct order is: 15, 10, 7, 6, 3, 2.
In this section you will learn how to write different fractions in ascending and descending order. To be
able to do this we need to compare different fractions and we do this through our knowledge of equivalent
fractions (see Section 4D).
Remember a fraction is greater than another fraction if it lies to the right of that fraction on a number line.
3
4
1
2
>
0 1 1
2
3
4
let’s start: The order of fi ve
• As a warm-up activity, ask fi ve volunteer
students to arrange themselves in alphabetical
order, then in height order and, fi nally, in
birthday order.
• Each of the fi ve students receives a large
fraction card and displays it to the class.
• The rest of the class must then attempt to order
the students in ascending order, according to
their fraction card. It is a group decision and
none of the fi ve students should move until the class agrees on a decision.
• Repeat the activity with a set of more challenging fraction cards.
4F
■■ To order (or arrange) positive fractions we must know how to compare different fractions. This
is often done by considering equivalent fractions.
■■ If the numerators are the same, the smallest fraction is the one with the biggest denominator, as
it has been divided up into the most pieces.
For example:
1
7
1
2
< .
■■ If the denominators are the same, the smallest fraction is the one with the smallest numerator.
For example:
3
10
7
10
< .
Key ideas
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Number and Algebra 163
■■ To order two fractions with different numerators and denominators, we can use our
knowledge of equivalent fractions to produce fractions with a common denominator and
then compare the numerators.
■■ The lowest common denominator (LCD) is the lowest common multiple of the different
denominators.
■■ Ascending order is when numbers are ordered going up, from smallest to largest.
■■ Descending order is when numbers are ordered going down, from largest to smallest.
Key ideas
Example 15 Comparing fractions
Place the correct mathematical symbol (i.e. <, = or >) in between the following pairs of fractions to
make true mathematical statements.
a 2
5
4
5
b 1
3
1
5
c 2
3
3
5
d 2
3
7
16
7
e − − 1
3
2
3
f − − 3
4
5
8
Solution Explanation
a
2
5
4
5
< Denominators are the same, therefore compare
numerators.
b
1
3
1
5
> Numerators are the same.
Smallest fraction has the biggest denominator.
c 2
3
3
5
10
15
9
15
> . Hence,
2
3
3
5
> .
LCD of 3 and 5 is 15.
Produce equivalent fractions.
Denominators now the same, therefore compare
numerators.
d 2
3
7
16
7
17
7
16
7
> . Hence, 2
3
7
16
7
> .
Convert mixed number to an improper fraction.
Denominators are the same, therefore compare
numerators.
e − > − 1
3
2
3
−1 0
2
3
1
3
− 1
3
is further to the right than − 2
3
.
f − − 3
4
5
8
− < − 6
8
5
8
−1 0
6
8
1
2 −
− 5
8 −
− 5
8
is further to the right than − 6
8
.
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Chapter 4 Understanding fractions, decimals 164 and percentages
Example 16 Ordering fractions
Place the following fractions in ascending order.
a 3
4
4
5
2
3
, , b 1
3
5
7
4
3
2
11
5
, , , 2
1
4
,
Solution Explanation
a
45
60
48
60
40
60
, ,
40
60
45
60
48
60
, ,
2
3
3
4
4
5
, ,
LCD of 3, 4 and 5 is 60. Produce equivalent fractions
with denominator of 60.
Order fractions in ascending order.
Rewrite fractions back in original form.
b
8
5
7
4
3
2
11
5
, , ,
9
4
,
32
20
35
20
30
20
44
20
, , ,
45
20
,
30
20
32
20
35
20
44
20
, , , ,
45
20
3
2
1
3
5
7
4
11
5
, , , , 2
1
4
Express all fractions as improper fractions.
LCD of 2, 4 and 5 is 20. Produce equivalent fractions with
a denominator of 20.
Order fractions in ascending order.
Rewrite fractions back in original form.
Is the fraction of balls which are red less than or greater than the fraction of dice which are white?
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number and algebra 165
1 Circle the largest fraction in each of the following lists.
a
3
7
2
7
5
7
, , ,
1
7
b , , ,
5
3
4
3
2
3
7
3
c
5
11
9
11
3
11
, , ,
4
11
d , , ,
7
5
8
5
4
5
6
5
2 State the lowest common multiple of the following sets of numbers.
a 2, 5 b 3, 7 c 5, 4 d 6, 5
e 3, 6 f 2, 10 g 4, 6 h 8, 6
i 2, 3, 5 j 3, 4, 6 k 3, 8, 4 l 2, 6, 5
3 State the lowest common denominator of the following sets of fractions.
a
1
3
3
5
, b 2
4
3
5
, c 4
7
2
3
, d 2
10
1
5
,
e
4
6
3
8
, f 5
12
2
5
, g 1
2
2
3
3
4
, , h 4
3
3
4
,
4 Fill in the gaps to produce equivalent fractions.
a
2
5 15
= b 2
3 12
= c 1
4 16
=
d 3
7 14
= e 3
8 40
= f 5
6 18
=
5 Place the correct mathematical symbol (i.e. <, = or >) in between the following pairs of
fractions to make true mathematical statements.
a
3
5
1
5
b
7
9
2
9
c
2
2
3
3
d
13
18
17
18
e
1
4
1
3
f
1
10
1
20
g
1
7
1
5
h
3
5
18
30
i
2
3
1
3
j
4
5
3
4
k
5
6
9
10
l
5
7
15
21
m
7
11
3
5
n 1
2
3
1
1
2
o 3
3
7
15
4
p
12
5
19
8
q − − 1
4
1
2
r − − 2
3
3
4
s − − 2
5
5
8
t − − 3
4
3
5
6 Place the following fractions in ascending order.
a
3
5
8
5
2
5
, , 1 b 5
9
1
3
2
9
, ,
c
2
5
3
4
4
5
, , d 5
6
3
5
2
3
, ,
Example 15
Example 16
Exercise 4F
WORK ING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 4 Understanding fractions, decimals 166 and percentages
e 2
1
4
11
4
5
2
, , , 3
1
3
f 15
8
11
6
7
4
, , ,
5
3
g 2
7
10
9
4
11
5
3
5
, , , 2
1
2
, 2 h 4
4
9
15
3
10
27
1
6
, , 4 , 4
2
3
, 4
7 Use a number line to place the following fractions in ascending order.
a − − − 5
4
2
1
2
3
2
, , b − − − 1
4
1
6
1
3
, ,
8 Place the following fractions in descending order, without fi nding common denominators.
a
1
3
1
5
1
4
1
2
, , , b 3
5
3
7
3
6
3
8
, , ,
c
7
2
7
5
7
8
7
7
, , , d
1
15
1
10
1
50
1
100
, , ,
e 7
1
11
3
5
4
9
2
3
, 8 , 5 , 10 f 2
1
3
1
9
1
6
1
5
, 2 , 2 , 2
9 Place the following cake fractions in decreasing order of size.
a sponge cake shared equally by four people = 1
4
cake
B chocolate cake shared equally by eleven people = 1
11
cake
C carrot and walnut cake shared equally by eight people = 1
8
cake
10 Four friends, Dean, David, Andrea and Rob, all competed in a marathon. Their respective
fi nishing times were 3
1
3
hours, 3
5
12
hours, 3
1
4
hours and 3
4
15
hours. Write down the correct
fi nishing order of the four friends.
11 Rewrite the fractions in each set with their lowest common denominator and then write the next
two fractions that would continue the pattern.
a 2
9
,
1
3
,
4
9
,
___
,
___
b 1
2
,
5
4
, 2,
___
,
___
c 11
6
,
3
2
,
7
6
,
___
,
___
d 1
2
,
4
7
,
9
14
,
___
,
___
12 Write a fraction that lies between the following pairs of fractions.
a
3
5
3
4
, b 1
4
1
2
, c 2
7
1
6
,
d
17
20
7
10
, e 2
1
3
1
5
, 2 f 8
7
10
3
4
, 8
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
4F
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Number and Algebra 167
13 Explain how to find a fraction that lies between two fractions with
different denominators.
14 Write the whole number values that ? can take so that
?
3
lies between:
a 2 and 3 b 5 and 5
1
2
15 Thomas and Nathan had a doughnut eating race to see who
could eat the most doughnuts in 1 minute. Before the race
started Thomas cut each of his doughnuts into fifths to make
them just the right bite-size. Nathan decided to cut each of his
doughnuts into quarters before the race. After 1 minute of
frenzied eating, the stop whistle blew. Thomas had devoured
28 fifths of doughnut and Nathan had munched his way
through 22 quarters of doughnut.
a Who won the doughnut eating race?
b What was the winning doughnut margin? Express your
answer in simplest form.
Enrichment: Shady designs
16 a For each of the diagrams shown, work out what fraction of the rectangle is coloured blue.
Explain how you arrived at each of your answers.
b Redraw the shapes in order from the most blue to least blue.
c Design and shade two more rectangle designs.
i ii
iii iv
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 4 Understanding fractions, decimals 168 and percentages
place value in decimals and ordering decimals
Some quantities change by whole number amounts, such as the
number of people in a room, but there are many quantities that
increase or decrease continuously, such as your height, weight
and age. Often we talk about age as a whole number (e.g. Mike
is 12 years old) but, in reality, our age is an ever-increasing
(continuous) quantity. For example, if Mike is 12 years, 4 months,
2 weeks, 3 days, 5 hours, 6 minutes and 33 seconds old, then Mike
is actually 12.38062147 years old!
There are many numbers in today’s society that are not whole
numbers. For example, it is unusual to buy an item in a supermarket
that is simply a whole number of dollars. The price of almost all
shopping items involves both dollars and cents. A chocolate bar may
cost $1.95, which is an example of a decimal number.
let’s start: Split-second timing
Organise students into pairs and use a digital stopwatch.
• Try to stop the stopwatch on exactly 10 seconds. Have two attempts each.
Were you able to stop it exactly on 10.00 seconds? What was the closest time?
• Try these additional challenges with your partner.
a Stop the watch exactly on: i 12 5
6
seconds ii 8.37 seconds
iii 9 7
10
seconds iv 14.25 seconds
b How quickly can you start and stop the stopwatch?
c How accurately can you time 1 minute without looking at the stopwatch?
4G
If a chocolate bar costs $2, would the price
still be a decimal number?
■■ A decimal point is used to separate the whole number from the decimal or fraction part.
■■ When dealing with decimal numbers, the place value table must be extended to involve tenths,
hundredths, thousandths etc.
The number 428.357 means:
Hundreds tens units . tenths Hundredths thousandths
4 2 8 . 3 5 7
4 × 100 2 × 10 8 × 1 . 3 ×
1
10
5 ×
1
100
7 ×
1
1000
400 20 8 .
3
10
5
100
7
1000
Key ideas
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Number and Algebra 169
Example 17 Understanding decimal place value
What is the value of the digit 8 in the following numbers?
a 12.85 b 6.1287
Solution Explanation
a The value of 8 is
8
10
.
The 8 is in the first column after the decimal point, which
is the tenths column.
b The value of 8 is
8
1000
.
The 8 is in the third column after the decimal point, which
is the thousandths column.
Example 18 Changing to decimals
Express each of the following proper fractions and mixed numerals as decimals.
a 7
10
b 5
100
c 3 17
100
Solution Explanation
a
7
10
= 0.7
7
10
means seven-tenths, so put the 7 in the tenths column.
b
5
100
= 0.05
5
100
means five-hundredths, so put the 5 in the
hundredths column.
c 3
17
100
= 3.17 3
17
100
means 3 units and 17 one-hundredths.
17 hundredths is one-tenth and seven-hundredths.
Example 19 Arranging decimal numbers in order
Arrange the following decimal numbers in ascending order (i.e. smallest to largest).
3.72, 7.23, 2.73, 2.37, 7.32, 3.27
Solution Explanation
2.37, 2.73, 3.27, 3.72, 7.23, 7.32 The units column has a higher value than the tenths
column, and the tenths column has a higher value than
the hundredths column.
2.73 is bigger than 2.37 because it has seven-tenths,
which is bigger than three-tenths.
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Chapter 4 Understanding fractions, decimals 170 and percentages
1 For the number 58.237, give the value of the digit:
a 2 b 3 c 7
2 A stopwatch is stopped at 36.57 seconds.
a What is the digit displayed in the tenths column?
b What is the digit displayed in the units column?
c What is the digit displayed in the hundredths column?
d Is this number closer to 36 or 37 seconds?
3 What is the value of the digit 6 in the following numbers?
a 23.612 b 17.46 c 80.016 d 0.693
e 16.4 f 8.56813 g 2.3641 h 11.926
4 State whether each of the following is true or false.
a 7.24 < 7.18 b 21.32 < 20.89 c 4.61 > 4.57 d 8.09 > 8.41
e 25.8 ≤ 28.5 f 2.1118 ≤ 2.8001 g 7.93 ≥ 8.42
h 11.11 ≥ 11.109
i 3
10
30
100
=
j 7
10
70
100
= k 5
10
 5 l 2
10
20
1000

5 Express each of the following proper fractions as a decimal.
a 3
10
b 8
10
c 15
100
d 23
100
e 9
10
f 2
100
g 121
1000
h 74
1000
6 Express each of the following mixed numerals as a decimal.
a 6
4
10
b 5
7
10
c 212
3
10
d 1
16
100
e 14
83
100
f 7
51
100
g 5
7
100
h 18
612
1000
7 Write the following number phrases as decimals.
a seven and six-tenths b twelve and nine-tenths
c thirty-three and four-hundredths d twenty-six and fi fteen-hundredths
e eight and forty-two hundredths f ninety-nine and twelve-thousandths
8 How close are the following decimal numbers to their nearest whole number?
a 6.9 b 7.03 c 18.98 d 16.5
e 17.999 f 4.99 g 0.85 h 99.11
Example 17
Example 18a,b
Example 18c
Exercise 4G
WORK ING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
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Number and Algebra 171
9 Arrange these groups of numbers in ascending order (i.e. smallest to largest).
a 3.52, 3.05, 3.25, 3.55 b 30.6, 3.06, 3.6, 30.3
c 17.81, 1.718, 1.871, 11.87 d 26.92, 29.26, 29.62, 22.96, 22.69
10 The batting averages for five retired Australian
Cricket test captains are: Adam Gilchrist 47.60,
Steve Waugh 51.06, Mark Taylor 43.49, Allan
Border 50.56 and Kim Hughes 37.41.
a List the five players in descending order of
batting averages (i.e. largest to smallest).
b Ricky Ponting’s test batting average is 51.85.
Where does this rank him in terms of the
retired Australian test captains listed above?
11 The depth of a river at 9 a.m. on six consecutive days was:
Day 1: 1.53 m Day 2: 1.58 m Day 3: 1.49 m
Day 4: 1.47 m Day 5: 1.52 m Day 6: 1.61 m
a On which day was the river level highest?
b On which day was the river level lowest?
c On which days was the river level higher than the previous day?
12 a, b and c are digits and a > b > c. Write these numbers from smallest to largest. Note that the
dot represents the decimal point.
a a.b, b.c, a.c, c.c, c.a, b.a
b a.bc, b.ca, b.bb, c.ab, c.bc, ba.ca, ab.ab, a.aa, a.ca
13 Write as decimals, if a is a digit.
a a
10
b a
100
c a a
10 100
+ d a
a a + +
10 1000
Enrichment: Different decimal combinations
14 a Write as many different decimal numbers as you can and place them in ascending order
using:
i the digits 0, 1 and a decimal point. Each digit must be used only once.
ii the digits 0, 1, 2 and a decimal point. Each digit must be used only once.
iii the digits 0, 1, 2, 3 and a decimal point. Each digit must be used only once.
b Calculate the number of different decimal numbers that could be produced using the digits 0,
1, 2, 3, 4 and a decimal point.
Example 19
WORK ING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 4 Understanding fractions, decimals 172 and percentages
Rounding decimals
Decimal numbers sometimes contain more decimal places than we
need. It is important that we are able to round decimal numbers
when working with money, measuring quantities, including time and
distance, or writing answers to some division calculations.
For example, the distance around the school oval might be 0.39647 km,
which rounded to 1 decimal place is 0.4 km or 400 m. The rounded fi gure,
although not precise, is accurate enough for most applications.
Running events are electronically measured and rounded to
2 decimal places. Usain Bolt, the fastest man on Earth, has repeatedly
broken his own world records. In August 2009 he set a new world
record of 9.58 seconds over 100 m at the World Championships in
Germany, which was 5-hundredths (0.05) of a second faster than his
London Olympic Games (August 2012) record of 9.63 seconds.
let’s start: Rounding brainstorm
In a group of four, brainstorm occasions when it may be useful to
round or estimate decimal numbers. Aim to get more than 10 common applications.
■■ Rounding involves approximating a decimal number to fewer decimal places.
4H
Usain Bolt’s 100-metre sprint world
record is measured to 2 decimal
places.
■■ The mathematical symbol used for approximation is ≈. The two wavy parallel lines indicate the
answer is approximately equal to the value given.
■■ To round a decimal:
– Cut the number after the required decimal place; e.g. round to 2 decimal places.
– To determine whether you should round your answer up or down, consider only the digit
immediately to the right of the specifi ed place. For rounding purposes this can be referred to as
the critical digit.
‘cut’
15.63 27
2 is the critical
digit in this example
– If the critical digit is less than 5 (i.e. 0, 1, 2, 3 or 4), then you round down. This means write
the original number to the place required, leaving off all other digits. This can be referred to as
simply leaving the number as it is.
– If the critical digit is 5 or more (i.e. 5, 6, 7, 8 or 9), then you round up. This means write the
original number to the place required, but increase this digit by 1. Leave off all other digits.
Key ideas
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Number and Algebra 173
Example 20 Determining the critical digit
The following decimal numbers need to be rounded to 2 decimal places. Draw a line where the number
must be cut and then circle the critical digit.
a 23.5398 b 1.75137
Solution Explanation
a 2 3 . 5 3 9 8 A line is drawn directly after the specified number of
decimal places, in this case, 2.
b 1 . 7 5 1 3 7 The critical digit is always the number straight after the
specified number of decimal places.
Example 21 Rounding decimals to 1 decimal place
Round each of the following to 1 decimal place.
a 25.682 b 13.5458
Solution Explanation
a 25.7 The critical digit is 8 and therefore the tenths column
must be rounded up from a 6 to a 7.
b 13.5 The critical digit is 4 and therefore the tenths column
remains the same, in effect rounding the original number
down to 13.5.
Example 22 Rounding decimals to different decimal places
Round each of the following to the specified number of decimal places.
a Round 18.34728 to 3 decimal places. b Round 0.43917 to 2 decimal places.
c Round 7.59967 to 3 decimal places.
Solution Explanation
a 18.347 The critical digit is 2, therefore round down.
b 0.44 The critical digit is 9, therefore round up.
c 7.600 The critical digit is 6, therefore round up.
Rounding up has resulted in digits being carried over.
Remember to show the stated number of decimal places;
hence, the zeros must be displayed.
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Chapter 4 Understanding fractions, decimals 174 and percentages
1 For each of the following, select the closer alternative.
a Is 5.79 closer to 5.7 or 5.8? b Is 2.4 closer to 2 or 3?
c Is 83 closer to 80 or 90? d Is 6.777 closer to 6.77 or 6.78?
2 The following decimals need to be rounded, correct to 2 decimal places. Draw a line where the
number must be cut and then circle the critical digit that must be checked as to whether to round
up or down.
a 12.6453 b 4.81932 c 157.281 d 4 001 565.38471
e 0.06031 f 203.5791 g 66.6666 h 7.995123
3 To round correctly to a specifi ed number of places, you must know which digit is the critical
digit. Remember: The critical digit is always the digit immediately to the right of the specifi ed number
of places.
a State the critical digit in each of the following numbers.
i 25.8174 rounded to 1 decimal place. Critical digit = ____
ii 25.8174 rounded to 2 decimal places. Critical digit = ____
iii 25.8174 rounded to 3 decimal places. Critical digit = ____
iv 25.8174 rounded to the nearest whole number. Critical digit = ____
b State the correct rounded numbers for the numbers in parts i to iv above.
4 Round each of the following to 1 decimal place.
a 14.82 b 7.38 c 15.62 d 0.87
e 6.85 f 9.94 g 55.55 h 7.98
5 Write each of the following, correct to 2 decimal places.
a 3.7823 b 11.8627 c 5.9156 d 0.93225
e 123.456 f 300.0549 g 3.1250 h 9.849
i 56.2893 j 7.121999 k 29.9913 l 0.8971
6 Round each of the following to the specifi ed number of decimal places, given as the number in
the brackets.
a 15.913 (1) b 7.8923 (2) c 235.62 (0) d 0.5111 (0)
e 231.86 (1) f 9.3951 (1) g 9.3951 (2) h 34.71289 (3)
7 Round each of the following to the specifi ed number of decimal places.
a 23.983 (1) b 14.8992 (2) c 6.95432 (0) d 29.999731 (3)
8 Round each of the following to the nearest whole number.
a 27.612 b 9.458 c 12.299 d 123.72
e 22.26 f 117.555 g 2.6132 h 10.7532
9 Round each of the following amounts to the nearest dollar.
a $12.85 b $30.50 c $7.10 d $1566.80
e $120.45 f $9.55 g $1.39 h $36.19
Example 20
Example 21
Example 22a,b
Example 22c
Exercise 4H
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Number and Algebra 175
10 Some wise shoppers have the habit of rounding all items to the nearest dollar as they place
them in their shopping basket. They can then keep a running total and have a close
approximation as to how much their final bill will cost. Use this technique to estimate the
cost of the following.
a Jeanette purchases 10 items:
$3.25, $0.85, $4.65, $8.99, $12.30, $7.10, $2.90, $1.95, $4.85, $3.99
b Adam purchases 12 items:
$0.55, $3.00, $5.40, $8.90, $6.90, $2.19, $3.20, $5.10, $3.15, $0.30, $4.95, $1.11
c Jeanette’s actual shopping total is $50.83 and Adam’s is $44.75. How accurate were
Jeanette’s and Adam’s estimations?
11 Electronic timing pads are standard in National
Swimming competitions. In a recent National
Under 15 100 metre freestyle race, Edwina
receives a rounded time of 52.83 seconds and
Jasmine a time of 53.17 seconds.
a If the timing pads can calculate times only
to the nearest second, what will be the time
difference between the two swimmers?
b If the timing pads can calculate times only to
the nearest tenth of a second, what will be the
time difference between the two swimmers?
c What is the time difference between the two swimmers, correct to 2 decimal places?
d If the timing pads can measure to 3 decimal places, what would be the quickest time in
which Edwina could have swum the race?
12 Using a calculator, evaluate 15.735629 ÷ 7, correct to 2 decimal places. What is the least
number of decimal places you need to find in the quotient to ensure that you have rounded
correctly to 2 decimal places?
13 Samara believes 0.449999 should be rounded up to 0.5, but Cassandra believes it should be
rounded down to 0.4. Make an argument to support each of their statements, but then show the
flaw in one girl’s logic and clearly indicate which girl you think is correct.
Enrichment: Rounding with technology
14 Most calculators are able to round numbers correct to a specified number of places. Find out
how to do this on your calculator and check your answers to Questions 5 and 6.
15 Spreadsheet software packages can also round numbers correct to a specified number of places.
Find out the correct syntax for rounding cells in a spreadsheet program, such as Microsoft
Excel, and then check your answers to Questions 7 and 8.
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Chapter 4 Understanding fractions, decimals 176 and percentages
Decimal and fraction conversions
Decimals and fractions are both commonly used
to represent numbers that are not simply whole
numbers. It is important that we know how to
convert a decimal number to a fraction, and
how to convert a fraction to a decimal number.
In the photo opposite, we can see that:
• 1
4
or 0.25 of the cake has been eaten
and 3
4
or 0.75 of the cake remains.
let’s start: Match my call
• In pairs, nominate one student to be ‘Fraction kid’ and the other to be ‘Decimal expert’. ‘Fraction kid’
starts naming some common fractions and ‘Decimal expert’ tries to give the equivalent decimal value.
Start with easy questions and build up to harder ones.
• After 10 turns, swap around. This time ‘Decimal expert’ will name some decimal numbers and
‘Fraction kid’ will attempt to call out the equivalent fraction.
• Discuss the following question in pairs:
Which is easier, converting fractions to decimals or decimals to fractions?
4I
■■ Converting decimals to fractions
– Using your knowledge of place value, express the
decimal places as a fraction whose denominator is
a power of 10. Remember to simplify the fraction
whenever possible.
■■ Converting fractions to decimals
– When the denominator is a power of 10, we can
simply change the fraction to a decimal through
knowledge of place value.
– When the denominator is not a power of 10, try to
fi nd an equivalent fraction whose denominator has
a power of 10 and then convert to a decimal.
– A method that will always work for converting
fractions to decimals is to divide the numerator
by the denominator.
■■ Recurring decimals are decimals with a repeated pattern.
– A dot, dots or a bar above a number or numbers indicates a repeated pattern.
e.g.
1
3
= 0 33333 = 0 3 = =
13
11
. … . 1.181818…
i
1 18
1 714 285714285 1
.
.
i i
…
or 1.18
12
7
= = .714285 1.714285
i i
or Key ideas
3 numerator
4
denominator
e.g. 0 25
25
100
1
4
. = =
e.g.
37
100
= 0.37
e.g. 2
5
4
10
= = 0.4
e.g. 5
8
= 8)5 0 00
0 6 25
5 2 4 0 625
.
.
= .
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Number and Algebra 177
Example 23 Converting decimals to fractions
Convert the following decimals to fractions in their simplest form.
a 0.239 b 10.35
Solution Explanation
a 239
1000
0.239 = 239 thousandths
b 10
35
100
10
7
20
=
0.35 = 35 hundredths, which can be simplified further by
dividing the numerator and denominator by the highest
common factor of 5.
Example 24 Converting fractions to decimals
Convert the following fractions to decimals.
a 17
100
b 5
3
5
c
7
12
Solution Explanation
a
17
100
= 0.17
17 hundredths
b 5
3
5
5
6
10
= = 5.6
6
10
is an equivalent fraction of
3
5
, whose denominator
is a power of 10.
c
7 i
12
= 0.58333… or 0.583 12)7 0 0 0 0 0
0 5 83 3 3
7 10 4 4 4
.
. …
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Chapter 4 Understanding fractions, decimals 178 and percentages
1 Complete each of these statements, which convert common fractions to decimals.
a 1
2 10
= = 0.5 b
1
4
25
= = 0.25
c 3
4 100
= = 0. 5 d 2 4
10
= = 0.
2 Complete each of these statements, which convert decimals to fractions, in simplest form.
a 0 2
10
1
5
. = = b 0 15
100
3
. = =
c 0 8
8
5
. = = d 0 64
64
100 25
. = =
3 State whether each of the following is true or false. Use the examples in the Key Ideas to help.
a 0.333… = 0.3 b … i 0.1111 = 0.1 c … i 3.2222 = 3.2
d … = i 1.7272 1.72 e … i i 3.161616 = 3.16 f 4.216216…= 4.216
4 Convert the following decimals to fractions in their simplest form.
a 0.5 b 6.4 c 10.15 d 18.12
e 3.25 f 0.05 g 9.075 h 5.192
5 Convert each of these fractions to decimals.
a 7
10
b 9
10
c 31
100
d 79
100
e 121
100
f 3
29
100
g 123
1000
h 3
100
6 Convert the following fractions to decimals, by fi rst changing the fraction to an equivalent
fraction whose denominator is 10, 100 or 1000.
a 4
5
b 1
2
c
7
20
d
23
50
e 5
19
20
f 3
1
4
g 5
2
h 3
8
7 Convert the following fractions to decimals, by dividing the numerator by the denominator.
Use a calculator to check your answers.
a
1
2
b 3
6
c 3
4
d 2
5
e 1
3
f 3
8
g 5
12
h 3
7
i
1
6
j
2
3
k
1
7
l
5
9
Example 23a
Example 23b
Example 24a
Example 24b
Example 24c
Exercise 4I
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Number and Algebra 179
8 Copy and complete the following fraction ↔ decimal tables. The quarters table (part c)
has already been done for you. It’s well worth trying to memorise these fractions and their
equivalent decimal values.
a halves b thirds
Fraction
0
2
1
2
2
2
Decimal
Fraction
0
3
1
3
2
3
3
3
Decimal
c quarters d fifths
Fraction
0
4
1
4
2
4
3
4
4
4
Decimal 0 0.25 0.5 0.75 1
Fraction
0
5
1
5
2
5
3
5
4
5
5
5
Decimal
9 Arrange the following from smallest to largest.
a 1
2
0 75
5
8
0 4 0 99
1
4
, . , , . , . , b 3
7
0 13
1
9
0 58 0 84
4
5
, . , , . , . ,
10 Tan and Lillian are trying to work out who is the better chess player. They have both been
playing chess games against their computers. Tan has played 37 games and beaten the computer
11 times. Lillian has played only 21 games and has beaten the computer 6 times.
a Using a calculator and converting the appropriate fractions to decimals, determine who is
the better chess player.
b Lillian has time to play another four games of chess against her computer. To be classified as
a better player than Tan, how many of these four games must she win?
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Chapter 4 Understanding fractions, decimals 180 and percentages
11 To estimate the thickness of one sheet of A4 paper,
Christopher measures a ream of paper, which
consists of 500 sheets of A4 paper. He determines
that the pile is 55 mm thick. How thick is one sheet
of A4 paper? Express your answer as a decimal
number and also as a fraction.
12 a Copy and complete the following fraction ↔ decimal table.
Fraction 1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
1
10
Decimal
b Comment on the trend in the decimal values as the denominator increases.
c Try to explain why this makes sense.
13 a Copy and complete the following decimal ↔ fraction table.
Decimal 0.1 0.2 0.25 0.4 0.5 0.6 0.75 0.8 0.9
Fraction
b Comment on the trend in the fractions as the decimal value increases.
c Try to explain why this makes sense.
14 a Write 1
3
as a recurring decimal.
b Write 2
3
as a recurring decimal.
c Using your calculator, fi nd 2 ÷ 3.
d Is the calculator correct or incorrect to display the answer as 0.666666667. Explain.
15 Write three different fractions with different denominators that are between the decimal value of
2.4 and 2.5.
16 When 4
7
is expressed in decimal form, fi nd the digit in the 23rd decimal place. Give a reason for
your answer.
Enrichment: Design a decimal game for the class
17 Using the skill of converting decimals to fractions and vice versa, design an appropriate
game that students in your class could play. Ideas may include variations of Bingo, Memory,
Dominoes etc. Try creating a challenging set of question cards.
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4I
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number and algebra 181
Connecting percentages with fractions and decimals
Percentages are related closely to fractions. A percentage is a fraction in which the denominator is 100.
Per cent is Latin for ‘out of 100’. One dollar is equivalent to 100 cents and a century is 100 years.
Percentages are used in many everyday situations. Interest rates, discounts, test results and statistics
are usually described using percentages rather than fractions or decimals because it is easier to compare
two different results.
let’s start: Comparing performance
Consider these netball scores achieved by four students.
Annie scores 30 goals from 40 shots (i.e. 30 out of 40).
Bella scores 19 goals from 25 shots.
Cara scores 4 goals from 5 shots.
Dianne scores 16 goals from 20 shots.
• Discuss ways to compare the accuracy of their goal shooting.
• How might percentages be used?
The chart below might be useful.
Annie divided her 40 shots in the 10 boxes and then shaded the ones she scored.
4J
Annie
(40 shots)
Bella Cara Dianne
4
4
4
4
4
4
4
4
4
4
100%
90%
80%
70%
60%
50%
40%
30%
20%
10%
0%
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Chapter 4 Understanding fractions, decimals 182 and percentages
■ Percentages have been used for hundreds of years but the symbol we use today is fairly recent.
The symbol % means per cent. It comes from the Latin words per centum, which mean ‘out
of 100’.
For example: 35% means ‘35 out of 100’ or
35
100
or 35 ÷ 100 or 0.35.
■ Percentages are a useful way to compare fractions.
For example: 3
4
75
100
= = 75% and
18
25
72
100
= = 72%, therefore 3
4
18
25
> .
■ It is important to understand the relationships and connections between fractions, decimals and
percentages. The ‘fraction wall’ diagram below shows these very clearly.
percentages
decimals
one whole 1
halves
thirds
quarters
fifths
sixths
eighths
ninths
tenths
twelfths
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
10% 20% 30% 40% 50% 60% 70% 80% 90%
Key ideas
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Number and Algebra 183
■■ It is also important to memorise the most commonly used conversions. These are listed in
the table below.
Words Diagram Fraction Decimal Percentage
one whole 1 1 100%
one-half
one-third
one-quarter
one-fifth
one-tenth
one-hundredth
0.5 50%
0.25
0.333... or 0.3
25%
0.2 20%
0.1 10%
100 0.01 1%
1
10
1
5 1
4 1
3 1
3
33 1 %
2 1
■■ The number facts in the table can be used to do other conversions.
For example:
×3
×5
×9
1
4
= 0.25 = 25%
3
4
= 0.75 = 75%
5
4
= 1.25 = 125%
9
4
= 2.25 = 225%
Key ideas
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Chapter 4 Understanding fractions, decimals 184 and percentages
Example 25 Using a known number fact to make conversions
Given that = 1
5
20%, complete the following.
a = = 1
5
20%, so
3
5
____% b = = 1
5
20%, so
6
5
____%
c = = 1
5
20%, so
11
5
____%
SOLUTION EXPLANATION
= 1
5
20%
This should be a memorised number fact.
a = × = 3
5
20% 3 60% Multiply the number fact by 3.
b = × = 6
5
20% 6 120% Multiply the number fact by 6.
c = × = 11
5
20% 11 220% Multiply the number fact by 11.
Example 26 Using memorised number facts
Convert the following fractions to decimals and percentages.
a seventeen-tenths b nine-quarters c two-thirds
SOLUTION EXPLANATION
a 1
10
10
100
= = 0.10 = 10%
 = = = 17
10
170
100
1.70 170%
This should be a memorised number fact.
Multiply the number fact by 17.
b 1
4
25
100
= = 0.25 = 25%
 = = 9
4
2.25 225%
This should be a memorised number fact.
Multiply the number fact by 9.
c 1
3
0 333 33
1
3
= . ... = % This should be a memorised number fact.
 = = 2
3
0.666... 66
2
3
% Multiply the number fact by 2.
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number and algebra 185
1 The percentage equivalent to one-quarter is:
a 4% B 2.5% C 25% D 40%
2 The percentage equivalent to three-quarters is:
a 7.5% B 34% C 75% D 80%
3 The percentage equivalent to 0.1 is:
a 0.1% B 1% C 10% D 100%
4 Use the fraction wall (see page 182) to complete the following.
a three-quarters = = 0.___ = ___% b nine-tenths = = 0.___ = ___%
c two-fi fths = = 0.___ = ___% d four-fi fths = = 0.___ = ___%
5 Complete the following.
a 1
2
50
3
2
= %, so = ___% b 1
2
50
7
2
= %, so = ___%
c 1
4
25
3
4
= %, so = ___% d
1
4
25
7
4
= %, so = ___%
e 1
5
20
9
5
= %, so = ___% f = = 1
5
20%, so
11
5
___%
6 Convert the following fractions to decimals and percentages.
a three-tenths b three-fi fths
c fi ve-quarters d four-thirds
7 a Use the fraction wall (see page 182) to write down:
i six fractions that are equivalent to 50%
ii ten fractions that are greater than 25% but less than 50%
b Comparing your answers to i and ii, which fraction is closest to 50%?
8 Use the fraction wall (see page 182) to complete these computations. Give your answer as a
fraction in simplest form.
a 1
2
1
4
+ b + 1
4
1
4
c 1
8
1
8
+
d 1
2
1
4
1
4
+ + e + 1
6
1
6
f 2
3
2
3
+
Example 25
Example 26
Exercise 4J
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Chapter 4 Understanding fractions, decimals 186 and percentages
9 Use the fraction wall (see page 182) to do these computations.
a 1
2
1
4
− b 1
2
1
6
− c 1
4
1
8
−
d 1
2
1
4
1
4
− − e 2
3
1
6
− f 2
3
1
12
−
g 1
1
4
− h 1
1
5
− i 1
3
8
−
10 Use the fraction wall (see page 182) to complete these computations.
a 1 - 0.1 b 1 - 0.7 c 1 - 0.25
d 1 - 0.65 e 0.25 + 0.25 f 0.25 + 0.65
g 3 × 0.2 h 6 × 0.2 i 0.3 × 7
11 Rachel’s birthday cake is cut into two equal pieces, then four
equal pieces, then eight equal pieces. Rachel eats three pieces.
a What percentage of the cake did Rachel eat?
b What percentage of the cake remains?
12 Use the fraction wall (see page 182) to answer
the following questions.
a Which is bigger: three-quarters or two-thirds?
b Which is bigger: two-thirds or three-fi fths?
c What is half of one-half?
d What is half of one-quarter?
e What fraction is exactly halfway between one-half and one-quarter?
13 Sophie’s netball team wins six of their fi rst seven games. They have three more games to play.
a What is the highest percentage the team can achieve?
b What is the lowest percentage the team can achieve?
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14 Use the fractions in the fraction wall (see page 182) to solve these problems. C
a Two fractions with the same denominator add up to one-half. What could they be?
What else could they be? Write down all the possibilities from the fractions in the
fraction wall.
b Two fractions with different denominators add up to one-half. What could they be? What else
could they be? Write down all the possibilities from the fractions in the fraction wall.
c Three fractions with the same denominator add up to one-half. What could they be? What else
could they be? Write down all the possibilities from the fractions in the fraction wall.
d Three fractions with different denominators add up to one-half. What could they be?
What else could they be? Write down all the possibilities from the fractions in the
fraction wall.
e Two fractions with different denominators add up to one-half. One of them is one-tenth. What is
the other one?
4J
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Number and Algebra 187
15 Are the following statements true or false? Explain your answers, using the fraction wall.
a 0.5 = 0.50% b 1
3
= 33% c 1
5
= 15%
d 9
10
= 90% e 1
8
 12% f 2
3
> 66%
16 Which one of the following price reductions represents the greatest percentage discount?
Explain your answer.
Before discount After discount
A $40 $28
B $90 $60
C $100 $69
D $80 $60
Enrichment: A frog named Willy Makeit
17 Willy Makeit is a very small frog. He is on flat ground, 1 metre from his pond. He needs to get
back to the pond but he gets very tired when he jumps. His first jump is half a metre. Every
jump he makes after that is half the distance of the previous jump.
Use the fraction wall (see page 182) to answer these questions.
a After Willy has made his first jump, how far is he from the pond?
b In metres, how long is his second jump?
c After Willy has made his second jump, how far is he from the pond?
d In metres, how long is his third jump?
e Will Willy make it to the pond?
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Chapter 4 Understanding fractions, decimals 188 and percentages
Decimal and percentage conversions
Percentages are commonly
used in today’s society.
They give an idea of
proportion. For example,
if a newspaper states that
2000 people want a council
swimming pool constructed,
then we know how many
want a pool but we don’t
know what proportion of the
community that is. However,
if there are 2500 people in this
community, the newspaper can state that 80% want a swimming pool. This informs us that a majority of
the community (i.e. 80 out of every 100 people) want a swimming pool constructed.
let’s start: Creative shading
• Draw a square of side length 10 cm and shade exactly 20% or
0.2 of this fi gure.
• Draw a square of side length 5 cm and shade exactly 60% or
0.6 of this fi gure.
• Draw another square of side length 10 cm and creatively shade
an exact percentage of the fi gure. Ask your partner to work out
the percentage you shaded.
4K
We see percentages used around us in many different situations.
What percentage is shaded?
■■ The symbol % means per cent. It comes from the Latin words per centum, which translate to
‘out of 100’.
For example: 23% means 23 out of 100 or
23
100
or 0.23.
■■ To convert a percentage to a decimal, divide by 100. This is done by moving the decimal point
2 places to the left.
For example: 42% = 42 ÷ 100 = 0.42
■■ To convert a decimal to a percentage, multiply by 100. This is done by moving the decimal point
2 places to the right.
For example: 0.654 = 0.654 × 100% = 65.4%
Key ideas
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number and algebra 189
Example 27 Converting percentages to decimals
Express the following percentages as decimals.
a 30% b 240% c 12.5% d 0.4%
Solution Explanation
a 30% = 0.3 30 ÷ 100
b 240% = 2.4 240 ÷ 100 = 240. = 2.4
c 12.5% = 0.125 Decimal point moves 2 places to the left.
d 0.4% = 0.004 Decimal point moves 2 places to the left.
Example 28 Converting decimals to percentages
Express the following decimals as percentages.
a 0.045 b 7.2
Solution Explanation
a 0.045 × 100% = 4.5% Decimal point moves 2 places to the right.
b 7.2 × 100% = 720% 7.2 × 100% = 720.%
1 What percentage of each square has been shaded?
a b
Exercise 4K
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Chapter 4 Understanding fractions, decimals 190 and percentages
c d
2 72.5% is equivalent to which of the following decimals?
a 72.5 B 7.25 C 0.725 D 725.0
3 1452% is equivalent to which of the following decimals?
a 0.1452 B 14.52 C 145 200 D 145.20
4 0.39 is equivalent to which of the following percentages?
a 39% B 3.9% C 0.39% D 0.0039%
5 Prue answered half the questions correctly for a test marked out of 100.
a What score did Prue get on the test?
b What percentage did Prue get on the test?
c Find the score you would expect Prue to get if the test was out of:
i 10 ii 200 iii 40 iv 2
d Find the percentage you would expect Prue to get if the test was out of:
i 10 ii 200 iii 40 iv 2
6 Fill in the empty boxes.
a 58 58 58 100
58
%= out of = = = 0 58
b 35 100 35
100
%= out of = ÷ = = .35
c 126 126 100 100
126
%= = ÷ = = 1.
7 Express the following percentages as decimals.
a 32% b 27% c 68% d 54%
e 6% f 9% g 100% h 1%
i 218% j 142% k 75% l 199%
Example 27a,b
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8 Express the following percentages as decimals.
a 22.5% b 17.5% c 33.33% d 8.25%
e 112.35% f 188.8% g 150% h 520%
i 0.79% j 0.025% k 1.04% l 0.95%
9 Express the following decimals as percentages.
a 0.8 b 0.3 c 0.45 d 0.71
e 0.416 f 0.375 g 2.5 h 2.314
i 0.025 j 0.0014 k 12.7 l 1.004
10 Place the following values in order from highest to lowest.
a 86%, 0.5%, 0.6, 0.125, 22%, 75%, 2%, 0.78
b 124%, 2.45, 1.99%, 0.02%, 1.8, 55%, 7.2, 50
11 An ice-cream store is offering a discount of 15% on orders over $25. A single-scoop ice-cream
in a cone is $5.50 and a double-scoop ice-cream in a cone is $7.25. A family of six purchases
three single-scoop ice-creams and three double-scoop ice-creams.
a Will the family receive the discount?
b What percentage will they pay?
12 Last Saturday, Phil spent 24 hours of the day in the following way: 0.42 of the time was spent
sleeping, 0.22 was spent playing sport and 0.11 was spent eating. The only other activity Phil
did for the day was watch TV.
a What percentage of the day did Phil spend watching TV?
b What percentage of the day did Phil spend either sitting down or lying down?
13 Sugarloaf Reservoir has a capacity of 96 gigalitres. However, as a result of the drought it is only
25% full. How many gigalitres of water are in the reservoir?
Example 27c,d
Example 28
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Chapter 4 Understanding fractions, decimals 192 and percentages
14 The average daily energy intake for adolescent boys is 11 500 kJ. The average serving size
of a bowl of Rice Bubbles with 1
2
cup of reduced-fat milk provides 770 kJ. What
percentage of a boy’s daily intake is a bowl of Rice Bubbles with milk? Round your
answer to 1 decimal place.
15 a, b, c and d are digits. Write the following decimal numbers as percentages.
a 0.abcd b a.ac c ab.dc
d 0.0dd e c.dba f 0.cccddd
16 a, b, c and d are digits. Write the following percentages as decimal numbers.
a a.b % b bcd % c ac %
d 0.da % e abbb % f dd.d %
17 Trudy says that it is impossible to have more than 100%. She supports her statement by saying
that if you get every question correct in a test, then you get 100% and you cannot get any more.
a Do you agree with Trudy’s statement?
b Provide four examples of when it makes sense that you cannot get more than 100%.
c Provide four examples of when it is perfectly logical to have more than 100%.
Enrichment: aFl ladder
18 The Australian Rules football ladder has the following column headings.
a Using a calculator, can you determine how the percentage column is calculated?
b What do you think the ‘F’ and the ‘A’ column stand for?
c In their next match, Essendon scores 123 points for their team and has 76 points scored
against them. What will be their new percentage?
d By how much do Hawthorn need to win their next game to have a percentage of 100?
e If Port Adelaide plays Hawthorn in the next round and the fi nal score is Port Adelaide 124
beats Hawthorn 71, will Port Adelaide’s percentage become higher than Hawthorn’s?
19 Create your own AFL-style ladder using a spreadsheet program. After entering the results, the
program should automatically update the points column and the percentage column.
When carrying out a sort on the data, ensure that your program will automatically change any
team’s position on the ladder, if necessary.
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number and algebra 193
Fraction and percentage conversions
We come across percentages in many everyday
situations. Interest rates, discounts, test results and
statistics are just some of the common ways in
which we deal with percentages. Percentages are
closely related to fractions. A percentage is another
way of writing a fraction with a denominator of
100. Therefore, 87% means that if something is
divided into 100 pieces you would have 87 of them.
let’s start: Student ranking
Five students completed fi ve different Mathematics tests. Each of the tests was out of a different number
of marks. The results are shown below. Your task is to rank the fi ve students in descending order,
according to their test result.
• Matthew scored 15 out of a possible 20 marks.
• Mengna scored 36 out of a possible 50 marks.
• Maria scored 33 out of a possible 40 marks.
• Marcus scored 7 out of a possible 10 marks.
• Melissa scored 64 out of a possible 80 marks.
Change these test results to equivalent scores out of 100, and state the percentage test score for
each student.
4L
A fraction can be interpreted as a percentage of the total.
■■ We can write percentages as fractions by changing the % sign to a denominator of 100
(meaning out of 100).
For example: 37% =
37
100
■■ We can convert fractions to percentages through our knowledge of equivalent fractions. This
idea works well if the denominator is 2, 4, 5, 10, 20, 25 or 50.
For example:
1
4
25
100
= = 25%
– Other denominators are more diffi cult.
■ ■ For example: = ×
×
= = = 3
8
3 125
8 125
375
1000
37.5
100
37.5%
■ ■ This may be done by calculator:
■ ■ 3 ÷ 8 × 100 = 37.5
■ ■ ■■ ■ ■ ■ So
3
8
= 37.5%
Key ideas
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Chapter 4 Understanding fractions, decimals 194 and percentages
Example 29 Converting percentages to fractions
Express these percentages as fractions or mixed numerals in their simplest form.
a 17% b 36% c 140%
Solution Explanation
a 17% =
17
100
Change % sign to a denominator of 100.
b 36
36
100
9 4
25 4
9
25
%=
= ×
×
=
Change % sign to a denominator of 100.
Cancel HCF.
Answer is now in simplest form.
c 140
140
100
7 20
5 20
7
5
1
2
5
%=
= ×
×
= =
Change % sign to a denominator of 100.
Cancel HCF.
Convert answer to a mixed numeral.
■■ Common percentages and their equivalent fractions are shown in the table below. It is useful to
know these.
Fraction
1
2
1
3
1
4
1
5
1
8
2
3
3
4
Percentage 50% 33
1
3
% 25% 20% 12
1
2
% 66
2
3
% 75%
Key ideas
Example 30 Converting to percentages through equivalent fractions
Convert the following fractions to percentages.
a 5
100
b 11
25
c 5
8
d 3
3
5
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number and algebra 195
Solution Explanation
a
5
100
= 5%
Denominator is already 100, therefore simply write
number as a percentage.
b =
=
11
25
44
100
44%
Require denominator to be 100.
Therefore, multiply numerator and denominator by 4
to get an equivalent fraction.
c 5
8
5 125
8 125
625
1000
62 5
100
62 5
= ×
×
=
=
=
.
. %
Convert the fraction to thousandths by multiplying the
denominator and numerator by 125.
Convert the fraction to hundredths by dividing the
denominator and numerator by 10.
Write the fraction as a percentage.
d 3
3
5
3
6
10
3
60
100
360
100
360
=
=
=
= %
Convert the fraction to tenths by multiplying the
denominator and numerator by 2.
Convert the fraction to hundredths by multiplying the
denominator and numerator by 10.
Write the mixed numeral as an improper fraction.
Write the improper fraction as a percentage.
1 Change these test results to equivalent scores out of 100, and therefore state the percentage.
a 7 out of 10 = _______ out of 100 = _____%
b 24 out of 50 = _______ out of 100 = _____%
c 12 out of 20 = _______ out of 100 = _____%
d 1 out of 5 = _______ out of 100 = _____%
e 80 out of 200 = _______ out of 100 = _____%
f 630 out of 1000 = _______ out of 100 = _____%
2 Write these fraction sequences into your workbook and write beside each fraction the equivalent
percentage value.
a
1
4
2
4
3
4
4
4
, , , b 1
5
2
5
3
5
4
5
5
5
, , , , c 1
3
2
3
3
3
, ,
3 a If 14% of students in Year 7 are absent due to illness, what percentage of Year 7 students are
at school?
b If 80% of the Geography project has been completed, what percentage still needs to be fi nished?
Exercise 4L
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Chapter 4 Understanding fractions, decimals 196 and percentages
4 Express these percentages as fractions in their simplest form.
a 11% b 71% c 43% d 49%
e 25% f 30% g 15% h 88%
5 Express these percentages as mixed numerals in their simplest form.
a 120% b 180% c 237% d 401%
e 175% f 110% g 316% h 840%
6 Convert these fractions to percentages, using equivalent fractions.
a
8
100
b 15
100
c 97
100
d 50
100
e
7
20
f 8
25
g 43
50
h 18
20
i
56
50
j 27
20
k 20
5
l 16
10
7 Convert these fractions to percentages. Check your answers using a calculator.
a
1
8
b 1
3
c 4
15
d 10
12
e 1
3
20
f 4
1
5
g 2
36
40
h 13
40
8 A bottle of lemonade is only 25% full.
a What fraction of the bottle has been consumed?
b What percentage of the bottle has been consumed?
c What fraction of the bottle is left?
d What percentage of the bottle is left?
9 A lemon tart is cut into eight equal pieces. What percentage of the tart does each piece
represent?
10 Petrina scores 28 out of 40 on her Fractions test.
What is her percentage score?
11 The nutrition label on a particular brand of sliced bread
states that the average serving size of two slices is equal
to 55 grams. It also states that there are 2.2 grams of
sugar per serve. What percentage of the bread is sugar?
12 The Sydney Kings basketball team have
won 14 out of 18 games. They still have two
games to play. What is the smallest and the
largest percentage of games the Kings could
win for the season?
Example 29a,b
Example 29c
Example 30a,b
Example 30c,d
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Number and Algebra 197
13 Lee won his tennis match with the score 6-4, 6-2, 6-1.
a What fraction of games did he win?
b What percentage of games did he win?
14 Scott and Penny have just taken out a home loan, with an interest rate of 5
1
2
%. Write this
interest rate as a fraction.
15 Write each of the following percentages as fractions.
a 2
1
2
% b 8
1
4
% c 12
1
2
% d 33
1
3
%
Enrichment: Lottery research
16 Conduct research on a major lottery competition. If possible:
a Find out, on average, how many tickets are sold each week.
b Find out, on average, how many tickets win a prize each week.
c Determine the percentage chance of winning a prize.
d Determine the percentage chance of winning the various divisions.
e Work out the average profit the lottery competition makes each week.
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Chapter 4 Understanding fractions, decimals 198 and percentages
percentage of a quantity
A common application of percentages is to
fi nd a certain percentage of a given number.
Throughout life you will come across many
examples where you need to calculate
percentages of a quantity. Examples
include retail discounts, interest rates,
personal improvements, salary increases,
commission rates and more.
In this section we will focus on the mental
calculation of percentages.
let’s start: Percentages in your head
It is a useful skill to be able to quickly calculate percentages mentally.
Calculating 10% or 1% is often a good starting point. You can then multiply or divide these values to
arrive at other percentage values.
• In pairs, using mental arithmetic only, calculate these 12 percentages.
a 10% of $120 b 10% of $35 c 20% of $160 d 20% of $90
e 30% of $300 f 30% of $40 g 5% of $80 h 5% of $420
i 2% of $1400 j 2% of $550 k 12% of $200 l 15% of $60
• Check your answers with a classmate or your teacher.
• Design a quick set of 12 questions for a classmate.
• Discuss helpful mental arithmetic skills to increase your speed at calculating percentages.
4M
Retail sales may involve a percentage of the original price being
taken away.
■■ To fi nd the percentage of a number:
1 Express the required percentage as a fraction.
2 Change the ‘of’ to a multiplication sign.
3 Look for a written or mental strategy to complete
the multiplication.
Using a calculator, 25% of 60 = 25 ÷ 100 × 60.
Key ideas
25% of 60 =
1
4
of 60
= 60 ÷ 4
= 60 ÷ 2 ÷ 2
= 30 ÷ 2
= 15
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Number and Algebra 199
Example 31 Finding the percentage of a number
Find:
a 30% of 50 b 15% of 400
Solution Explanation
a 30 50
30
100
50
1
30
2
15
% of = ×
= =
Mental arithmetic:
10% of 50 = 1
10
of 50
= 50 ÷ 10
= 5
Hence, 30% of 50 = 15.
Write % as a fraction.
Cancel and simplify.
10% is the same as 1
10
.
Multiply by 3.
b 15 400
15
100
400
1
15 4
1
60
% of = ×
= × =
Mental arithmetic:
10% of 400 = 40, ∴ 5% of 400 = 20
Hence, 15% of 400 = 40 + 20 = 60.
Write % as a fraction.
Cancel and simplify.
5% is half of 10%.
10% + 5% gives 15%.
Example 32 Solving a worded percentage problem
Jacqueline has saved up $50 to purchase a new pair of jeans. She tries on many different pairs but only likes
two styles, Evie and Next. The Evie jeans are normally $70 and are on sale with a 25% discount. The Next
jeans retail for $80 and have a 40% discount for the next 24 hours. Can Jacqueline afford either pair of jeans?
Solution Explanation
Evie jeans
Discount = 25% of $70
= 70 ÷ 4
= 70 ÷ 2 ÷ 2
= $17.50
Sale price = $70 – $17.50
= $52.50
To calculate the discount on the Evie jeans, find 25%
of $70.
Find the sale price by subtracting the discount.
Next jeans
Discount = 40% of $80
= $8 × 4
= $32
Sale price = $80 – $32
= $48
Jacqueline can afford the Next jeans.
Calculate the discount on the Next jeans.
10% of $80 = $8
∴ 40% of 80% = $32
Find the sale price by subtracting the discount.
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Chapter 4 Understanding fractions, decimals 200 and percentages
1 Copy and complete the following sentences.
a Finding 10% of a quantity is the same as dividing the quantity by ______.
b Finding 1% of a quantity is the same as dividing the quantity by ______.
c Finding 50% of a quantity is the same as dividing the quantity by ______.
d Finding 100% of a quantity is the same as dividing the quantity by ______.
e Finding 20% of a quantity is the same as dividing the quantity by ______.
f Finding 25% of a quantity is the same as dividing the quantity by ______.
2 Without calculating the exact values, determine which alternative (i or ii) has the highest value.
a i 20% of $400 ii 25% of $500
b i 15% of $3335 ii 20% of $4345
c i 3% of $10 000 ii 2% of $900
d i 88% of $45 ii 87% of $35
3 Find the following percentages, using a mental strategy. Check your answers with a calculator.
a 50% of 140 b 10% of 360 c 20% of 50 d 30% of 90
e 25% of 40 f 25% of 28 g 75% of 200 h 80% of 250
i 5% of 80 j 4% of 1200 k 5% of 880 l 2% of 9500
m 11% of 200 n 21% of 400 o 12% of 300 p 9% of 700
4 Find:
a 120% of 80 b 150% of 400 c 110% of 60 d 400% of 25
e 125% of 12 f 225% of 32 g 146% of 50 h 3000% of 20
5 Without using a calculator, match the questions with their correct answer.
Question Answer
10% of $200 $8
20% of $120 $16
10% of $80 $20
50% of $60 $24
20% of $200 $25
5% of $500 $30
30% of $310 $40
10% of $160 $44
1% of $6000 $60
50% of $88 $93
6 Without using a calculator, fi nd:
a 30% of $140 b 10% of 240 millimetres c 15% of 60 kilograms
d 2% of 4500 tonnes e 20% of 40 minutes f 80% of 500 centimetres
g 5% of 30 grams h 25% of 12 hectares i 120% of 120 seconds
Example 31
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Number and Algebra 201
7 Harry scored 70% on his Percentages test. If the test is out of 50 marks, how many marks did
Harry score?
8 Grace wants to purchase a new top and has $40 to spend. She really likes a red top that was
originally priced at $75 and has a 40% discount ticket on it. At another shop, she also likes a
striped hoody, which costs $55. There is 20% off all items in the store on this day. Can Grace
afford either of the tops?
9 In a student survey, 80% of students said they received too much homework. If 300 students
were surveyed, how many students claimed that they get too much homework?
10 25% of teenagers say their favourite fruit is watermelon. In a survey of 48 teenagers, how many
students would you expect to write watermelon as their favourite fruit?
11 At Gladesbrook College, 10% of students walk to school, 35% of students catch public transport
and the remainder of students are driven to school. If there are 1200 students at the school, find
how many students:
a walk to school
b catch public transport
c are driven to school
12 Anthea has just received a 4% salary increase. Her wage before the increase was $2000 per week.
a How much extra money does Anthea receive due to her salary rise?
b What is Anthea’s new salary per week?
c How much extra money does Anthea receive per year?
13 Sam has 2 hours of ‘free time’ before dinner is ready. He spends 25% of that time playing
computer games, 20% playing his drums, 40% playing outside and 10% reading a book.
a How long does Sam spend
doing each of the four
different activities?
b What percentage of time
does Sam have remaining
at the end of his four
activities?
c Sam must set the table
for dinner, which takes
5 minutes. Does he still
have time to get this done?
Example 32
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Chapter 4 Understanding fractions, decimals 202 and percentages
14 Gavin mows 60% of the lawn in 48 minutes. How long will it take him to mow the entire
lawn if he mows at a constant rate?
15 Find:
a 20% of (50% of 200) b 10% of (30% of 3000)
c 5% of (5% of 8000) d 80% of (20% of 400)
16 Write a survey question for the students in your class, such as ‘What is your favourite colour?’
or ‘How many days last week did you catch a bus to school?’. Provide fi ve different answer
options. Survey your class and calculate the percentage of students who chose each option. Use
a sector graph (i.e. a pie chart) to display your fi ndings.
17 Which is larger: 60% of 80 or 80% of 60?
18 Tom does the following calculation: 120 ÷ 4 ÷ 2 × 3. What percentage of 120 does he fi nd?
Enrichment: Waning interest
19 When someone loses interest or motivation in a task, they can be described as having a ‘waning
interest’. Jill and Louise are enthusiastic puzzle makers, but they gradually lose interest when
tackling very large puzzles.
a Jill is attempting to complete a 5000-piece jigsaw puzzle in 5 weeks. Her interest drops off,
completing 100 fewer pieces each week.
i How many pieces must Jill complete in the fi rst week to ensure that she fi nishes the
puzzle in the 5-week period?
ii What percentage of the puzzle does Jill complete during each of the 5 weeks?
iii What is the percentage that Jill’s interest wanes each week?
b Louise is attempting to complete an 8000-piece jigsaw puzzle in 5 weeks. Her interest wanes
at a constant rate of 5% per week.
i What percentage of the puzzle must Louise complete in the fi rst week to ensure she
fi nishes the puzzle in the 5-week period?
ii Record how many pieces
of the puzzle Louise
completes each week and the
corresponding percentage of
the puzzle.
iii Produce a table showing the
cumulative number of pieces
completed and the cumulative
percentage of the puzzle
completed over the 5-week
period.
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number and algebra 203
using fractions and percentages to
compare two quantities
Sometimes we want to know the proportion of a certain
quantity compared to a given total or another quantity.
This may be done using a fraction, percentage or ratio.
The Earth’s surface, for example, is about 70% ocean.
So, the proportion of land could be written as 30% (as
a percentage) or 3
10
(as a fraction). The ratio of land to
ocean could be described as 30 parts of land to 70 parts
of ocean. Alternatively, the ratio could be expressed as 3
parts of land to 7 parts of ocean.
let’s start: Tadpole proportion
Scientists Hugh and Jack take separate samples of
tadpoles, which include green and brown tadpoles, from
their local water channels. Hugh’s sample contains 3 green tadpoles and 15 brown tadpoles, whereas
Jack’s sample contains 27 green tadpoles and 108 brown tadpoles.
• Find the proportion of green tadpoles in each of Hugh and Jack’s samples.
• Use both fractions and percentages to compare the proportions.
• Which sample might be used to convince the local council that there are too many brown tadpoles in
the water channels?
■■ To express one quantity as a fraction of another:
4N
■ Fraction =
amount
total
■■ To express one quantity as a percentage of another, fi nd an equivalent fraction in which the
denominator is 100.
This idea works well when the denominator is 2, 4, 5, 10, 20, 25 or 50.
■■ A ratio compares parts of a total.
■ ■ ■ ■ ■ ■ ■ ■ Red fraction = = =
2
5
4
10
40
100
■ ■ ■ ■ ■ ■ ■ ■
Red percentage =
40
100
= 40%
■ ■ ■ ■ ■ ■ ■ ■ Ratio = 2 parts red to 3 parts yellow
– This can also be done using a calculator.
2 ÷ 5 × 100 = 40
So 2
5
= 40%
Key ideas
The proportion of land to sea in this photo of the
Whitsunday Islands, Queensland, could be expressed
as a fraction, percentage or ratio.
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Chapter 4 Understanding fractions, decimals 204 and percentages
Example 33 Expressing as a proportion
Express the following as both a fraction and percentage of the total.
a $40 out of a total of $200 b 24 green ducks out of a total of 30 ducks
Solution Explanation
a Fraction
40
200
4
20
1
5
= = = Write the given amount over the total. Then simplify
the fraction.
Percentage = = =
40
200
20
100
20% Convert to hundredths.
b Fraction = =
24
30
4
5
There is a total of 24 brown ducks out of a total of 30.
Simplify the fraction.
Percentage = = =
4
5
8
10
80
100
= 80%
Convert to hundredths.
Example 34 Using ratios
A glass of cordial is 1 part syrup to 9 parts water.
a Express the amount of syrup as a fraction of the total.
b Express the amount of water as a percentage of the total.
Solution Explanation
a Fraction =
1
10
There is a total of 10 parts, including 1 part syrup.
b Percentage = = =
9
10
90
100
90% There is a total 9 parts water in a total of 10 parts.
9
10
90
100
=
1 2 3 4 5 6
water 7
8
9
10
syrup
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number and algebra 205
Note: The numbers in this exercise have been chosen carefully so that mental strategies may be
used, rather than a calculator.
1 This square shows some coloured triangles and some white triangles.
a How many triangles are coloured?
b How many triangles are white?
c What fraction of the total is coloured?
d What percentage of the total is coloured?
e What fraction of the total is white?
f What percentage of the total is white?
2 A farmer’s pen has two black sheep and eight white sheep.
a How many sheep are there in total?
b What fraction of the sheep are black?
c What fraction of the sheep are white?
d What percentage of the sheep are black?
e What percentage of the sheep are white?
3 Express the following as both a fraction and a percentage of the total. Check your answers
using a calculator.
a 30 out of a total of 100 b 3 out of a total of 5
c $10 out of a total of $50 d $60 out of a total of $80
e 2 kg out of a total of 40 kg f 14 g out of a total of 28 g
g 3 L out of a total of 12 L h 30 mL out of a total of 200 mL
4 Write each coloured area as both a fraction and percentage of the total area.
a b c
d e f
5 A jug of lemonade is made up of 2 parts of lemon juice to 18 parts of water.
a Express the amount of lemon juice as a fraction of the total.
b Express the amount of lemon juice as a percentage of the total.
6 A mix of concrete is made up of 1 part of cement to 4 parts of sand.
a Express the amount of cement as a fraction of the total.
b Express the amount of cement as a percentage of the total.
c Express the amount of sand as a fraction of the total.
d Express the amount of sand as a percentage of the total.
Example 33
Example 34
Exercise 4N
WORK ING
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Chapter 4 Understanding fractions, decimals 206 and percentages
7 A pair of socks is made up of 3 parts of wool to 1 part of nylon.
a Express the amount of wool as a fraction of the total.
b Express the amount of wool as a percentage of the total.
c Express the amount of nylon as a fraction of the total.
d Express the amount of nylon as a percentage of the total.
8 Gillian pays $80 tax out of her income of $1600. What percentage of her income does
she keep?
9 Over summer, a dam’s water volume reduces from 20 megalitres to 4 megalitres. What fraction
of the water in the dam has been lost?
10 Express the following as a fraction and percentage of the total.
a 20 cents of $5
b 14 days out of 5 weeks
c 15 centimetres removed from a total length of 3 metres
d 3 seconds taken from a world record time of 5 minutes
e 180 grams of a total of 9 kilograms
f 1500 centimetres from a total of 0.6 kilometres
11 Of 20 students, 10 play sport and 12 play a musical instrument,
with some of these students playing both sport and music. Two
students do not play any sport or musical instrument.
a What fraction of the students play both sport and a musical
instrument?
b What percentage of the students play a musical instrument but
not a sport?
12 An orchard of 80 apple trees is tested for diseases. 20 of the trees have blight disease, 16 have
brown rot disease and some trees have both. A total of 48 trees have neither blight nor
brown rot.
a What percentage of the trees has both diseases?
b What fraction of the trees has blight but does not have brown rot?
13 For a recent class test, Ross scored 45 out of 60 and Maleisha scored 72 out of 100. Use
percentages to show that Ross obtained the higher mark.
14 The prices of two cars are reduced for sale. A hatch priced at $20 000 is now reduced by $3000
and a 4WD priced at $80 000 is now reduced by $12 800. Determine which car has the largest
percentage reduction, giving reasons.
15 A yellow sports drink has 50 grams of sugar dissolved in fl uid and weighs a total of 250 grams.
A blue sports drink has 57 grams of sugar dissolved in fl uid and weighs a total of 300 grams.
Which sports drink has the least percentage of sugar? Give reasons.
? ? ?
?
music sport
WORKING
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WORKING
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WORKING
MATHEMATICALLY
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4N
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Number and Algebra 207
16 A room contains a girls and b boys.
a Write an expression using the pronumerals a and b for the fraction of:
i boys in the room ii girls in the room
b Write an expression using the pronumerals a and b for the percentage of:
i boys in the room ii girls in the room
17 A mixture of dough has a parts
of flour to b parts of water.
a Write an expression for the
fraction of flour.
b Write an expression for the
percentage of water.
Enrichment: Transport turmoil
18 A class survey of 30 students reveals that the students use three modes of transport to get to
school: bike, public transport and car. All of the students used at least one of these three modes
of transport in the past week.
Twelve students used a car to get to school and did not use any of the other modes of
transport. One student used all three modes of transport and one student used only a bike for
the week. There were no students who used both a bike and a car but no public transport. Five
students used both a car and public transport but not a bike. Eight students used only public
transport.
Use this diagram to help answer the following.
a How many students used both a bike and public
transport but not a car?
b What fraction of the students used all three modes of
transport?
c What fraction of the students used at least one mode
of transport, including a bike?
d What fraction of the students used at least one mode
of transport, including public transport?
e What percentage of students used public transport and a car during the week?
f What percentage of students used either public transport or a car or both during the week?
bike
public
transport
car
WORKING
MATHEMATICALLY
U F
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Chapter 4 Understanding fractions, decimals 208 and percentages
Egyptian fractions
The fractions in the ancient Egyptian Eye of
Horus were used for dividing up food and
land, as well as portions of medicine. They
are called unitary fractions because all the
numerators are 1.
Clearly, the ancient Egyptians had no
calculators or precise measuring
instruments; nevertheless, by repeatedly
dividing a quantity in half, the fractions
1
2
1
4
1
8
1
16
1
32
, , , or were combined to estimate any other fraction.
Imagine that you are an ancient Egyptian baker and wish to share your last three loaves of bread
equally between four people.
First, you cut two loaves in half and give
half a loaf to each of your four customers.
You have one loaf remaining and you
can cut that into quarters (i.e. half and then
half again).
So each of your four customers now receives half a loaf and one-quarter of a loaf, which is
3
4
( )
of a loaf.
1/8
1/16 1/4 1/2
1/32
1/64
investigation
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Number and Algebra 209
Using ancient Egyptian fractions, how could three loaves be divided equally between five people?
First, cut the loaves in half and give each customer
1
2
( ) a loaf. The remaining half loaf can be cut
into eight parts and each person is given
1
8
of
1
2
1
16
= th ( ) of a loaf. There is a small portion left
3
1
16
portions of  
 
, so these portions can be divided in half and each customer given
1
2
of
1
16
1
32
=
( ) of a loaf.
12
12
loaf 1
customer 1 customer 2
12
12
loaf 2
customer 3 customer 4
1
16
1
32
1
32 left over
loaf 3
shared
between
customers
customer 5
Each customer has an equal share 1
2
1
16
1
32
+ + ( ) of the loaf and the baker will have
the small
1
32
( ) of a loaf
left over.
12
1
16
1
32
If each loaf is divided exactly into five parts, the three loaves would have 15 equal parts altogether
and each customer could have three parts of the 15;
3
15
1
5
= th
of the total or
3
5
th
of one loaf.
3
5
= 0.6 and
1
2
1
16
1
32
+ + = 0.59375  0.6
(≈ means approximately equal).
So even without calculators or sophisticated measuring instruments, the ancient Egyptian method
of repeated halving gives quite close approximations to the exact answers.
Task
Using diagrams, explain how the following portions can be divided equally using only the ancient
Egyptian unitary fractions of
1
2
1
4
1
8
1
16
1
32
, , , and .
a three loaves of bread shared between eight people
b one loaf of bread shared between five people
c two loaves of bread shared between three people
Include the Egyptian Eye of Horus symbols for each answer, and determine the difference between
the exact answer and the approximate answer found using the ancient Egyptian method.
w
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Chapter 4 Understanding fractions, decimals 210 and percentages
1 These cities are known as India’s Golden Triangle. To fi nd the names of these cities, complete
the puzzle.
Match each of the fractions in the middle row with the equivalent fraction in the bottom row.
Place the letter in the code below.
puzzles and challenges
2 7 9 4 5 3 1 6 3 3 12 2 10 3 5 8 11 6
1 2 3 4 5 6 7 8 9 10 11 12
4
24
28
35
100
120
5
7
21
36
1
2
22
77
2
3
4
2
5
81
90
25
3
43
9
U= 8
1
3
A = 5
6
H= 15
21
D= 4
5
G= 1
6
N= 4
7
9
I= 7
12
E = 2
7
P = 18
27
J= 9
10
R = 48
96
L = 22
5
2 At the end of each practice session, Coach Andy rewards his swim team by distributing 30 pieces
of chocolate according to effort. Each swimmer receives a different number of whole pieces of
chocolate. Suggest possible numbers (all different) of chocolate pieces for each swimmer attending
practise when the chocolate is shared between:
a four swimmers b fi ve swimmers
c six swimmers d seven swimmers
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Number and Algebra 211
3 Forming fractions
Make a set of cards that look like these shown below.
1 2 3 4 5 6 7 8 9
In the following questions, any two of these cards can be used to make a fraction.
One card is used as the numerator and the other is used as the denominator.
a What fraction with the least value can you make?
b What fraction with the greatest value can you make?
c How many fractions can be simplified to give whole numbers?
d What is the fraction you can make with these cards that is closest to 1 but less than 1?
e What fractions can you make that are equal to 0.5?
f What fractions can you make that are equal to 75%?
g How many fractions can you make that are greater than 0.5 but less than 1?
h Use four different cards to make two fractions that add together to give 1. In how many ways
can this be done?
i Use six different cards to make three fractions that add together to give 1. In how many ways
can this be done?
j How many fractions can you make that are greater than 1 but less than 1.5?
4 When a $50 item is increased by 20%, the final price is $60. Yet, when a $60 item is reduced by
20% the final price is not $50. Explain.
5 The length and width of a rectangular projector screen in a small theatre is 200% more than the
length and width of a television screen in the same room. How much bigger is the area of the
screen than that of the television? Give your answer as a percentage.
6 Find the missing number.
= +
+
7
4
1
1
1
1
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Chapter 4 Understanding fractions, decimals 212 and percentages
Chapter summary
Fractions
Decimals
numerator
8 is the lowest common denominator
(LCD), which is the lowest common
multiple (LCM) of 4 and 8.
or
Simplify
Comparing fractions
<
?
Recall
2 parts selected denominator
5 parts in the whole
proper fraction
improper fraction
mixed numeral 5
or
Equivalent fractions
simplest form
= = = = 1
2
50
100
30
60
21
42
8
16
100% = 100
100 = 1
1
100 1% =
25% = =
20% = =
10% = 10 =
100
20
100
25
100
1
4
1
5
1
10
50% = 50 =
100
1
2
75% = 75 =
100
3
4
80% = 80 =
100
4
5
7 = 7 ÷ 7 = 1
7
42 = =
63
21 × 2
21 × 3
2
3
42
63
3 × 2
3 × 3
7 × 6
7 × 9 = = = = 2
3
6
9
2
5
3
4
10
7
3
4
5
8
5
8
6
8
3
4
= 5 × 2
5 × 3
40 = 2 =
15
40 =
15
10
15 2 2 2
3
= = 2 2
3
8
3
mixed improper
= 10 +
5
13
5 3 =
5 2 3
5
improper mixed
5 × 8
5 × 3
HCF of 42 and
63 is 21.
Changing
decimals to fractions
0.16 = 4
25 16 =
100
2.008 = 2 2 1
125 8 =
1000
Changing fractions
to decimals
4 = 0.4
10 2 =
5
35 = 0.35
100 7 =
20
Comparing decimals
12.3 > 12.1
6.72 < 6.78
0.15  0.105
284.7  284.7
0.184 = 8
100 + 4
1000 1 +
10
Decimals as fractions
= 184
1000
Decimals and percentages
0.63 = 63 ÷ 100 = 63%
8% = 8 ÷ 100 = 0.08
240% = 240 ÷ 100 = 2.4
Place value of digits
1 tenth
8 hundredths
4 thousandths
0.184
The critical digit is circled.
Rounding
2.34 | 2
2.34 | 5
2.34 | 6
2.89 | 9
2.99 | 7
2.34
2.35
2.35
2.90
3.00
Fractions to decimals
13 = 0.13
100
3 = 0.375
8
= 0.2222...
= 0.2
2
9
0.375
8) 3.000
0.222...
9) 2.000...
 means is approximately equal to
Facts about fractions, decimals
and percentages
= 0.5 = 50% 





1
2
1 = 0.3 = 33 %
3
1
3
1 = 0.25 = 25%
4
1 = 0.2 = 20%
5
1 = 0.1 = 10%
10
1 = 0.01 = 1%
100
3 = 1.5 = 150%
2
2 = 0.6 = 66 %
3
2
3
3 = 0.75 = 75%
4
2 = 0.4 = 40%
5
3 = 0.3 = 30%
10
12 = 0.12 = 12%
100
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Number and Algebra 213
Multiple-choice questions
1 Which set of fractions corresponds to each of the different shapes positioned on the number line?
0 1 2
A 3
8
, , ,
6
8
1
3
8
12
8
B
3
8
, , ,
3
4
1
1
4
12
8
C
1
2
, , ,
3
4
9
8
1
5
8
D
2
8
, , ,
3
4
1
3
8
1
1
2
E
3
8
, , ,
3
4
1
1
2
14
8
2 Which of the following statements is not true?
A 3
4
9
12
= B
6
11
18
33
= C
3
10
15
40
=
D
13
14
39
42
= E
2
7
16
56
=
3 Which set of mixed numbers corresponds to the letters written on the number line?
1 2
A B C D
0 3
A 1 1
3
5
2
2
5
3
1
5
1
5
, , , B 1 1
3
5
2
3
5
3
1
5
2
5
, , ,
C 1 1
2
5
2
2
5
3
2
5
1
5
, , , D 1 1
4
5
2
2
5
3
2
5
2
5
, , ,
E 1 1
3
5
2
3
5
3
1
5
1
5
, , ,
4 Which is the lowest common denominator for this set of fractions: 7
12
, ,
11
15
13
18
?
A 60 B 120 C 180 D 3240 E 90
5 Which of the following fraction groups is in correct descending order?
A 1
5
, ,
1
3
2
2
B 3
4
, , ,
3
5
3
8
3
7
C
5
8
, , ,
4
5
3
8
2
3
D
1
10
, , ,
1
20
1
50
1
100
E 2 2
8
15
2
2
3
2
3
4
1
5
, , ,
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Chapter 4 Understanding fractions, decimals 214 and percentages
6 Which statement is incorrect?
a + = 1
2
1
2
1 B + = 1
2
1
2
2
2
C + = 1
2
1
2
2
4
D + = 1
2
1
2
1.0 E + = 1
2
1
2
4
4
7 Three friends share a pizza. Kate eats
1
4
of the pizza, Archie eats 1
3
of the pizza and Luke eats the rest.
What fraction of the pizza does Luke eat?
a
4
12
B
2
3
C
14
15
D
7
15
E
5
12
8 Which list is in correct ascending order?
a 0 68
3
4
0 76 77
13
40
. , , . , %, B
7
8
, 82%, 0.87,
12
15
, 88%
C 21%, 0.02, 0.2, 0.22,
22
10
D 14
40
0 3666 0 36 37
93
250
, . , . , %,
i
E 0.76, 72%,
3
4
, 0.68,
13
40
9 60
14
can be written as:
a 4
2
7
B 2
4
7
C 4
2
14
D 7
4
7
E 5
1
7
10
17
25
of a metre of material is needed for a school project. How many centimetres is this?
a 65 cm B 70 cm C 68 cm D 60 cm E 75 cm
Short-answer questions
1 List the shaded fractions in correct ascending order.
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Number and Algebra 215
2 Write four fractions equivalent to
3
5
and write a sentence to explain why they are equal in value.
3 Write the following fractions in simplest form.
a
18
30
b
8
28
c
35
49
4 Convert each of the following to a mixed number in simplest form.
a
15
10
b
63
36
c
45
27
d
56
16
5 Place the correct mathematical symbol (i.e. <, = or >) in between the following pairs of fractions
to make true mathematical statements.
a
2
7
4
7
b
3
8
1
8
c 1 1
3
5
2
3
d 3
29
9
1
9
6 State the largest fraction in each list.
a
3
7
, , ,
2
7
5
7
1
7
b
3
8
, , ,
2
8
5
8
1
8
7 State the lowest common multiple for each pair of numbers.
a 2, 5 b 3, 7 c 8, 12
8 State the lowest common denominator for each set of fractions.
a
1
2
,
3
5
b 2
3
,
3
7
c 3
8
,
5
12
9 Rearrange each set of fractions in descending order.
a 1
9
5
2
1
5
3
5
, , b 14
8
, , ,
11
6
9
4
5
3
c 5
48
9
5
7
18
5
2
9
5
1
3
2
3
, , , ,
10 Determine the simplest answer for each of the following.
a
3
8
1
8
+ b
1
3
1
2
+ c + + 1
2
1
2
1
2
d + 1
3
1
6
e
7
8
3
8
− f + 1
10
3
10
11 Copy the table into your workbook and complete.
Percentage form 36% 140% 18%
Fraction 2
1
5
5
100
11
25
12 Determine which alternative (i or ii ) is the larger discount.
a i 25% of $200 ii 20% of $260
b i 5% of $1200 ii 3% of $1900
13 Express the following as both a fraction and percentage of the total.
a 6 out of 10 b $4 out of 20 c 50 cents out of $8 d 600 mL out of 2 L
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Chapter 4 Understanding fractions, decimals 216 and percentages
14 What is the place value of the digit 3 in the following numbers?
a 12.835 b 6.1237 c 13.5104
15 Write each fraction as a decimal and percentage.
a 81
10
b 81
100
c 801
100
d
801
1000
16 List all the possible numbers with 3 decimal places that, when rounded to 2 decimal places, result
in 45.27.
17 Write down the factors of these numbers.
a 24 b 32 c 36 d 64
e 100 f 144 g 72 h 75
18 Use your answers to Question 17 to find the highest common factor of:
a 24 and 32 b 32 and 36 c 144 and 72 d 75 and 100
19 Find the lowest common multiple of these number pairs.
a 4 and 6 b 2 and 5 c 3 and 6 d 3 and 2
e 5 and 12 f 4 and 2 g 3 and 5 h 2 and 6
i 4 and 8 j 4 and 10 k 5 and 10 l 8 and 10
Extended-response questions
1 Copy the table into your workbook and complete.
Simplified fraction Decimal Percentage
35%
3
5
1
1
10
2.75
750%
5
3
5
6
0.375
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Number and Algebra 217
2 Of the following, for which subject did Keira obtain the best test result?
English: 17 out of 25
Mathematics: 20 out of 30
Science: 15 out of 20
History: 7 out of 10
3 Five students, Penny, Philip, Jay, Tiger and Marie, share the winnings of a $100 fête prize. They
share the prize in the following way.
Penny 25%
Philip
1
5
Jay 10%
Tiger 0.15
Marie ?
a Write Penny’s share as a:
i fraction ii decimal
b Write Tiger’s share as a:
i percentage ii fraction
c How much money do these students receive?
i Philip ii Tiger
d Write Marie’s share as:
i a percentage ii a fraction iii a decimal iv an amount of money
e Philip and Marie combine their share.
i How much is this, in dollars?
ii Write this value as a fraction, a decimal and as a percentage.
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Chapter 218 5 Probability
Chapter Probability 5
What you will learn
Describing probability
Theoretical probability in single-step experiments
Experimental probability in single-step experiments
Compound events in single-step experiments
Venn diagrams and two-way tables
Probability in two-step experiments EXTENSION
5A
5B
5C
5D
5E
5F
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Gambling problem or problem
gambling?
Would you like to give away $4000 a year for no gain?
That is what the average gaming machine player
loses in NSW every year. This contributes to a total
gambling loss to Australians of about $20 billion each
year. The social cost of this is an extra $5 billion per
year as a result of people who become addicted to
gambling and become a fi nancial burden on their
families and the community.
Gambling activities include lotteries, online gaming,
gaming machines, sports betting and table games.
The people who invent and run these activities
calculate the mathematical probabilities so that, in the
long run, the players lose their money.
It is worth thinking about Probability before
becoming involved in gambling activities.
nSW Syllabus
for the australian
Curriculum
Strand: Statistics and Probability
Substrand: PRoBaBilitY
Outcome
A student represents probabilities of
simple and compound events.
(MA4–21SP)
219
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Chapter 220 5 Probability
1 Write these fractions in simplest form.
a 10
20
b 20
30
c 21
28
d 12
48
2 Consider the set of numbers 4, 2, 6, 5, 9.
a How many numbers are in the set?
b How many of the numbers are even?
c What fraction of the numbers are odd?
3 Write the following values as decimals.
a 2 ÷ 4 b 20 ÷ 50 c 12 ÷ 60 d 11 ÷ 55
4 Order these events from least likely to most likely.
a Rolling a die and it landing on the number 3.
B Flipping a coin and it landing with ‘tails’ showing.
C The Prime Minister of Australia being struck by lightning tomorrow.
D The internet being used by somebody in the next 20 minutes.
5 a List three events that have a low chance of occurring.
b List two events that have an even chance (i.e. 50-50) of occurring.
c List three events that have a high chance of occurring.
6 Copy this table into your workbook and complete.
Fraction Decimal Percentage
1
2
1
3
1
4
1
5
1
10
1
100
Pre-test
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Statistics and Probability 221
Describing probability
Often, there are times when you may wish to describe how likely it is that
an event will occur. For example, you may want to know how likely it is
that it will rain tomorrow, or how likely it is that your sporting team will
win this year’s premiership, or how likely it is that you will win a lottery.
Probability is the study of chance.
let’s start: Likely or unlikely?
Try to rank these events from least likely
to most likely. Compare your answers with
other students in the class and discuss any
differences.
• It will rain tomorrow.
• Australia will win the soccer World Cup.
• Tails landing uppermost when a 20-cent
coin is tossed.
• The Sun will rise tomorrow.
• The king of spades is at the top of a
shuffl ed deck of 52 playing cards.
• A diamond card is at the bottom of a
shuffl ed deck of 52 playing cards.
5A
The probability of winning fi rst prize in a lottery is close to zero.
This topic involves the use of sophisticated terminology.
terminology Example Defi nition
chance experiment rolling a fair 6-sided die A chance experiment is an activity that may
produce a variety of different results which occur
randomly. The example given is a single-step
experiment.
trials rolling a die 50 times When an experiment is performed one or more
times, each occurrence is called a trial. The
example given indicates 50 trials of a single-step
experiment.
Key ideas
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Chapter 222 5 Probability
outcome rolling a 5 An outcome is one of the possible results of a
chance experiment.
equally likely
outcomes
rolling a 5
rolling a 6
Equally likely outcomes are two or more results
that have the same chance of occurring.
sample space {1, 2, 3, 4, 5, 6} The sample space is the set of all possible
outcomes of an experiment. It is usually written
inside braces, as shown in the example.
event e.g. 1: rolling a 2
e.g. 2: rolling an even number
An event is either one outcome or a collection of
outcomes. It is a subset of the sample space.
compound event rolling an even number A compound event is a collection of two or more
outcomes from the sample space of a chance
experiment.
mutually exclusive
events
rolling a 5
rolling an even number
Two or more events are mutually exclusive if they
share no outcomes.
non-mutually
exclusive events
rolling a 5
rolling an odd number
Events are non-mutually exclusive if they share
one or more outcomes. In the given example, the
outcome 5 is shared.
complementary
events
rolling a 2 or 3
rolling a 1, 4, 5 or 6
If all the outcomes in the sample space are
divided into two events, they are complementary
events.
complement Rolling 2, 3, 4 or 5 is an event.
Rolling a 1 or 6 is the complement.
If an experiment was performed and an event
did not occur, then the complement definitely
occurred.
favourable
outcome(s)
In some games, you must roll a 6 before you
can start moving your pieces.
Outcomes are favourable if they are part of some
desired event.
theoretical
probability
or
likelihood
or
chance
The probability of rolling an even number is
written as:
Theoretical probability is the actual chance
or likelihood that an event will occur when an
experiment takes place.
Probabilities can be expressed as fractions,
decimals and percentages.
Probabilities range from 0 to 1 or 0% to 100%.
experimental
probability
A die is rolled 600 times and shows a 5 on
99 occasions. The experimental probability
of rolling a 5 on this die is:
P (5)
99
600
 = 0.165 = 16.5%
Sometimes it is difficult or impossible to calculate
a theoretical probability, so an estimate can
be found using a large number of trials. This is
called the experimental probability. If the number
of trials is large, the experimental probability
should be very close to that of the theoretical.
P (even)
3
6
1
2
= = = 0.5 = 50% P (event)
number of favourable outcomes
total number of outcomes
=
Key ideas
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Statistics and Probability 223
certain rolling a number less than 7 The probability is 100% or 1.
likely rolling a number less than 6
even chance rolling a 1, 2 or 3 The probability is 50% or 0.5 or 1
2
.
unlikely rolling a 2
impossible rolling a 7 The probability is 0% or 0.
the sum of all
probabilities in an
experiment
P P P
P P P
(1)
1
6
(2)
1
6
(3)
1
6
(4)
1
6
(5)
1
6
(6)
1
6
1
6
1
6
1
6
1
6
1
6
1
6
6
6
1 100%
= = =
= = =
+ + + + + = = =
The sum of the probabilities of all the outcomes
of a chance experiment is 1 (or 100%).
the sum of the
probabilities of
an event and its
complement
P
P
(rolling 1 or 6)
2
6
(rolling 2, 3, 4 or 5)
4
6
2
6
4
6
6
6
1 100%
=
=
+ = = =
The sum of the probabilities of an event and its
complement is 1 (or 100%).
P (event) + P (complementary event) = 1
Key ideas
Example 1 Describing chance
Classify each of the following statements as either true or false.
a It is likely that children will go to school next year.
b It is an even chance for a fair coin to display tails.
c Rolling a 3 on a 6-sided die and getting heads on a coin are equally likely.
d It is certain that two randomly chosen odd numbers will add to an even number.
Solution Explanation
a true Although there is perhaps a small chance that the laws might
change, it is (very) likely that children will go to school next year.
b true There is a 50-50, or an even chance, of a fair coin displaying tails.
It will happen, on average, half of the time.
c false These events are not equally likely. It is more likely to flip heads
on a coin than to roll a 3 on a 6-sided die.
d true No matter what odd numbers are chosen, they will always add to
an even number.
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Chapter 224 5 Probability
1 Match each of the events a to d with a description of how likely they are to occur (a to D).
a A tossed coin landing heads up. a unlikely
b Selecting an ace fi rst try from a fair deck of 52 playing cards. B likely
c Obtaining a number other than 6 if a fair 6-sided die is rolled. C impossible
d Obtaining a number greater than 8 if a fair 6-sided die is rolled. D even chance
2 Fill in the blanks, using the appropriate terminology.
a If an event is guaranteed to occur, we say it is __________.
b An event that is equally likely to occur or not occur has an __________ __________.
c A rare event is considered __________.
d An event that will never occur is called __________.
Exercise 5A
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3 Consider a fair 6-sided die with the numbers 1 to 6 on it. Answer true or false to each of
the following.
a Rolling a 3 is unlikely.
b Rolling a 5 is likely.
c Rolling a 4 and rolling a 5 are equally likely events.
d Rolling an even number is likely.
e There is an even chance of rolling an odd number.
f There is an even chance of rolling a multiple of 3.
4 Match up each of the events a to d with an equally likely event a to D.
a rolling a 2 on a 6-sided die
b selecting a heart card from a fair deck of
52 playing cards
c fl ipping a coin and tails landing face up
d rolling a 1 or a 5 on a 6-sided die
a selecting a black card from a fair deck
of 52 playing cards
B rolling a number bigger than 4 on a
6-sided die
C selecting a diamond card from a fair
deck of 52 playing cards
D rolling a 6 on a 6-sided die
Example 1
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Statistics and Probability 225
5 Consider the spinner shown, which is spun and could land with the arrow pointing to any
of the three colours. (If it lands on a boundary, it is re-spun until it lands on a colour.)
a State whether each of the following is true or false.
i There is an even chance that the spinner will point
to green.
ii It is likely that the spinner will point to red.
iii It is certain that the spinner will point to purple.
iv It is equally likely that the spinner will point to red
or blue.
v Green is twice as likely as blue.
b Use the spinner to give an example of:
i an impossible event
ii a likely event
iii a certain event
iv two events that are equally likely
blue
red
green
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6 Three spinners are shown below. Match each spinner with the description.
red blue
green
red
blue
green red
red blue
spinner 1 spinner 2 spinner 3
a Has an even chance of red, but blue is unlikely.
b Blue and green are equally likely, but red is unlikely.
c Has an even chance of blue, and green is impossible.
7 Explain why in Question 6 red is twice as likely to occur as blue in spinner 3 but equally likely to
occur in spinner 2 even though both spinners have equally-sized sectors.
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Chapter 226 5 Probability
8 Draw spinners to match each of the following descriptions, using blue, red and green as the
possible colours.
a Blue is likely, red is unlikely and green is impossible.
b Red is certain.
c Blue has an even chance, red and green are equally likely.
d Blue, red and green are all equally likely.
e Blue is twice as likely as red, but red and green are equally likely.
f Red and green are equally likely and blue is impossible.
g Blue, red and green are all unlikely, but no two colours are equally likely.
h Blue is three times as likely as green, but red is impossible.
9 For each of the following spinners, give a description of the chances involved so that someone could
determine which spinner is being described. Use the colour names and the language of chance (i.e.
‘likely’, ‘impossible’ etc.) in your descriptions.
a
green
red
blue
b
green red
red green
c
green
red
blue
d
blue
e
red
red
green
blue
blue
blue
f
red blue
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10 A coin consists of two sides that are equally likely to occur when tossed. It is matched up
with a spinner that has exactly the same chances, as shown below.
heads red blue
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5A
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Statistics and Probability 227
Tossing the coin with heads landing uppermost is equally likely to spinning red on the spinner.
Tossing the coin with tails landing uppermost is equally likely to spinning blue on the spinner.
Hence, we say that the coin and the spinner are equivalent.
a Draw a spinner that is equivalent to a fair 6-sided die. (Hint:
The spinner should have six sections of different colours.)
b How can you tell from the spinner you have drawn that it is
equivalent to a fair die?
c A die is ‘weighted’ so that that there is an even chance of
rolling a 6, but rolling the numbers 1 to 5 are still equally likely.
Draw a spinner that is equivalent to such a die.
d How could you make a die equivalent to the spinner shown in
the diagram?
e Describe a spinner that is equivalent to selecting a card from a fair deck of 52 playing cards.
1
2
4 3
6
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Enrichment: Spinner proportions
11 The language of chance is a bit vague. For example, for each of the following spinners it is ‘unlikely’
that you will spin red, but in each case the chance of spinning red is different.
red
blue
green
red
blue green
red
blue
spinner 1 spinner 2 spinner 3
Rather than describing this in words, we could give the fraction (or decimal or percentage) of the
spinner occupied by a colour.
a For each of the spinners above, give the fraction of the spinner occupied by red.
b What fraction of the spinner would be red if it has an even chance?
c Draw spinners for which the red portion occupies:
i 100% of the spinner ii 0% of the spinner
d For the sentences below, fill in the gaps with appropriate fraction or percentage values.
i An event has an even chance of occurring if that portion of the spinner occupies _________ of
the total area.
ii An event that is impossible occupies _________ of the total area.
iii An event is unlikely to occur if it occupies more than _________ but less than _________ of
the total area.
iv An event is likely if it occupies more than _________ of the total area.
e How can the fractions help determine if two events are equally likely?
f Explain why all the fractions occupied by a colour must be between 0 and 1.
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Chapter 228 5 Probability
theoretical probability in single-step experiments
The probability of an event occurring is a number between 0 and 1. This number states precisely how likely it
is for an event to occur. It is often written as a fraction and can indicate how frequently the event would occur
over a large number of trials. For example, if you toss a fair coin many times, you would expect heads to
come up half the time, so the probability is 1
2. If you roll a fair 6-sided die many times, you should roll a 4
about one-sixth of the time, so the probability is 16
.
To be more precise, we should list the possible outcomes of rolling the die: 1, 2, 3, 4, 5, 6. Doing this shows
us that there is a 1 out of 6 chance that you will roll a 4 and there is a 0 out of 6 (= 0) chance of rolling a 9.
let’s start: Spinner probabilities
Consider the three spinners shown below.
red blue
green
red
blue
green red
red blue
• What is the probability of spinning blue for each of these spinners?
• What is the probability of spinning red for each of these spinners?
• Try to design a spinner for which the probability of spinning green is
4
7
and the probability of spinning
blue is 0.
5B
■ Many key ideas relevant to this section can be found in the list of terminology that begins on
page 221.
■ Some examples of single-step experiments are:
– tossing a coin once – spinning a spinner once
– rolling a die once – choosing one prize in a raffl e
– choosing one card from a deck of playing cards
■ Theoretical probability is the actual chance or likelihood that an event will occur when an
experiment takes place.
P(event) =
number of favourable outcomes
total number of outcomes
For example: The chance of rolling a fair die once and getting a 2.
P(rolling a 2) =
1
6
■ Probabilities can be expressed as:
– fractions or decimals
between 0 and 1
– percentages between 0%
and 100% Key ideas
0
0%
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
50% 100%
impossible unlikely even likely
chance
(50-50)
certain
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Statistics and Probability 229
Example 2 Calculating probability
A fair 6-sided die is rolled.
a List the sample space.
b Find the probability of rolling a 3, giving your answer as a fraction.
c Find the probability of rolling an even number, giving your answer as a decimal.
d Find the probability of rolling a number less than 3, giving your answer as a percentage.
Solution ExPlanation
a Sample space = {1, 2, 3, 4, 5, 6} For the sample space, we list all the possible outcomes.
Technically, the sample space is {roll a 1, roll a 2, roll a 3,
roll a 4, roll a 5, roll a 6}, but we do not usually include the
additional words.
b P(3)
1
6
= The event can occur in one way (rolling a 3) out of six possible
outcomes.
c P(even) =
1
2
= 0.5 = 50%
The event can occur in three ways (i.e. 2, 4 or 6). So the
probability is 3
6
1
2 = or 0.5 or 50%.
d P( less than 3) = = 1
3
0.3
•
= 33 %
1
3
The event can occur in two ways (1 or 2). So the probability
is 2
6
1
3 = or 0 3 33
1
3
. %
•
or .
place.
1 Match up each event a to d with the set of possible outcomes a to D.
a tossing a coin
b rolling a die
c selecting a suit from a fair deck of
52 playing cards
d spinning the spinner shown at right
a {1, 2, 3, 4, 5, 6}
B {red, green, blue}
C {heads, tails}
D {hearts, diamonds, clubs, spades}
green
red
blue
Exercise 5B
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Chapter 230 5 Probability
2 Complete the following sentences.
a The _________ _________ is the set of possible outcomes.
b An impossible event has a probability of _________.
c If an event has a probability of 1, then it is _________.
d The higher its probability, the _________ likely the event will occur.
e An event with a probability of 1
2
has an _____ ____ of occurring.
3 Consider a fair 6-sided die.
a List the sample space.
b List the odd numbers on the die.
c State the probability of throwing an even number.
Example 2a
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4 Consider the spinner shown.
a How many outcomes are there? List them.
b Find P(red); i.e. fi nd the probability of the spinner pointing
to red.
c Find P(red or green).
d Find P(not red).
e Find P(yellow).
5 A spinner with the numbers 1 to 7 is spun. The numbers are
evenly spaced.
a List the sample space.
b Find P(6).
c Find P(8).
d Find P(2 or 4).
e Find P(even).
f Find P(odd).
g Give an example of an event having the probability of 1.
6 The letters in the word MATHS are written on fi ve cards and then one is drawn from a hat.
a List the sample space.
b Find P(T), giving your answer as a decimal.
c Find P(consonant is chosen), giving your answer as a decimal.
d Find the probability that the letter drawn is also in the word TAME, giving your answer
as a percentage.
Example 2b–d
green
red
blue
1
2
3
4
5
6
7
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5B
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Statistics and Probability 231
7 The letters in the word PROBABILITY are written on 11 cards and then one is
drawn from a hat.
a Find P(P).
b Find P(P or L).
c Find P(letter chosen is in the word BIT).
d Find P(not a B).
e Find P(a vowel is chosen).
f Give an example of an event with the probability of
3
11
.
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8 A bag of marbles contains 3 red marbles, 2 green marbles and 5 blue marbles. They are all
equal in size and weight. A marble is chosen at random.
a What is the probability that a red marble is chosen? (Hint: It is not
1
3
because the colours
are not all equally likely.) Give your answer as a percentage.
b What is the probability that a blue marble is chosen? Give your answer as a percentage.
c What is the probability that a green marble is not chosen? Give your answer as a percentage.
9 Consider the spinner opposite, numbered 2 to 9.
a List the sample space.
b A number is prime if it has exactly two factors. Therefore,
5 is a prime number but 6 is not. Find the probability that a
prime number will be spun, giving your answer as a decimal.
(Remember that 2 is a prime number.)
c Giving your answers as decimals, state the probability of
getting a prime number if each number in the spinner
opposite is:
i increased by 1 ii increased by 2
iii doubled
(Hint: It will help if you draw the new spinner.)
d Design a new spinner for which the P(prime) = 1.
10 A bag contains various coloured marbles – some are red, some are blue, some are yellow and
some are green. You are told that P(red) = 1
2
, P(blue) = 14
and P(yellow) = 1
6
. You are not told
the probability of selecting a green marble.
a If there are 24 marbles:
i Find how many marbles there are of each colour.
ii What is the probability of getting a green marble?
b If there are 36 marbles:
i Find how many marbles there are of each colour.
ii What is the probability of getting a green marble?
c What is the minimum number of marbles in the bag?
d Does the probability of getting a green marble depend
on the actual number of marbles in the bag? Justify
your answer.
2
3
4
6 5
7
8
9
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Chapter 232 5 Probability
11 a State the values of the letters in the following table.
Event P(event occurs) P(event does not occur)
Sum of two
numbers
rolling a die, get a 3
1
6
5
6
a
tossing a coin, get H
1
2
b c
rolling a die, get 2 or 5 d
2
3
e
selecting letter from
‘HEART’, get a vowel
f g h
b If the probability of selecting a vowel in a particular word is
3
13
, what is the probability of
selecting a consonant?
c If the probability of spinning blue with a particular spinner is 4
7
, what is the probability of
spinning a colour other than blue?
12 A box contains different coloured counters, with P(purple) = 10%, P(yellow) =
•
0.6 and
P(orange) = 1
7
.
a Is it possible to obtain a colour other than purple, yellow or orange? If so, state the probability.
b What is the minimum number of counters in the box?
c If the box cannot fi t more than 1000 counters, what is the maximum number of counters
in the box?
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Enrichment: Designing spinners
13 For each of the following, design a spinner using only red, green and blue sectors to obtain the desired
probabilities. If it cannot be done, then explain why.
a P(red)= P(green) , P(blue)
1
2
1
4
1
4
, = =
b P(red) = P(green) = , P(blue) = 1
2
1
2
1
2
,
c P(red) = P(green) = , P(blue) = 1
4
1
4
1
4
,
d P(red) = 0.1, P(green) = 0.6, P(blue) = 0.3
5B
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Statistics and Probability 233
Experimental probability in single-step experiments
Although the probability of an event tells us how often
an event should happen in theory, we will rarely fi nd
this being exactly right in practice. For instance, if you
toss a coin 100 times, it might come up heads
53 times out of 100, which is not exactly 1
2
of the
times you tossed it. Sometimes we will not be able to
fi nd the exact probability of an event, but we can carry
out an experiment to estimate it.
let’s start: Tossing coins
For this experiment, each class member needs a fair
coin that they can toss.
• Each student should toss the coin 20 times and
count how many times heads occurs.
• Tally the total number of heads obtained by the
class.
• How close is this total number to the number
you would expect that is based on the probability
of 1
2
? Discuss what this means.
5C
A fair coin tossed 100 times might not show heads 50 times,
but it is reasonable to expect approximately 50 heads.
■ The experimental probability of an event occurring based on a particular experiment is
defi ned as:
number of times the event occurs
total number of trials in the experiment
■ The expected number of occurrences = probability × number of trials.
■ If the number of trials is large, then the experimental probability should be close to the
theoretical probability of an event.
Key ideas
Example 3 Working with experimental probability
When playing with a spinner with the numbers 1 to 4 on it, the following numbers come up:
1, 4, 1, 3, 3, 1, 4, 3, 2, 3.
a What is the experimental probability of getting a 3?
b What is the experimental probability of getting an even number?
c Based on this experiment, how many times would you expect to get a 3 if you spin 1000 times?
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Chapter 234 5 Probability
Solution ExPlanation
a
2
5
or 0.4 or 40%
number of 3s
number of trials
= = 4
10
2
5
b
3
10
number of times with even result
number of trials
= 3
10
c 400 times probability × number trials = × = 2
5
1000 400
1 A 6-sided die is rolled 10 times and the following numbers come up: 2, 4, 6, 4, 5, 1, 6, 4, 4, 3.
a What is the experimental probability of getting a 3?
b What is the experimental probability of getting a 4?
c What is the experimental probability of getting an odd number?
d Is the statement ‘rolling an even number and rolling a 5 are complementary events’ true or false?
2 When a coin is tossed 100 times, the results are 53 heads and 47 tails.
a What is the experimental probability of getting a head?
b What is the experimental probability of getting a tail?
c What is the theoretical probability of getting a tail if the coin is fair?
d If ‘tossing a head’ is an event, what is the complementary event?
Example 3a,b
Exercise 5C
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3 A survey is conducted on people’s television viewing habits.
a How many people participated in the survey?
b What is the probability that a
randomly selected participant
watches less than 5 hours of
television?
c What is the probability that a randomly selected participant watches 20–30 hours of television?
d What is the probability that a randomly selected participant watches between 5 and 20 hours
of television?
e Based on this survey, the experimental probability of watching 30+ hours of television is 0.
Does this mean that watching 30+ hours is impossible?
4 A fair coin is tossed.
a How many times would you expect it to show tails in 1000 trials?
b How many times would you expect it to show heads in 3500 trials?
c Initially, you toss the coin 10 times to fi nd the probability of the coin showing tails.
i Explain how you could get an experimental probability of 0.7.
ii If you toss the coin 100 times, are you more or less likely to get an experimental
probability close to 0.5?
number of hours per week 0–5 5–10 10–20 20–30 30+
number of people 20 10 15 5 0
Example 3c
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Statistics and Probability 235
5 A fair 6-sided die is rolled.
a How many times would you expect to get a 3 in 600 trials?
b How many times would you expect to get an even number in 600 trials?
c If you roll the die 600 times, is it possible that you will get an even number 400 times?
d Are you more likely to obtain an experimental probability of 100% from two throws or to
obtain an experimental probability of 100% from 10 throws?
6 Each time a basketball player takes a free throw there is a 4 in 6 chance that the shot will go in.
This can be simulated by rolling a 6-sided die and using numbers 1 to 4 to represent ‘shot goes in’
and numbers 5 and 6 to represent ‘shot misses’.
a Use a 6-sided die over 10 trials to find the experimental probability that the shot goes in.
b Use a 6-sided die over 50 trials to find the experimental probability that the shot goes in.
c Working with a group, use a 6-sided die over 100 trials to find the experimental probability
that the shot goes in.
d Use a 6-sided die over just one trial to find the experimental probability that the shot goes in.
(Your answer should be either 0 or 1.)
e Which of the answers to parts a to d above is closest to the theoretical probability of 66.67%?
Justify your answer.
f Is this statement true or false? ‘Shot goes in’ and ‘shot misses’ are complementary events.
7 The colour of the cars in a school car park is recorded.
Colour red black white blue purple green
Number of cars 21 24 25 20 3 7
Based on this sample:
a What is the probability that a randomly chosen car is white?
b What is the probability that a randomly chosen car is purple?
c What is the probability that a randomly chosen car is green or black?
d How many purple cars would you expect to see in a shopping centre car park with 2000 cars?
e If ‘red or black’ is an event, what is the complementary event?
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Chapter 236 5 Probability
8 The number of children in some families is recorded in the table shown.
a How many families have no children?
b How many families have an even number
of children?
c How many families participated in the survey?
d Based on this experiment, what is the probability that a
randomly selected family has 1 or 2 children?
e Based on this experiment, what is the probability that a randomly selected family has an even
number of children?
f What is the total number of children considered in this survey?
g If ‘no children’ is an event, what is the complementary event?
number of children 0 1 2 3 4
number of families 5 20 32 10 3
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9 A handful of 10 marbles of different colours is placed into a bag. A marble is selected at
random, its colour recorded and then returned to the bag. The results are presented in
the table.
a Based on this experiment, how many marbles of each
colour do you think there are? Justify your answer in
a sentence.
b For each of the following, state whether or not they
are possible outcomes for the 10 marbles.
i 3 red, 3 green, 4 blue ii 2 red, 4 green, 4 blue
iii 1 red, 3 green, 6 blue iv 2 red, 3 green, 4 blue, 1 purple
v 2 red, 0 green, 8 blue
10 Match each of the experiment results a to d with the most likely
spinner that was used (a to D).
Red marble
chosen
Green marble
chosen
Blue marble
chosen
21 32 47
red green blue
a 18 52 30
b 27 23 0
c 20 23 27
d 47 0 53
a
blue
red
B
red green
blue
C
red
green
blue
D
green
red
red green
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5C
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Statistics and Probability 237
11 Assume that any baby has a 50% chance of being a boy or a girl, and use a coin to simulate a
family with four children. Toss the coin four times, using heads to represent boys and tails to
represent girls. Count the number of girls in the family. Repeat this experiment 20 times and
present your results in a table like the one below.
Number of girls 0 1 2 3 4 Total
N umber of families 20
a Based on your simulation, what is the experimental probability that a family will have just
one girl?
b Based on your simulation, what is the experimental probability that a family will have four girls?
c Explain why you might need to use simulations and experimental probabilities to find the answers
to parts a and b above.
d If you had repeated the experiment only 5 times instead of 20 times, how might the accuracy of
your probabilities be affected?
e If you had repeated the experiment 500 times instead of 20 times, how might the accuracy of your
probabilities be affected?
12 Classify the following statements as true or false. Justify each answer in a sentence.
a If the probability of an event is 1
2
, then it must have an experimental probability of 1
2
.
b If the experimental probability of an event is 1
2
, then it must have a theoretical probability of 1
2
.
c If the experimental probability of an event is 0, then the theoretical probability is 0.
d If the probability of an event is 0, then the experimental probability is also 0.
e If the experimental probability is 1, then the theoretical probability is 1.
f If the probability of an event is 1, then the experimental probability is 1.
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Enrichment: Improving estimates
13 A spinner is spun 500 times. The table below shows the tally for every 100 trials.
red green blue
First set of 100 trials 22 41 37
Second set of 100 trials 21 41 38
Third set of 100 trials 27 39 34
Fourth set of 100 trials 25 46 29
Fifth set of 100 trials 30 44 26
a Give the best possible estimate for P(red), P(green) and P(blue) based on these trials.
b If your estimate is based on just one set of trials, which one would cause you to have the most
inaccurate results?
c Design a spinner that could give results similar to those in the table. Assume you can use up
to 10 sectors of equal size.
d Design a spinner that could give results similar to those in the table if you are allowed to use
sectors of different sizes.
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Chapter 238 5 Probability
Compound events in single-step experiments
When solving probability problems, it
is important to read the question very
carefully, especially when dealing with
compound events. Terminology such
as at least and more than may seem the
same but they are not. Even simple words
like and, or and not require your careful
attention.
let’s start: What is in a standard deck of 52 playing cards?
Do you know what is in a deck of cards?
• When and where was this standard deck of cards fi rst used?
• How many cards are there in a standard deck? Why that number?
• How many cards are red? How many cards are black? How many cards are aces?
• What are ‘suits’? How many are there and what are they called?
• How many cards are there in each suit?
• What are ‘court cards’? How many are there?
• What are ‘jokers’? What are some card games that involve the use of the jokers?
• Why is the fi rst card in every suit called an ace, not a 1?
• Are the decks of cards used in other countries different from this one?
• In how many ways can you choose a card that is red or a 7? Is it 26, 28 or 30?
5D
Some of the following key ideas are repeated from earlier pages and some are new.
In the following table, an ace = 1, jack = 11, queen = 12 and a king = 13, but this is not the case in
every card game.
terminology Example Defi nition/Explanation
chance experiment randomly choosing one card
from a standard deck
A chance experiment is an activity which may
produce a variety of different results that occur
randomly. The example given is a single-step
experiment.
event e.g. 1: choosing the 5 of
clubs
e.g. 2: choosing a 5
An event is either one outcome or a collection of
outcomes. It is a subset of the sample space.
Key ideas
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Statistics and Probability 239
compound event choosing a court card A compound event is a collection of two or more
outcomes from the sample space of a chance
experiment.
mutually exclusive events choosing a 5
choosing a 6
Two or more events are mutually exclusive if they
share no outcomes.
non-mutually exclusive
events
choosing a 5
choosing a red card
Events are non-mutually exclusive if they share one
or more outcomes. In the example, there are four
cards numbered 5, of which two are also red.
‘more than’
or
‘greater than’
choosing a card greater
than 10
In this example, the cards numbered 10 are not
included.
There are 12 cards in this compound event.
‘at least’
or
‘greater than or equal to’
choosing a 10 at least In this example, the cards numbered 10 are
included.
There are 16 cards in this compound event.
‘less than’ choosing a card less than 10 In this example, the cards numbered 10 are not
included.
There are 36 cards in this compound event.
‘at most’
or
‘less than or equal to’
choosing a 10 at most In this example, the cards numbered 10 are
included.
There are 40 cards in this compound event.
‘not’ choosing a 10 that is not red There are four cards numbered 10.
Only two of them are not red.
There are two cards in this compound event.
exclusive ‘or’ choosing a card that is either
red or a 10, but not both
There are 26 red cards.
There are four cards numbered 10 but two of them
are also red.
There are 26 cards in this compound event.
inclusive ‘or’ choosing a card that is red or
a 10 or both
There are 26 cards that are red.
There are two black cards that show 10.
There are 28 cards in this compound event.
‘and’ choosing a card that is red
and a 10
There are 26 red cards but only two of them are
numbered 10.
There are two cards in this compound event.
Key ideas
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Chapter 240 5 Probability
Example 4 Choosing one card from a standard deck
One card is chosen randomly from a standard deck of cards. What is the probability that it is:
a red? b not red? c a club? d not a club?
e a 7? f neither a 7 nor 8? g a red ace? h a red card or an ace?
i a red card that is not an ace?
Solution ExPlanation
a P(red) = = 26
52
1
2
There are 52 cards in the deck of which 26 are red.
b P(not red) = 1− =
1
2
1
2
Red and not red are complementary events.
c P(club) = = 13
52
1
4
There are 52 cards in the deck of which 13 are clubs.
d P(not a club) = 1− =
1
4
3
4
Club and not club are complementary events.
e P(7)
4
52
1
13
= = There are 52 cards in the deck of which four show a 7.
f P(neither a 7 nor 8) = 1− = =
8
52
44
52
11
13
There are 52 cards in the deck of which eight show a
7 or 8. That leaves 48 cards that do not show a 7 or 8.
g P(red ace) = = 2
52
1
26
There are four aces but only two of them are red.
h P(red or ace) = = 28
52
7
13
There are 26 red cards, including two red aces.
There are also two black aces.
i P(red but not ace) = = 24
52
6
13
There are 26 red cards, including two red aces.
So there are only 24 red cards that are not aces.
1 Use the terminology given in the fi rst column of the table on page 221 in this chapter to
fi ll in the blanks. You may use some of the terminology more than once.
Consider the following chance experiment.
These discs are identical except for their colour and their number.
1 3 5 2 4
Exercise 5D
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Statistics and Probability 241
They are placed in a bag and shaken. One disc is chosen randomly from the bag.
a ‘Choosing a blue disc’ is an example of an e _ _ _ t or o _ _ _ _ _ e.
b ‘Choosing a blue disc’ and ‘choosing a green disc’ are e _ _ _ _ _ y
l _ _ _ _ y o _ _ _ _ _ _ s. They are also m _ _ _ _ _ _ _
e _ _ _ _ _ _ _ _ e _ _ _ _ _.
c The p _ _ _ _ _ _ _ _ _ y of ‘choosing a red disc’ is 60%.
d The chosen number will be a _ l _ _ _ _ 1.
e It is c_ _ _ _ _ _ that the chosen number will be less than 6.
f It is c_ _ _ _ _ _ that the chosen disc will be red or even.
g The probability of ‘choosing a number 1 _ _ _ t _ _ _ 5’ is 80%.
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2 Complete the following, using the experiment in Question 1. Give your answers
as percentages.
What is the probability that the disc:
a is red or blue? b is red and blue?
c is red or shows the number 4? d is red and shows the number 3?
e shows a number of 2 or more? f shows a number greater than 3?
3 A standard die is rolled once. What is the probability (as a simple fraction) that the number rolled is:
a even or a 5? b even and a 5? c at least 5?
d greater than 5? e less than 5? f at most 5?
g not 5? h odd but not 5? i less than 4 and even?
j less than 4 or even? k less than 4 or even but not both?
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4 Sophie has randomly chosen a card from a standard deck and placed it in her pocket.
She is going to randomly choose a second card from the deck.
a What is the probability that she chooses the same card as the one in her pocket?
b What is the probability that the second card has the same suit as the fi rst card?
c What is the probability that the second card’s suit is different from that of the fi rst card?
5 Rachel has eight socks in her sock drawer. They are not joined together. Two are red, two are
green, two are yellow and two are blue. She has randomly chosen one sock and can see its
colour. She is now going to randomly choose another sock.
a What is the probability that it is the same colour as the fi rst sock?
b What is the probability that it is not the same colour as the fi rst sock?
Example 4
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Chapter 242 5 Probability
6 In this exercise you get a chance to be the teacher and make up the questions. You are
required to use the terminology in the fi rst column of the table below to write questions
for another student in your class. The answers to your questions must not be 0 or 1.
The other student fi lls in the answers in the probability column.
A chance experiment
Every domino tile in the picture shows two numbers. The fi rst tile shows a 5 and a 6.
The six tiles are placed face down and shuffl ed. One of them is chosen at random.
terminology Your question Probability answer
greater than
at least
less than
at most
not
exclusive or
inclusive or
and
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Enrichment: Combinations on your calculator
7 Scientifi c calculators have a button called nCr that is useful for combinations. Examples of a
combination are given below.
a There are fi ve people in a room (A, B, C, D, E). You must choose two.
Write down all the possibilities. How many possibilities are there?
b Enter 5C2. This should confi rm your answer to part a.
c Now there are 10 people in the room and you must choose two. How many combinations
are there?
d i There are 40 balls in a barrel and you must draw 6. How many combinations are there?
ii Four extra balls are placed in the barrel. How many combinations are there now?
5D
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Statistics and Probability 243
Venn diagrams and two-way tables
When two events are being considered, Venn diagrams and two-way tables give another way to
view the probabilities. They are especially useful when survey results are being considered and
converted to probabilities.
let’s start: Are English and Mathematics enemies?
Conduct a poll among students in the class, asking whether they like English and whether they like
Maths. Use a tally like the one shown.
like Maths Do not like Maths
like English |||| | |||| ||||
Do not like English |||| |||| | |||| ||
Use your survey results to debate these questions.
• Are the students who like English more or less likely to enjoy Maths?
• If you like Maths, does that increase the probability that you will like English?
• Which is the more popular subject within your class?
5E
A two-way table lists the number of outcomes or people in different categories, with the fi nal
row and column being the total of the other entries in that row or column. For example:
like Maths Do not like Maths total
like English 28 33 61
Do not like English 5 34 39
total 33 67 100
■ A two-way table can be used to fi nd probabilities.
e.g. P(like Maths) = 33
100
P(like Maths and not English) =
5
100
= 1
20
■ A Venn diagram is a pictorial
representation of a two-way table
without the total row and column.
The two-way table above can be
written as shown.
■ Mutually exclusive events cannot both
occur at the same time; e.g. rolling
an even number and rolling an odd number.
5 28 33
34
Maths English
like Maths
and English
like
English only
like
Maths only
like neither
Maths nor English
3
even
number
odd
number
There is nothing in both circles,
so the events are mutually exclusive.
3
Key ideas
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Chapter 244 5 Probability
Example 5 Constructing Venn diagrams and two-way tables
A survey is conducted of 50 people, asking who likes coffee and who likes tea. It was found that
20 people liked both, 15 people liked coffee but not tea, and 10 people liked tea but not coffee.
a How many people liked neither tea nor coffee?
b Represent the survey fi ndings in a Venn Diagram.
c How many people surveyed like tea?
d How many people like both coffee and tea?
e How many people like coffee or tea (or both)?
f Represent the survey fi ndings in a two-way table.
Solution ExPlanation
a 5 50 – 20 – 15 – 10 = 5 people who do not like either.
b
15 20 10
5
coffee tea
The Venn diagram includes four numbers,
corresponding to the four possibilities.
For example, the number 15 means that 15 people
like coffee but not tea.
c 20 + 10 = 30 10 people like tea but not coffee, but 20 people like
both. In total, 30 people like tea.
d 20 20 out of 50 people like both coffee and tea.
e 45 15 + 20 + 10 = 45 people like either coffee or tea
or both.
f
like
coffee Dislike coffee total
like tea 20 10 30
Dislike
tea
15 5 20
total 35 15 50
The two-way table has the four numbers from the
Venn diagram and also a ‘total’ column
(e.g. 20 + 10 = 30, 15 + 5 = 20) and a ‘total’ row.
Note that 50 in the bottom corner is both 30 + 20
and 35 + 15.
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Statistics and Probability 245
Example 6 using two-way tables to calculate probabilities
Consider the two-way table below showing the eating and sleeping preferences of different animals at
the zoo.
Eats meat no meat total
Sleeps during day 20 12 32
only sleeps at night 40 28 68
total 60 40 100
a For a randomly selected animal, fi nd:
i P(sleeps only at night) ii P(eats meat or sleeps during day)
b If an animal is selected at random and it eats meat, what is the probability that it sleeps during
the day?
c What is the probability that an animal that sleeps during the day does not eat meat?
Solution ExPlanation
a i P(sleeps only at night)
=
68
100
=
17
25
ii P(eats meat or sleeps during day)
=
72
100
=
18
25
The total number of animals that sleep at night
is 68.
So
68
100
17
25
= .
20 + 12 + 40 = 72 animals eat meat or sleep
during the day (or both).
72
100
18
25
=
b P(sleeps during day and eats meat)
=
20
60
=
1
3
Of the 60 animals that eat meat, 20 sleep during
the day, so the probability is
20
60
1
3
= .
c P(sleeps during day and does not eat meat)
=
12
32
=
3
8
Of the 32 animals that sleep during the day,
12 do not eat meat. The probability is
12
32
3
8
= .
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Chapter 246 5 Probability
1 a Copy and complete the two-way table by writing in the missing totals.
like bananas Dislike bananas total
like apples 30 15 45
Dislike apples 10 20
total 35 75
b How many people like both apples and bananas?
c How many people dislike both apples and bananas?
d How many people participated in the survey?
e It is not possible to like apples and dislike apples. These two events are ______ ______.
2 Consider the Venn diagram representing cat and dog ownership.
a State the missing number (1, 2, 3 or 4) to make the following
statements true.
i The number of people surveyed who own a cat and a
dog is ____.
ii The number of people surveyed who own a cat but do
not own a dog is ____.
iii The number of people surveyed who own neither a cat nor a dog is ____.
iv The number of people surveyed who own a dog but do not own a cat is ____.
b Is owning a cat and owning a dog a mutually exclusive event? Why/why not?
4 2 3
1
own a cat own a dog
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3 In a group of 30 students it is found that 10 play both cricket and soccer, 5 play only C
cricket and 7 play only soccer.
a How many students do not play cricket or soccer?
b Represent the survey fi ndings in a Venn diagram.
c How many of the students surveyed play cricket?
d How many of the students surveyed play cricket or soccer or both?
e How many of the students surveyed play either cricket or soccer but not both?
4 Consider this Venn diagram, showing the number of people surveyed
who have a university degree and the number of those surveyed who
are employed.
a What is the total number of people surveyed who are
employed?
Example 5a–e
3 10 5
2
university employed
Example 5f
Exercise 5E
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Statistics and Probability 247
b Copy and complete the two-way table shown below.
Employed unemployed total
university degree
no university degree
total
c If the 10 in the centre of the Venn diagram is changed to 11, which cells in the two-way table
would change?
5 The two-way table below shows the results of a poll conducted of a group of students who
own mobile phones to fi nd out who pays their own bills.
Boys Girls total
Pay own bill 4 7 11
Do not pay own
bill
8 7 15
total 12 14 26
a How many students participated in this poll?
b How many participants were boys?
c How many of the students surveyed pay their own bill?
d Find the probability that a randomly selected participant:
i is a boy who pays his own bill ii is a girl who pays her own bill
iii is a girl iv does not pay their own bill
e There are four events shown in the table above (i.e. being a boy, being a girl, paying own
bill, not paying own bill). Which pair(s) of events are mutually exclusive?
6 Forty men completed a survey about home ownership and car ownership. The results are shown
in the two-way table below.
own car Do not own car total
own home 8 2 10
Do not own home 17 13 30
total 25 15 40
a Represent the two-way table above as a Venn diagram.
b If a survey participant is chosen at random, give the probability that:
i he owns a car and a home
ii he owns a car but not a home
iii he owns a home
c If a survey participant is selected at random and he owns a car, what is the probability that he
also owns a home?
d If a survey participant is selected at random and he owns a home, what is the probability that
he also owns a car?
Example 6a
Example 6b,c
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Chapter 248 5 Probability
7 The Venn diagram shows the number of people surveyed who like juice and/or soft drinks.
10 2 14
4
juice soft drink
1
a What is the total number of people surveyed who like juice?
b What is the probability that a randomly selected
survey participant likes neither juice nor soft drink?
c What is the probability that a randomly selected survey
participant likes juice or soft drink or both?
d What is the probability that a randomly selected survey
participant likes juice or soft drink but not both?
e Explain the difference between inclusive or used in
part c and exclusive or used in part d. Make two copies
of the Venn diagram and use shading to illustrate the
difference.
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8 A car salesperson notes that among 40 cars for sale, there are 15 automatic cars and 10 sports
cars. Only two of the sports cars are automatic.
a Create a two-way table of this situation.
b What is the probability that a randomly selected car will be a sports car that is not
automatic?
c What is the probability that a randomly selected car will be an automatic car that is not a
sports car?
d If an automatic car is chosen at random, what is the probability that it is a sports car?
9 A page of text is analysed and, of the 150 words on it, 30 are nouns, 10 of which start with a
vowel. Of the words that are not nouns, 85 of them do not start with vowels.
a If a word on the page is chosen at random, what is the probability that it is a noun?
b How many of the words on the page start with vowels?
c If a word on the page starts with a vowel, what is the probability that it is a noun?
d If a noun is chosen at random, what is the probability that it starts with a vowel?
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10 In a two-way table, there are nine spaces to be fi lled with numbers.
a What is the minimum number of spaces that must be fi lled before the rest of the table
can be determined? Explain your answer.
b If you are given a two-way table with fi ve spaces fi lled, can you always determine the
remaining spaces? Justify your answer.
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Statistics and Probability 249
c Explain why the following two-way table must contain an error.
B Not B Total
A 20
Not A 29
Total 62 81
11 In this Venn diagram, w, x, y and z are all unknown positive integers.
Copy and complete this two-way table.
B Not B Total
A x
Not A z
Total x + y
A
w x y
z
B
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Enrichment: Triple Venn diagrams
12 A group of supermarket shoppers is surveyed on their age,
gender and whether they shop using a trolley or a basket.
This Venn diagram summarises the results.
a How many shoppers participated in the survey?
b How many of the participants are aged 40 or over?
c Give the probability that a randomly selected
survey participant:
i uses a trolley ii is female
iii is aged 40 or over
iv is male and uses a trolley
v is female and younger than 40
vi is younger than 40 and uses a trolley
d If a female survey participant is chosen at random, what is the probability that she:
i uses a trolley? ii is aged 40 or over?
e If a survey participant that uses a trolley is chosen at random, what is the probability that they:
i are male? ii are under 40?
f Describe what you know about the four participants outside of the three circles in the diagram.
g If all you know about a survey participant is that they use a trolley, are they more likely to be male
or female? Justify your answer.
h If a female survey participant is shopping, are they more likely to use a trolley or a basket?
female
use
trolley
age
40+
22
5 20
10
30 15 14
4
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Chapter 250 5 Probability
Probability in two-step experiments
Sometimes an experiment consists of two independent components, such as when a coin is tossed and
then a die is rolled. Or perhaps a card is pulled from a hat and then a spinner is spun. We can use tables to
list the sample space.
Consider the following example in which a coin is fl ipped and then a die is rolled.
Die
1 2 3 4 5 6
Coin
Heads H1 H2 H3 H4 H5 H6
tails T1 T2 T3 T4 T5 T6
There are 12 outcomes listed in the table. So the probability of getting a ‘tail’ combined with the
number 5 is
1
12
.
let’s start: Dice dilemma
In a board game, two dice are rolled and
the player moves forward according to
their sum.
• What are the possible values that
the sum could have?
• Are some values more likely than
others? Discuss.
• How likely is it that the numbers
showing on the two dice will add
to 5?
5F
Are you as likely to roll a 9 on two dice as any other number?
■ If two independent events occur, the outcomes can be listed as a table.
■ The probability is still given by
P(event) =
number of outcomes in which the event occurs
total number of possible outcomes
Key ideas
Example 7 using a table for multiple events
A spinner with the numbers 1, 2 and 3 is spun, and then a card is chosen at random from the letters ATHS.
a Draw a table to list the sample space of this experiment.
b How many outcomes does the experiment have?
c Find the probability of the combination 2S.
d Find the probability of an odd number being spun and the letter H being chosen.
EXTENSION
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Statistics and Probability 251
Solution ExPlanation
a a t H S
1 1A 1T 1H 1S
2 2A 2T 2H 2S
3 3A 3T 3H 3S
The sample space of the spinner {1, 2, 3} is put into
the left column.
The sample space of the cards {A, T, H, S} is put into
the top row.
b There are 12 outcomes. The table has 4 × 3 = 12 items in it.
c P(2S) =
1
12
All 12 outcomes are equally likely. Spinning 2 and
choosing an S is one of the 12 outcomes.
d P(odd, H) = 2
12
1
6
=
Possible outcomes are 1H and 3H, so
probability = 2 ÷ 12.
1 A coin is fl ipped and then a spinner is spun. The possible outcomes are listed in the table
below.
1 2 3 4 5
H H1 H2 H3 H4 H5
T T1 T2 T3 T4 T5
a How many outcomes are possible?
b List the four outcomes in which an even number is displayed on the spinner.
c Hence, state the probability that an even number is displayed.
d List the outcomes for which tails is fl ipped and an odd number is on the spinner.
e What is P(T, odd number)?
2 Two coins are tossed and the four possible outcomes are shown below.
20-cent coin
H t
50-cent coin
H HH HT
t TH TT
a What is the probability that the 50-cent coin will be heads and the 20-cent coin will be tails?
b For which outcomes are the two coins displaying the same face?
c What is the probability of the two coins displaying the same face?
Exercise 5F E X T E N S I O N
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 252 5 Probability
3 A coin is fl ipped and then a die is rolled.
a Draw a table to list the sample space of this experiment.
b How many possible outcomes are there?
c Find the probability of the pair H3.
d Find the probability of fl ipping ‘heads’ and rolling an odd number.
4 A letter is chosen from the word LINE and another is chosen from the word RIDE.
a Draw a table to list the sample space.
b How many possible outcomes are there?
c Find P(NR); i.e. the probability that N is chosen from LINE and R is chosen from RIDE.
d Find P(LD).
e Find the probability that two vowels are chosen.
f Find the probability that two consonants are chosen.
g Find the probability that the two letters chosen are the same.
5 The spinners shown below are each spun.
a Draw a table to list the sample space. Use R for red, P for purple and so on.
b Find the probability that spinner 1 will display red and spinner 2 will display blue.
c Find the probability that both spinners will display red.
d What is the probability that spinner 1 displays red and spinner 2 displays purple?
e What is the probability that one of the spinners displays red and the other displays blue?
f What is the probability that both spinners display the same colour?
6 A letter from the word EGG is chosen at random and then a letter from ROLL is chosen at random.
The sample space is shown below.
R o l l
E ER EO EL EL
G GR GO GL GL
G GR GO GL GL
a Find P(ER). b Find P(GO).
c Find P(both letters are vowels). d Find P(both letters are consonants).
Example 7
spinner 1 spinner 2
WORKING
MATHEMATICALLY
U F
R PS
C
red green
purple blue
red purple
blue
5F
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Statistics and Probability 253
7 Two dice are rolled for a board game. The numbers showing are then added together to get
a number between 2 and 12.
a Draw a table to describe the sample space.
b Find the probability that the two dice add to 5.
c Find the probability that the two dice add to an even number.
d What is the most likely sum to occur?
e What are the two least likely sums to occur between 2 and 12?
8 In Rosemary’s left pocket she has two orange marbles and one white marble. In her right pocket she
has a yellow marble, a white marble and three blue marbles. She chooses a marble at random from
each pocket.
a Draw a table to describe the sample space. (Hint: The left-pocket outcomes are W, O, O.)
b Find the probability that she will choose an orange marble and a yellow marble.
c What is the probability that she chooses a white marble and a yellow marble?
d What is the probability that she chooses a white marble and an orange marble?
e Find the probability that a white and a blue marble are selected.
f What is the probability that the two marbles selected are the same colour?
9 In a game show a wheel is spun to determine the
prize money and then a die is rolled. The prize
money shown is multiplied by the number on the
die to give the total winnings.
a What is the probability that a contestant will
win $6000?
b What is the probability that they will win more
than $11 000?
$5000
$2000
$3000 $1000
WORKING
MATHEMATICALLY
U F
R PS
C
10 Two different experiments are conducted simultaneously. The fi rst has seven possible outcomes
and the second has nine outcomes. How many outcomes are there in the combined experiment?
11 In a standard deck of 52 playing cards there are four suits (diamonds, hearts, clubs and spades)
and 13 cards in each suit (A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K).
a What is the probability that a randomly chosen card is of the diamond suit?
b If a card is chosen at random, what is the probability that it will be 3♦?
c What is the probability of selecting a card that is red and a king?
d If two cards are chosen at random from separate decks,
what is the probability that:
i they are both diamonds? (Hint: Do not draw a
52 × 52 table.)
ii they are both red cards?
iii 3♦ is chosen from both decks?
e How would your answers to part d change if
the two cards were drawn from the same deck?
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 254 5 Probability
Enrichment: Spinners with unequal areas
12 Consider the spinners below.
a Find the following probabilities for spinner 2.
i P(red) ii P(blue)
b Find the probability of the following occurring when spinner 2 is spun twice.
i two reds ii two blues
iii a red, then a green iv a red and a green (in either order)
c Spinner 3 has P(orange) = 1
6
, P(yellow) =
1
3
and P(blue) =
1
2
.
What six letters could be used to describe the six equally likely outcomes when spinner 3 is
spun?
d If spinner 3 is spun twice, fi nd the probability of obtaining:
i yellow twice ii the same colour twice
iii orange and then blue iv orange and blue (either order)
v at least one orange vi at least one blue
e Spinners 2 and 3 are both spun. Find the probability of obtaining:
i red then orange ii green then blue
iii orange and not blue iv both blue
v neither blue vi neither red
spinner 1
outcomes: R, G, B, B
spinner 2
outcomes: R, G, B
spinner 3
red
green
blue red
yellow
blue blue
green
blue
orange
5F
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Statistics and Probability 255
Develop a spreadsheet simulation of two dice rolling
1 Set up a table in a spreadsheet to randomly generate 500 outcomes for tossing two dice.
• Include a column for each die.
• Include a column to show the sum of each pair of outcomes.
2 Using the spreadsheet software, count how many times each sum from 2 to 12 is achieved. Plot
the data as a histogram (or a 2D column graph if your spreadsheet software does not include
histograms). Comment on whether your distribution is symmetrical or skewed.
3 Use a table to calculate the theoretical probability of each of the dice sums, and compare the results
with the experimental probability.
Considering other dice sums
Using the spreadsheet software, conduct a large simulation (1000 or more rows) of rolling three dice
and noting the sums. Use the spreadsheet software to generate a frequency column graph of your
results and comment on how this graph looks compared to the simulation of two dice being rolled.
investigation
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Chapter 256 5 Probability
Monopoly risk
In the game of Monopoly, two fair 6-sided dice are rolled to work out how far a player should go
forward. For this investigation, you will need two 6-sided dice or a random number simulator that
simulates numbers between 1 and 6.
a Roll the two dice and note what they add up to. Repeat this 100 times and complete this table.
Dice sum 2 3 4 5 6 7 8 9 10 11 12 Total
Tally 100
b Represent the results in a column graph. Describe the shape of the graph. Do you notice
any patterns?
c Use the results of your experiment to give the experimental probability of two dice adding to:
i 3 ii 6 iii 8 iv 12 v 15
d What is the most likely sum for the dice to add to, based on your experiment?
e If the average Monopoly game involves 180 rolls, find the expected number of times, based on your
experiment, that the dice will add to:
i 3 ii 6 iii 8 iv 12 v 15
f Why do you think that certain sums happen more often than others? Explain why this might happen
by comparing the number of times the dice add to 2 and the number of times they add to 8.
g What is the mean dice sum of the 100 trials you conducted above?
To conduct many experiments, a spreadsheet can be used. For example, the spreadsheet below can
be used to simulate rolling three 6-sided dice. Drag down the cells from the second row to row 1000
to run the experiment 1000 times.
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Statistics and Probability 257
1 At the local sports academy, everybody plays netball or tennis. Given that half the tennis players
also play netball and one-third of the netballers also play tennis, what is the probability that a
randomly chosen person at the academy plays both?
2 For each of the following, fi nd an English word that matches the description.
a P(vowel) =
1
2
b P(F) =
2
3
c P(vowel) =
1
4
and P(D) =
1
4
d P(I) =
2
11
and P(consonant) =
7
11
e P(M) =
1
7
and P(T) =
1
7
and P(S) =
1
7
f P(vowel) = 0 and P(T) =
1
3
3 In a particular town, there are 22 women who can cook and 18 men who cannot cook. Given that
half the town is male and 54% of the town can cook, how many men in the town can cook?
4 In the following game, the player fl ips a fair coin each turn to move a piece. If the coin shows
‘heads’ the piece goes right, and if it is ‘tails’ the coin goes left. What is the approximate
probability that the player will win this game?
WIN START LOSE
5 If a person guesses all the answers on a 10-question true or false test, what is the probability that
they will get them all right?
6 A bag contains eight counters that are red, blue or yellow. A counter is selected from the bag, its
colour noted and the counter replaced. If 100 counters were selected and 14 were red, 37 were blue
and 49 were yellow, how many counters of each colour are likely to be in the bag?
7 Each of the eight letters of a word is written on a separate card. Given the following probabilities,
what is the word?
P(letter P) = P(letter R) = 12.5%, P(letter B) = 1
4
, P(vowel) = 0.375
Puzzles and challenges
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Chapter 258 5 Probability
8 What is the capital city of Iceland?
Find the answer to this question by looking at the pie chart
and finding the answers below. You’ll need a protractor to
measure each angle in the graph. Match up the letter with
the correct numerical answer given below.
A school of 1080 students asks its students to nominate
their favourite sport offered by the school program.
A the probability that a randomly chosen student
prefers golf
E the number of students who prefer swimming
I the probability of a student choosing basketball
J the number of students who nominate table tennis
K the probability that a randomly chosen student nominates soccer
R If golf and table tennis are cut from the school program, how many students must choose
a different sport?
V the probability that a student does not choose swimming or surfing
Y the probability of a student being a keen surfer
180 240
1
6
1
9
150
1
36
11
18
1
12
1
9
swimming
soccer
surfing
running
golf
basketball
table
tennis
tennis
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Statistics and Probability 259
Chapter summary
Probability
e.g. Roll a fair die
Sample space: {1, 2, 3, 4, 5, 6}
P(roll a 5) =
P(roll odd number) = =
16
36
12
An experiment can be used
if the exact probability
cannot be calculated.
e.g. Spin spinner
Sample space: {red,
green, blue}
P(spin red) =
P(don’t spin blue) =
13
23
green
red
blue
P(event) × number of trials
e.g. Flip coin 100 times,
expected number of heads
× 100 = 50
× 36 = 6
e.g. Roll die 36 times,
expected number of 5s
12
16
Probabilities can be
given as fractions,
decimals or percentages.
14
7
10
Experimental probability
Use an experiment or survey
or simulation to estimate
probability.
e.g. Spinner lands on blue 47
times out of 120
Experimental probability = 47
120
Expected number is
12
even chance
more likely
certain
unlikely likely
impossible
0 1
Probability: How likely an event will occur
=
=
e.g. 25%, , 0.25
e.g. 70%, , 0.7
Chance experiment
e.g. Select a playing card and
note its suit.
=
14
24
12
Sample space: {spades, diamonds,
hearts, clubs}
P(diamonds) =
P(hearts or clubs) =
Theoretical probability
Trial: Select a playing card
and note its suit.
Sample space: {spade, diamond
club, heart}
P(black) =
P(heart) =
P(not spade) =
P(either red or a spade) =
P(red ace) =
Theoretical probabilities
26 =
52
12
13 =
52
14
39 =
52
34
39 =
52
34
2 =
52
1
26
Trial: Roll a fair die
Sample space (possible outcomes):
{1, 2, 3, 4, 5, 6}
P(odd number) = = 12
36
Experimental probability
Playing card selected and replaced
20 times, and its suit noted.
Outcome Frequency
Experimental
probability
Heart
Diamond
Club
Spade
4
5
4
7
4
20
5
20
4
20
7
20
n = 20
Expected number of outcomes
Outcome Frequency
Experimental
probability
Heart
Diamond
Club
Spade
× 20 = 5 = 1414
14
× 20 = 5 14
× 20 = 5 14
× 20 = 5 14
13
52
13 =
52
= 14
13
52
= 14
13
52
Playing cards
Spades
Diamonds
Clubs
Hearts
P(event) =
total number of outcomes
number of favourable
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Chapter 260 5 Probability
Multiple-choice questions
1 The results of a survey are shown below. Note that each student learns only one instrument.
instrument learned piano violin drums guitar
number of students 10 2 5 3
Based on the survey, the experimental probability that a randomly selected survey participant is
learning the guitar is:
a
1
4
B
1
2
C 3 D
3
5
E
3
20
2 Which of the following events has the same probability as rolling an odd number on a fair
6-sided die?
a rolling a number greater than 4 on a fair 6-sided die
B choosing a vowel from the word CAT
C tossing a fair coin and getting heads
D choosing the letter T from the word TOE
E spinning an odd number on a spinner numbered 1 to 7
3 Each letter of the word APPLE is written separately on fi ve cards. One card is then chosen at
random. P(letter P) is:
a 0 B 0.2 C 0.4 D 0.5 E 1
4 A fair 6-sided die is rolled 600 times. The expected number of times that the number rolled is
either a 1 or a 2 is:
a 100 B 200 C 300 D 400 E 600
5 The letters of the word STATISTICS are placed on 10 different cards and placed into a hat. If a
card is drawn at random, the probability that it will show a vowel is:
a 0.2 B 0.3 C 0.4 D 0.5 E 0.7
6 A fair die is rolled and then the spinner shown at right is spun.
The probability that the die will display the same number as the
spinner is:
a
1
36
B
1
18
C
1
6
D
1
2
E 1
7 A coin is tossed three times. The probability of obtaining at least
two tails is:
a
2
3
B 4 C
1
2
D
3
8
E
1
8
1
2
3
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Statistics and Probability 261
8 An experiment is conducted in which three dice are rolled and the sum of the faces is added. In 12 of
the 100 trials, the sum of the faces is 11. Based on this, the experimental probability of having three
faces add to 11 is:
a
11
100
B
12
111
C
3
25
D 12 E
1
2
9 Rachel has a fair coin. She has tossed ‘heads’ fi ve times in a row. Rachel tosses the coin one more
time. What is the probability of tossing ‘tails’?
a 0 B 1 C
1
2
D less than 1
2
E more than 1
2
10 When a fair die is rolled, what is the probability that the number is even but not less than 3?
a 0 B 1
6
C 1
3
D 1
2
E 2
3
Short-answer questions
1 For each of the following descriptions, choose the probability from the set
0, 1
8
, 3
4
, 1, 19
20
that matches best.
a certain b highly unlikely c highly likely
d likely e impossible
2 List the sample space for each of the following experiments.
a A fair 6-sided die is rolled.
b A fair coin is tossed.
c A letter is chosen from the word DESIGN.
d Spinning the spinner shown opposite.
3 Vin spins a spinner with nine equal sectors, which are numbered 1 to 9.
a How many outcomes are there?
b Find the probability of spinning:
i an odd number ii a multiple of 3
iii a number greater than 10 iv a prime number less than 6
v a factor of 8
blue
yellow
green
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Chapter 262 5 Probability
4 One card is chosen at random from a standard deck of 52 playing cards.
Find the probability of drawing:
a a red king
b a king or queen
c a jack of diamonds
d a picture card (i.e. king, queen or jack)
5 A coin is tossed 100 times, with the outcome 42 heads and 58 tails.
a What is the experimental probability of getting heads? Give your answer as a percentage.
b What is the actual probability of getting heads if the coin is fair? Give your answer as a
percentage.
6 Consider the spinner shown.
a State the probability that the spinner lands in the green section.
b State the probability that the spinner lands in the blue section.
c Grace spins the spinner 100 times. What is the expected
number of times it would land in the red section?
d She spins the spinner 500 times. What is the expected number
of times it would land in the green section?
Extended-response questions
1 The Venn diagram shows how many numbers between 1 and 100 are
odd and how many are prime.
Consider the numbers 1 to 100.
a How many are odd?
b How many prime numbers are there?
c What is the probability that a randomly selected number will be
odd and prime?
d What is the probability that a randomly selected number will be prime but not odd?
e If an odd number is chosen, what is the probability that it is prime?
f If a prime number is chosen, what is the probability that it is odd?
blue
green
red
26 24 1
49
odd prime
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Statistics and Probability 263
2 The two-way table below shows the results of a survey on car ownership and public transport usage.
You can assume the sample is representative of the population.
uses public transport
Does not use public
transport total
owns a car 20 80
Does not own a car 65 35
total
a Copy and complete the table.
b How many people participated in the survey?
c What is the probability that a randomly selected person will have a car?
d What is the probability that a randomly selected person will use public transport even though
they own a car?
e What is the probability that someone owns a car given that they use public transport?
f If a car owner is selected, what is the probability that they will catch public transport?
g In what ways could the survey produce biased results if it had been conducted:
i outside a train station?
ii in regional New South Wales?
3 A spinner is made using the numbers 1, 3, 5 and 10 in four sectors. The spinner is spun 80 times, and
the results obtained are shown in the table.
number on spinner Frequency
1
3
5
10
30
18
11
21
80
a Which sector on the spinner occupies the largest area? Explain.
b Two sectors of the spinner have the same area. Which two numbers do you think have equal
areas, and why?
c What is the experimental probability of obtaining a 1 on the next spin?
d Draw an example of what you think the spinner might look like, in terms of the area covered by
each of the four numbers.
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Chapter
Computation
with decimals
and fractions
6with What you will learn
Adding and subtracting decimals
Adding fractions
Subtracting fractions
Multiplying fractions
Multiplying and dividing decimals by 10, 100, 1000 etc.
Multiplying by a decimal
Dividing fractions
Dividing decimals
Computation with negative fractions EXTENSION
6A
6B
6C
6D
6E
6F
6G
6H
6I
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Swimming records and decimal places
NSW Syllabus
for the Australian
Curriculum
Strand: Number and Algebra
Substrand: FRACtioNS, DECiMAlS AND
pERCENtAGES
Outcome
A student operates with fractions,
decimals and percentages.
(MA4–5NA)
265
Swimming times are electronically measured and
recorded, with the seconds given to 2 decimal
places. Accuracy is very important so that the
right decision is made for placegetters and
records. Sometimes winners are separated by as
little as one-hundredth of a second. Ian Thorpe,
who set 13 world records, had his 400 m freestyle
record (3:40.06) broken in 2009 by German
swimmer Paul Biedermann, who swam only 0.01
of a second faster than Thorpe.
Australian swimmers Melanie Schlanger,
Brittany Elmslie, Alicia Coutts and Cate Campbell
won the Women’s 4 × 100 m Freestyle Relay at
the London 2012 Olympic Games in an Olympic
record time of 3 minutes and 33.15 seconds.
Second and third placegetters were The
Netherlands (3:33.79) and USA (3:34.24).
The decimal system allows us to express
quantities with great accuracy. There are many
instances in our everyday lives in which accuracy
of measurement is highly important. For example,
bicycle, car and aeroplane parts must be
manufactured to precise measurements; medicine
production requires measurement of chemicals
in precise quantities; and investment rates, stock
market prices and values are measured to many
decimal places (8% per year equals 0.153846%
per week).
Can you think of any other quantities that
require decimal measurement?
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266 Chapter 6 Computation with decimals and fractions
1 Complete the following.
a 1
10
= 0.__ b 3
10
= 0.__ c 17
10
= 1.__
d 1
100
= 0.0 __ e 1
1000
= 0.__ __ __ f 47
10
= __.__
2 Write the decimal for:
a one-half b one-quarter c three-quarters
3 Write the following cents as dollars.
a 70 cents b 85 cents c 100 cents
d 5 cents e 105 cents f 3 cents
4 Find how many cents are in:
a half a dollar b one-quarter of $1
c three-quarters of $1 d half of $5
5 Find the cost of:
a six labels at 45 cents each b 12 pears at $1.05 each
c 1 1
2
boxes of mangoes at $15 a box d seven pens at 27 cents a pen
6 Tom paid $50 for 200 photos to be printed. What was the cost of each print?
7 $124 is shared between eight people. If each share is the same amount, how much does
each person receive?
8 Complete:
a $8.50 × 10 = ______ b $6 − $5.90 = ______ c $10 − $7.30 = ______
d $70 ÷ 100 = ______ e $6.90 + $4.30 = ______ f $20 − $19.76 = ______
9 Petrol is 124 cents a litre. Calculate how much change from $100 Calvin receives when he
buys:
a 10 litres b 50 litres c 70 litres
10 Find the total of these amounts: $7, $5.50, $4.90, $12, $56, $10.10 and $9.15.
11 Complete these computations.
a 329
+194
b 1024
− 185
c 104
× 13
d 5)6185
pre-test
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Number and Algebra 267
Adding and subtracting decimals
The addition and subtraction of decimals follows similar procedures to those for whole numbers. To add
or subtract whole numbers you must line up the units, tens, hundreds and so on, and then you add or
subtract each column. When dealing with the addition or subtraction of decimals the routine is the same.
Consider how similar the following two sums are:
5 1 4
2 7 2
1 0 6
8 9 2
1
5 1 . 4
2 7 . 2
1 0 . 6
8 9 . 2
1
let’s start: What’s the total?
Each student thinks of three coins (gold or
silver) and writes their total value on a sheet
of paper. Each student in the class then
estimates the total value of the amounts
written down in the classroom. Record each
student’s estimated total.
• Each student then writes the value of the
three coins they thought of on the board
(e.g. $2.70, $0.80 etc.).
• Students copy down the values into
their workbooks and add the decimal
numbers to determine the total value of
the coins in the classroom.
• Which student has the closest
estimation?
6A
Accounting involves the addition and subtraction of decimal numbers.
■■ When adding or subtracting decimals, the decimal points and each of the decimal places
must be aligned under one another.
■■ The location of the decimal point in the answer is directly in line with the location of each of
the decimal points in the question.
■■ Once the numbers are aligned correctly, proceed as if completing whole number addition or
subtraction.
■■ If the numbers of decimal places in the numbers being added or subtracted are different, it is
helpful to place additional zeros in the ‘shorter’ number(s) to prevent calculation errors.
Key ideas
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Chapter 6 Computation with decimals 268 and fractions
Example 1 Adding decimals
Find:
a 8.31 + 5.93 b 64.8 + 3.012 + 5.94
SolutioN ExplANAtioN
a 18 31
5 93
14 24
.
.
.
+
Make sure all decimal points and decimal places are
correctly aligned directly under one another.
b 1614 8
3 012
5 94
7 3 752
.
.
.
.
00
+ 0
Align decimal points directly under one another. Fill in
missing decimal places with additional zeros.
Carry out addition, following the same procedure as that
for addition of whole numbers.
Example 2 Subtracting decimals
Find:
a 5.83 − 3.12 b 146.35 − 79.5
SolutioN ExplANAtioN
a 5 83
3 12
2 71
.
.
.
−
Make sure all decimal points and decimal places are
correctly aligned directly under one another.
b
13 15 1
1 4 6 35
7 9 5
6 6 85
0
.
.
.
−
Align decimal points directly under one another and fi ll
in missing decimal places with additional zeros.
Carry out subtraction, following the same procedure as
that for subtraction of whole numbers.
Exercise 6A
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1 The values 7.12, 8.5 and 13.032 must be added together. Which of the following is the best C
way to prepare these numbers ready for addition?
A 7 12
8 5
13 032
.
.
+ .
B 7 12
8 5
13 032
.
.
+ .
C 7 120
8 500
13 032
.
.
+ .
D 7 12
8 5
13 032
.
.
+ .
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Number and Algebra 269
2 Which of the following is the correct way to present and solve the subtraction problem
77.81 − 6.3?
A 77 81
6 3
84 11
.
.
.
−
B 77 81
6 30
71 51
.
.
.
−
C 7 7 81
6 3
14 81
.
.
.
−
D 77 8 1
6 3
77 18
.
.
.
−
3 Find each of the following.
a 13 25
14 72
.
+ .
b 7 23
16 31
2 40
.
.
+ .
c 210 0
22 3
15 1
.
.
+ .
d 47 81
6 98
3 52
.
.
+ .
4 Find each of the following.
a 12.45 + 3.61 b 5.37 + 13.81 + 2.15
c 312.5 + 31.25 d 1.567 + 3.4 + 32.6
e 5.882 + 3.01 + 12.7 f 323.71 + 3.4506 + 12.9
5 Find:
a 17 2
5 1
.
− .
b 128 63
14 50
.
− .
c 23 94
17 61
.
− .
d 158 32
87 53
.
− .
6 Find:
a 14.8 − 2.5 b 234.6 − 103.2
c 25.9 − 3.67 d 31.657 − 18.2
e 412.1 − 368.83 f 5312.271 − 364.93
7 Find, using a calculator.
a 46.189 + 23.85 - 7.816 b 282.375 - 159.483 - 72.689
8 Find the missing numbers in the following sums. Check your final answers using a calculator.
a 3 .
+ 4 . 6
. t3
b 8. 9
+ . 7 5
4. 4
c 1. 1
+ . 1 1
1 1. 1
d . 3 6
2. 4 3
+ 1. 8 9
1. 3 9 5
9 How much greater is 262.5 than 76.31?
10 Stuart wants to raise $100 for the Rainbow Club charity. He already has three donations of
$30.20, $10.50 and $5.00. How much does Stuart still need to raise?
11 Daily rainfalls for four days over Easter were 12.5 mm, 3.25 mm, 0.6 mm and 32.76 mm. What
was the total rainfall over the four-day Easter holiday?
Example 1
Example 2
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Chapter 6 Computation with decimals 270 and fractions
12 Complete the addition table below.
+ 0.05
0.3 1.72
0.75 1.13
1.21 1.58
3.03
13 Michelle earned $3758.65 working part-time
over a 1-year period. However, she was required
to pay her parents $20 per week for board for
52 weeks. Michelle also spent $425.65 on
clothing and $256.90 on presents for her family
and friends during the year. She placed the rest
of her money in the bank. How much did
Michelle bank for the year?
14 If a = 2.8, b = 1.31 and c = 3.928, fi nd:
a a + b + c b a + b − c c c + b − a d c – (b + b)
15 a Write down three numbers between 1 and 10, each with 2 decimal places, that would add
to 11.16.
b Can you fi nd a solution to part a that uses each digit from 1 to 9 exactly once each?
Enrichment: Money, money, money…
16 Investigate the following procedures and share your fi ndings with a friend.
a Choose an amount of money that is less than $10.00 (e.g. $3.25).
b Reverse the order of the digits and subtract the smaller number from the larger number (e.g.
$5.23 − $3.25 = $1.98).
c Reverse the order of the digits in your new answer and now add this number to your most
recent total (e.g. $1.98 + $8.91 = $10.89).
Did you also get $10.89? Repeat the procedure using different starting values.
Try to discover a pattern or a rule. Justify your fi ndings.
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6A
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Number and Algebra 271
Adding fractions
Fractions with the same denominator can be added together easily.
+ =
3
8
2
8
5
8
+ =
Fractions with different denominators cannot be added together so easily.
+ =
1
3
1
4
+ = ?
But with a common denominator it is possible.
+ =
1
3
1
4
4
12
3
12
7
12
+ =
+ =
?
let’s start: ‘Like’ addition
Pair up with a classmate and discuss the following.
Part A
Which of the following pairs of numbers can be simply added together without having to carry out any
form of conversion?
a 6 goals, 2 goals b 11 goals, 5 behinds c 56 runs, 3 wickets
d 6 hours, 5 minutes e 21 seconds, 15 seconds f 47 minutes, 13 seconds
g 15 cm, 3 m h 2.2 km, 4.1 km i 5 kg, 1680 g
j 2
7
3
7
, k 1
4
1
2
, l 2
5
12
1
1
3
,
Does it become clear that we can only add pairs of numbers that have the same unit? In terms of
fractions, we need to have the same ______________?
Part B
By choosing your preferred unit (when necessary), work out the answer to each of the problems above.
6B
Note:
1
3
1
4
1
7
1
3
1
4
2
7
+ 
+ 
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Chapter 6 Computation with decimals 272 and fractions
Example 3 Adding fractions with the same denominators
Add the following fractions together.
a 1
5
3
5
+ b 3
11
5
11
6
11
+ +
Solution Explana tion
a
1
5
3
5
4
5
+ =
The denominators are the same; i.e. ‘like’, therefore
simply add the numerators.
b
3
11
5
11
6
11
14
11
+ + =
= 1
3
11
Denominators are the same, so add numerators.
Simplify answer by converting to a mixed numeral.
■■ Fractions can be simplified using addition only if they are ‘like’ fractions; that is, they must have
the same denominator. This means they have been divided up into the same number of pieces.
Same denominators
■■ If two or more fractions have the same denominator, to add them together simply add the
numerators and keep the denominator. This allows you to find the total number of divided pieces.
Different denominators
■■ If the denominators are different, we must use our knowledge of equivalent fractions to convert
them to fractions with the same lowest common denominator (LCD).
To do this, carry out these steps.
1 Find the LCD (often, but not always, found by multiplying denominators).
2 Convert fractions to their equivalent fractions with the LCD.
3 Add the numerators and write this total above the LCD.
■■ After adding fractions, always look to see if your answer needs to be simplified.
Key ideas
Example 4 Adding fractions with different denominators
Add the following fractions together.
a 1
5
1
2
+ b 3
4
5
6
+
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Number and Algebra 273
Solution Explana tion
a
1
5
1
2
2
10
5
10
+ = +
= 7
10
LCD is 10.
Write equivalent fractions with the LCD.
Denominators are the same, so add numerators.
b 3
4
5
6
9
12
10
12
+ = +
= 19
12
= 1
7
12
LCD is 12.
Write equivalent fractions with the LCD.
Denominators are the same, so add numerators.
Simplify answer to a mixed numeral.
Example 5 Adding mixed numerals
Simplify:
a 3
2
3
4
2
3
+ b 2
5
6
3
3
4
+
Solution Explana tion
a Method 1
3 4
2
3
2
3
7
4
3
+ + + = +
= 8
1
3
Add the whole number parts together.
Add the fraction parts together.
Noting that
4
3
1
1
3
= , simplify the answer.
Method 2
11
3
14
3
25
3
+ =
= 8
1
3
Convert mixed numerals to improper fractions. Have
the same denominators, so add numerators.
Convert improper fraction back to a mixed numeral.
b Method 1
2 3
5
6
3
4
5
10
12
9
12
5
19
12
6
7
12
+ + +
= + +
= +
=
Add the whole number parts together.
LCD of 6 and 4 is 12.
Write equivalent fractions with LCD.
Add the fraction parts together.
Noting that
19
12
1
7
12
= , simplify the answer.
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Chapter 6 Computation with decimals 274 and fractions
Method 2
17
6
15
4
34
12
45
12
79
12
6
7
12
+ = +
=
=
Convert mixed numbers to improper fractions.
Write equivalent fractions with LCD.
Add the numerators together.
Simplify answer back to a mixed numeral.
Exercise 6B
1 Copy the following sentences into your workbook and fi ll in the gaps.
a To add two fractions together, they must have the same ______________.
b When adding fractions together, if they have the same ______________, you simply add the
______________.
c When adding two or more fractions where the ______________ are different, you must fi nd
the ___________ ___________ ___________.
d After carrying out the addition of fractions, you should always ______________ your
answer to see if it can be ______________.
2 Copy the following sums into your workbook and fi ll in the empty boxes.
a
3
8
2
8 8
+ = c 1
3
1
4
12 12
12
+
= +
=
d 2
5
3
4
20 20
20
1
20
+
= +
= =
b
4
7
1
7 7
+ =
3 State the LCD for the following pairs of ‘incomplete’ fractions.
a
5 3
+ b
4 5
+ c
2 3
+ d
6 3
+
e
2 8
+ f
5 10
+ g
7 11
+ h
3 9
+
i
12 8
+ j
2 18
+ k
15 10
+ l
12 16
+
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Number and Algebra 275
4 The following sums have been completed, but only six of them are correct. Copy them into
your workbook, then place a tick beside the six correct answers and a cross beside the six
incorrect answers.
a
1
6
3
6
4
6
+ = b
1
3
1
4
2
7
+ = c
2
5
4
5
6
10
+ = d
1
11
3
11
4
11
+ =
e
3
5
4
5
1
2
5
+ = f
2
7
2
7
2
7
+ = g
7
12
4
12
11
12
+ = h
4
9
4
5
4
14
+ =
i
3
10
4
10
7
10
+ = j
1
2
2
5
3
7
+ = k 2
2
7
3
1
7
5
3
7
+ = l 1
2
3
2
1
5
3
3
8
+ =
5 Add the following fractions.
a
1
8
4
8
+ b
2
7
3
7
+ c
1
5
3
5
+ d
3
11
6
11
+
e
5
8
2
8
+ f
1
12
6
12
+ g
3
15
4
15
+ h
3
9
2
9
+
i
6
7
3
7
+ j
7
10
6
10
+ k
2
5
3
5
4
5
+ + l
12
19
3
19
8
19
+ +
6 Add the following fractions.
a
1
2
1
4
+ b
1
3
3
5
+ c
1
2
1
6
+ d
1
4
1
3
+
e
2
5
1
4
+ f
1
5
3
4
+ g
2
7
1
3
+ h
3
8
1
5
+
i
3
5
5
6
+ j
4
7
3
4
+ k 8
11
2
3
+ l 2
3
3
4
+
7 Simplify:
a 1
1
5
2
3
5
+ b 3
2
7
4
1
7
+ c 11
1
4
1
2
4
+ d 1
3
9
4
2
9
+
e 5
2
3
4
2
3
+ f 8
3
6
12
4
6
+ g 9
7
11
9
7
11
+ h 4
3
5
7
4
5
+
8 Simplify:
a 2
2
3
1
3
4
+ b 5
2
5
1
5
6
+ c 3
1
2
8
2
3
+ d 5
4
7
7
3
4
+
e 8
1
2
6
3
5
+ f 12
2
3
6
4
9
+ g 17
8
11
7
3
4
+ h 9
7
12
5
5
8
+
Example 3a
Example 3b
Example 4a
Example 4b
Example 5a
Example 5b
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Chapter 6 Computation with decimals 276 and fractions
9 Simplify, using a calculator.
a 15 +
3
5
8
2
7
b 62 +
1
9
17
3
4
c 143 +
1
3
56
7
8
d 125 +
3
10
134
17
100
10 Myles, Liza and Camillus work at a busy
cinema complex. For a particular movie,
Myles sells
3
5
of all the tickets and Liza
sells 1
3
.
a What fraction of movie tickets are
sold by Myles and Liza, together?
b If all of the movie’s tickets are sold, what
is the fraction sold by Camillus?
11 Martine loves to run and play. Yesterday, she ran for 2
1
4
kilometres, walked for 5
2
5
kilometres
and skipped for 1
2
a kilometre. What was the total distance that Martine ran, walked and
skipped?
12 Jackson is working on a 1000-piece jigsaw puzzle. All the pieces are the same size. After
1 week, he has completed
1
10
of the puzzle. After 2 weeks he has completed another
2
5
of the
puzzle. In the third week, Jackson completed another 1
4
of the puzzle.
a By the end of the third week, what
fraction of the puzzle has Jackson
completed?
b How many pieces of the puzzle does
Jackson place in the second week?
c What fraction of the puzzle is still
unfi nished by the end of the third week?
How many pieces is this?
13 A survey of Year 7 students’ favourite sport is carried out. A total of 180 students participate in
the survey. One-fi fth of students reply that netball is their favourite, one-quarter reply rugby and
one-third reply soccer. The remainder of students leave the question unanswered.
a What fraction of the Year 7 students answered the survey question?
b What fraction of the Year 7 students left the question unanswered?
c How many students did not answer the survey question?
WORK ING
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6B
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Number and Algebra 277
14 Fill in the empty boxes to make the following fraction sums correct.
a
1 1 7
10
+ = b
1 1 1 7
8
+ + =
c 3
4
17
20
+ = d
2
3
4
+ + = 1
15 Four students each read a portion of the same English novel over two nights, for homework.
The table shows what fraction of the book was read on each of the two nights.
Student First night Second night
Mikhail
2
5
1
4
Jim
1
2
1
10
Vesna*
1
4
1
5
Juliet
7
12
1
20
*Vesna woke up early on the third morning and read another
1
6
of the novel before leaving for
school.
Place the students in order, from least to most, according to what fraction of the book they had
read by their next English lesson.
Enrichment: Raise it to the max, lower it to the min
16 a Using the numbers 1, 2, 3, 4, 5 and 6 only once, arrange them in the boxes below to, first,
produce the maximum possible answer, and then the minimum possible answer. Work out
the maximum and minimum possible answers.
+ +
b Repeat the process for four fractions using the digits 1 to 8 only once each. Again, state the
maximum and minimum possible answers.
c Investigate maximum and minimum fraction statements for other sets of numbers and
explain your findings.
d Explain how you would arrange the numbers 1 to 100 into 50 different fractions if you were
trying to achieve the maximum or minimum sum.
WORKING
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Chapter 6 Computation with decimals 278 and fractions
Subtracting fractions
Subtracting fractions is very similar to adding fractions. You must establish the lowest common
denominator (LCD) if one does not exist and this is done through producing equivalent fractions. Then,
instead of adding numerators at the fi nal step, you simply carry out the correct subtraction.
Complications can arise when subtracting mixed numerals and Example 7b shows the available
methods that can be used to overcome such problems.
let’s start: Alphabet subtraction
0 1 1
12
2
12
3
12
4
12
5
12
6
12
7
12
8
12
9
12
10
12
11
12
• Copy into your workbook the number line above.
• Place the following letters in the correct position on the number line.
A =
2
3
B =
5
12
C =
1
2
D =
11
12
E =
1
12
F =
1
4
G =
0
12
H =
1
3
I =
7
12
J =
5
6
K =
12
12
L =
3
4
M =
1
6
• Complete the following alphabet subtractions, giving your answer as a fraction and also the
corresponding alphabet letter.
a J – F b A – G c D – F – M d C – B
e K – C f L – H – E g K – J – E h L – I – M
• What does A + B + C + D + E + F + G + H + I – J – K – L – M equal?
6C
■■ Fractions can be simplifi ed using subtraction only if they are ‘like’ fractions.
■■ The process for subtracting fractions is the same as adding fractions, until the fi nal step. At the
fi nal step you follow the operation and subtract the second numerator from the fi rst numerator.
■■ When subtracting mixed numerals, you must have a fraction part that is large enough to allow
the other proper fraction to be subtracted from it. If this is not the case at the start of the problem,
you may choose to borrow a whole.
For example:
7
1
2
2
3
4
− 1
2
is not big enough to have 3
4
subtracted from it.
6
3
2
2
3
4
− Therefore, we choose to borrow a whole from the 7.
■■ A fail-safe method for subtracting mixed numerals is to convert to improper fractions right from
the start.
For example: 7
1
2
2
3
4
− = 15
2
11
4
−
Key ideas
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Number and Algebra 279
Example 6 Subtracting fractions
Simplify:
a 7
9
2
9
− b 5
6
1
4
−
Solution Explana tion
a
7
9
2
9
5
9
− = Denominators are the same, therefore we are ready to
subtract the second numerator from the first.
b 5
6
1
4
10
12
3
12
7
12
− = −
=
Need to find the LCD, which is 12.
Write equivalent fractions with the LCD.
We have the same denominators now, so subtract the
second numerator from the first.
Example 7 Subtracting mixed numerals
Simplify:
a 5
2
3
3
1
4
− b 8
1
5
4
3
4
−
Solution Explana tion
Method 1: Borrowing a whole number
a 5
2
3
3
1
4
5
2
3
3
1
4
5 3
2
3
1
4
− = +  
 
− +  
 
= − ( ) + −  
 
= + −  
 
=
2
8
12
3
12
2
5
12
Understand that a mixed numeral is the addition of a
whole number and a proper fraction.
Group whole numbers and group proper fractions.
Simplify whole numbers; simplify proper fractions.
Borrowing a whole was not required.
b 8
1
5
4
3
4
8
1
5
4
3
4
7
6
5
4
− = +  
 
− +  
 
= +    −
+
3
4
7 4
6
5
3
4
3
24
20
15
20
 
 
= − + −  
 
= + −  

( )

= 3
9
20
3
4
cannot be taken away from 1
5
easily.
Therefore, we must borrow a whole.
Group whole numbers and group proper fractions.
Simplify whole numbers; simplify proper fractions.
Borrowing a whole was required.
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Chapter 6 Computation with decimals 280 and fractions
Method 2: Converting to an improper fraction
a 5
2
3
3
1
4
17
3
13
4
68
12
39
12
29
12
2
5
12
− = −
= −
=
=
Convert mixed numerals to improper fractions.
Need to fi nd the LCD, which is 12.
Write equivalent fractions with the LCD.
We have the same denominators now, so subtract second
numerator from the fi rst and convert back to improper
fraction.
b 8
1
5
4
3
4
41
5
19
4
164
20
95
20
69
20
3
9
20
− = −
= −
=
=
Convert mixed numerals to improper fractions.
Need to fi nd the LCD, which is 20.
Write equivalent fractions with the LCD.
We have the same denominators now, so subtract second
numerator from the fi rst and convert back to improper
fraction.
Exercise 6C
1 Copy the following sentences into your workbook and fi ll in the blanks.
a To subtract one fraction from another, you must have a common ______________.
b One fail-safe method of producing a common denominator is to simply ______________
the two denominators.
c The problem with fi nding a common denominator that is not the lowest common
denominator is that you have to deal with larger numbers and you also need to ___________
your answer at the fi nal step.
d To fi nd the LCD you can ______________ the denominators and then divide by the HCF of
the denominators.
2 State the LCD for the following pairs of ‘incomplete’ fractions.
a
4 6
− b
2 10
− c
15 5
− d
6 9
−
e
8 12
− f
12 20
− g
14 8
− h
9 21
−
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Number and Algebra 281
3 Copy these equations into your workbook, and fill in the empty boxes.
a
3
7
2
7 7
− = c
1
3
1
4
12 12
12
−
= −
=
d
4
5
2
3
15 15
15
−
= −
=
b
8
13
5
13 13
− =
4 The following equations have been completed, but only six of them are correct. Copy them
into your workbook, then place a tick beside the six correct answers and a cross beside the six
incorrect answers.
a
8
10
5
10
3
10
− = b
3
5
2
3
1
2
− = c
5
12
5
10
5
2
− = d
3
4
1
4
2
4
− =
e
8
11
8
10
0
1
− = = 0 f
12
15
3
15
9
15
− = g 2
3
2
3
− = 0 h
5
7
2
7
2
7
− =
i
3
20
2
20
1
20
− = j 2
5
9
1
4
9
1
1
9
− = k 2
8
14
5
14
2
3
0
− = l
12
21
7
11
5
10
1
2
− = =
5 Simplify:
a
5
7
3
7
− b
4
11
1
11
− c
12
18
5
18
− d
2
3
1
3
−
e
3
5
3
5
− f
6
9
2
9
− g
5
19
2
19
− h
17
23
9
23
−
i
84
100
53
100
− j
41
50
17
50
− k
23
25
7
25
− l
7
10
3
10
−
6 Simplify:
a
2
3
1
4
− b
3
5
1
2
− c
3
5
3
6
− d
4
7
1
4
−
e
1
2
1
3
− f
3
4
1
9
− g
8
11
1
3
− h
4
5
2
3
−
i
3
4
5
8
− j
11
20
2
5
− k
5
12
7
18
− l
7
9
2
3
−
7 Simplify:
a 3
4
5
2
1
5
− b 23
5
7
15
2
7
− c 8
11
14
7
9
14
− d 3
5
9
3
9
−
e 6
2
3
4
1
4
− f 5
3
7
2
1
4
− g 9
5
6
5
4
9
− h 14
3
4
7
7
10
−
Example 6a
Example 6b
Example 7a
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Chapter 6 Computation with decimals 282 and fractions
8 Simplify:
a 5
1
3
2
2
3
− b 8
2
5
3
4
5
− c 13
1
2
8
5
6
− d 12
2
9
7
1
3
−
e 8
5
12
3
3
4
− f 1
3
5
7
9
− g 11
1
11
1
1
4
− h 6
3
20
3
2
3
−
9 Simplify, using a calculator.
a 15 −
1
5
8
4
5
b 27 −
1
3
9
3
4
c − 85
7
10
2
9
d 421 −
1
5
213
3
11
10 Tiffany poured herself a large glass of cordial. She noticed that the cordial jug has 3
4
of a litre
in it before she poured her glass and only 1
5
of a litre in it after she fi lled her glass. How much
cordial did Tiffany pour into her glass?
11 A family block of chocolate is made up of 60 small squares of chocolate. Marcia eats 10 squares,
Jon eats 9 squares and Holly eats 5 squares. What fraction of the block of chocolate is left?
12 Three friends split a restaurant bill. One pays 1
2
of the bill and one pays 1
3
of the bill. What
fraction of the bill must the third friend pay?
13 Patty has 23 1
4
dollars, but owes her parents 15 1
2
dollars. How much money does Patty have left
after she pays back her parents? Repeat this question using decimals and dollars and cents. Do
you get the same answer?
14 Three cakes were served at a birthday party: an ice-cream cake, a chocolate cake and a sponge
cake. 3
4
of the ice-cream cake was eaten. The chocolate cake was cut into 12 equal pieces,
of which 9 were eaten. The sponge cake was divided into 8 equal pieces, with only 1 piece
remaining.
a What fraction of each cake was
eaten?
b What fraction of each cake was left
over?
c What was the total amount of cake
eaten during the party?
d What was the total amount of cake
left over after the party?
Example 7b
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6C
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Number and Algebra 283
15 Fill in the empty boxes to make the following fraction sums correct.
a
1 1 1
12
− = b
5 2
1
10
− =
c 2
3
1
3
2
3
− = d 8
1
6
4
1
1
2
− =
16 Today David’s age is one-seventh of Felicity’s age.
Felicity is a teenager.
a In 1 year’s time David will be one-fifth of
Felicity’s age. What fraction of her age will he be
in 2 years’ time?
b How many years must pass until David is onethird
of Felicity’s age?
c How many years must pass until David is half
Felicity’s age?
17 Simplify:
a Example 7 shows two possible methods for
subtracting mixed numerals: ‘Borrowing a
whole number’ and ‘Converting to an improper
fraction’. Simplify the following two expressions
and discuss which method is the most suitable
for each question.
i 2
1
5
1
2
3
− ii 27
5
11
23
4
5
−
b If you have an appropriate calculator, work out
how to enter fractions and check your answers to
parts i and ii above.
Enrichment: Letter to an absent friend
18 Imagine that a friend in your class is absent for this lesson on the subtraction of fractions. They
were present yesterday and understood the process involved when adding fractions. Your task
is to write a letter to your friend, explaining how to subtract mixed numerals. Include some
examples, discuss both possible methods but also justify your favourite method.
Finish off with three questions for your friend to attempt and include the answers to these
questions on the back of the letter.
WORK ING
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Chapter 6 Computation with decimals 284 and fractions
Multiplying fractions
What does it mean to multiply two fractions together?
Do you end up with a smaller amount or a larger amount when you multiply two proper fractions
together?
What does 1
3
2
3
× equal?
‘Strip’ method
Imagine you have a strip of paper.
You are told to shade
2
3
of the strip.
You are now told to shade in a darker colour
1
3
of your
2
3
strip.
The fi nal amount shaded is your answer.
‘Number line’ method
Consider the number line from 0 to 1 (shown opposite).
It is divided into ninths.
Locate
2
3
.
Divide this position into three equal pieces (shown as ).
To locate
1
3
2
3
× you have only one of the three pieces.
The fi nal location is your answer (shown as ); i.e.
2
9
.
‘Shading’ method
Consider 1
3
of a square multiplied by
2
3
of a square.
‘The rule’ method
When multiplying fractions, multiply the numerators together 1
3
2
3
1 2
3 3
2
9
× = ×
×
=
and multiply the denominators together.
Cutting materials to fi t a purpose may involve multiplying a fraction by a fraction.
6D
2
9
1
1
0
0
1
9
2
9
3
9
4
9
5
9
6
9
7
9
8
9
2
3
1
3
29
× = =
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Number and Algebra 285
Let’s start: ‘Clock face’ multiplication
Explain and discuss the concept of fractions of an hour on the
clock face.
In pairs, students match up the following 10 ‘clock face’
multiplication questions with their correct answer. You may like to
place a time limit of 5 minutes on the activity.
Discuss answers at the end of the activity.
Questions Answers
1
1
2
of 4 hours A 25 minutes
2
1
3
of 2 hours B 1
1
2
hours
3
1
4
of 6 hours C 5 minutes
4
1
3
of
1
4
hour D
1
4
hour
5
1
4
of
1
3
hour E 2 hours
6
1
3
of
3
4
hour F 2 hours 40 minutes
7
1
10
of
1
2
hour G
1
12
hour
8
1
5
of
1
2
hour H 40 minutes
9
2
3
of 4 hours I
1
10
hour
10
5
6
of
1
2
hour J 3 minutes
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Chapter 6 Computation with decimals 286 and fractions
■■ Fractions do not need to have the same denominator to be multiplied together.
■■ To multiply fractions, multiply the numerators together and multiply the denominators together.
–– In symbols: a
b
c
d
a c
b d
× = ×
×
■■ If possible, ‘simplify’, ‘divide’ or ‘cancel’ fractions before multiplying.
–– Cancelling can be done vertically or diagonally.
–– Cancelling can never be done horizontally.
× 3
5
4
8
1
2
cancelling vertically ✓
× 3
5
4
6
1
2
cancelling diagonally ✓
1 2
3
5
6
7
× cancelling horizontally ✕
■■ A whole number can be written as a fraction with a denominator of 1.
■■ ‘of ’, ‘×’, ‘times’, ‘lots of’ and ‘product’ all refer to the same mathematical operation of multiplying.
■■ Mixed numerals must be changed to improper fractions before multiplying.
■■ Final answers should be written in simplest form.
Key ideas
Example 8 Finding a simple fraction of a quantity
Find:
a 2
3
of 15 bananas b 3
10
of 50 lollies
Solution Explana tion
a
2
3
of 15 bananas
1
3
2 10 of 15  
 
× =
Divide 15 bananas into 3 equal groups.
Therefore, 5 in each group.
Take 2 of the groups.
Answer is 10 bananas.
b 3
10
of 50 lollies
1
10
of 50 3 15

 

 
× =
Divide 50 into 10 equal groups.
Therefore, 5 in each group.
Take 3 of the groups.
Therefore, answer is 15 lollies.
Never do this!
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Number and Algebra 287
Example 9 Multiplying proper fractions
Find:
a
2
3
1
5
× b
3
4
8
9
× c
4
8
3
6
of
Solution Explana tion
a 2
3
1
5
2 1
3 5
2
15
× = ×
×
=
Multiply the numerators together.
Multiply the denominators together.
The answer is in simplest form.
b 3
4
8
9
3 8
4 9
2
3
1 2
1 3 × = ×
×
=
Cancel first.
Then multiply numerators together and denominators
together.
c = ×
= ×
×
=
4
8
of
3
6
4
8
3
6
4 3
8 6
1
4
1 1
2 2
Change ‘of’ to multiplication sign.
Cancel and then multiply the numerators and the
denominators.
The answer is in simplest form.
Example 10 Multiplying proper fractions by whole numbers
Find:
a 1
3
× 21 b 2
5
of 32
Solution Explana tion
a 1
3
21
1
3
21
1
7
1
7
1
7
× = ×
=
=
Rewrite 21 as a fraction with a denominator equal to 1.
Cancel and then multiply numerators and denominators.
7 ÷ 1 = 7
b = ×
=
=
2
5
of 32
2
5
32
1
64
5
12
4
5
Rewrite ‘of’ as a multiplication sign.
Write 32 as a fraction.
Multiply numerators and denominators.
Convert answer to a mixed numeral.
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Chapter 6 Computation with decimals 288 and fractions
Example 11 Multiplying improper fractions
Find:
a 5
3
7
2
× b 8
5
15
4
×
Solution Explana tion
a 5
3
7
2
5 7
3 2
35
6
5
5
6
× = ×
×
= =
Multiply the numerators together.
Multiply the denominators together.
Convert the answer to a mixed numeral.
b 8
5
15
4
8 15
5 4
6
1
6
2 3
1 1 × = ×
×
= =
Cancel first.
Multiply ‘cancelled’ numerators together and ‘cancelled’
denominators together.
Write the answer in simplest form.
Example 12 Multiplying mixed numerals
Find:
a 2
1
3
1
2
5
× b 6
1
4
2
2
5
×
Solution Explana tion
a 2
1
3
1
2
5
7
3
7
5
49
15
3
4
15
× = ×
=
=
Convert mixed numerals to improper fractions.
Multiply numerators together.
Multiply denominators together.
Write the answer in simplest form.
b 6
1
4
2
2
5
25
4
12
5
15
1
15
5
1
3
1 × = ×
=
=
Convert to improper fractions.
Simplify fractions by cancelling.
Multiply numerators and denominators together.
Write the answer in simplest form.
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Number and Algebra 289
1 Copy these sentences into your workbook and fi ll in the blanks.
a A proper fraction has a value that is between _______ and ________.
b An improper fraction is always greater than _________.
c A mixed numeral consists of two parts, a __________ ___________ part and a ___________
______________ part.
2 When multiplying a whole number by a proper fraction, do you get a smaller or larger answer when
compared with the whole number? Explain your answer.
3 Copy into your workbook the grid shown opposite.
a On your diagram, shade in blue 1
3
of the grid.
b Now shade in red 1
4
of the shaded blue.
c You have now shaded 1
4
of 1
3
. What fraction is this of the original grid?
4 Use drawings to show the answers to these problems.
a
1
3
of 12 lollies b
1
5
of 10 pencils c
2
3
of 18 donuts
d
3
4
of 16 boxes e
3
8
of 32 dots f
3
7
of 21 triangles
5 One of the following four methods is the correct solution to the problem
1
2
1
5
× . Find the correct
solution and copy it into your workbook.
A 1
2
1
5
1 1
2 5
2
7
×
= +
+
=
B 1
2
1
5
1 1
2 5
2
10
×
= ×
×
=
C
1
2
1
5
5
10
2
10
7
20
×
= ×
=
D
1
2
1
5
1 1
2 5
1
10
×
= ×
×
=
6 Evaluate:
a
3
4
1
5
× b
2
7
1
3
× c
2
3
5
7
× d
4
9
2
5
×
e
2
3
3
5
× f
4
7
1
4
× g
3
4
1
3
× h
5
9
9
11
×
i
3
6
5
11
× j
2
3
4
8
× k
8
11
3
4
× l
2
5
10
11
×
m
2
7
of
3
5
n
3
4
of
2
5
o
5
10
of
4
7
p
6
9
of
3
12
Example 8
Example 9
Exercise 6D
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WORKING
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Chapter 6 Computation with decimals 290 and fractions
7 Find:
a 1
3
of 18 b 1
5
of 45 c 2
3
of 24 d 3
5
of 25
e 2
7
of 42 f 1
4
of 16 g 4
5
of 100 h 3
7
of 77
8 Find:
a 5
2
7
3
× b 6
5
11
7
× c 6
4
11
5
× d 9
6
13
4
×
e 8
5
10
3
× f 21
4
8
6
× g 10
7
21
5
× h 14
9
15
7
×
9 Find:
a 1
3
5
2
1
3
× b 1
1
7
1
2
9
× c 3
1
4
2
2
5
× d 4
2
3
5
1
7
×
10 Find:
a
6
5
8
3
× b
1
2
3
8
× c
3
4 3
of 5
1
d 7
1
2
4
2
5
×
e 3
7
of
2
3
f 1
1
2
2
1
4
× g 8
9
6
20
× h 15
4
8
5
×
11 Repeat Question 10, using a calculator. Don’t forget to estimate your answer fi rst, so that you can
pick up any calculator keystroke errors.
Example 10
Example 11
Example 12
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WORKING
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12 At a particular secondary college,
2
5
of the Year 7 students are boys.
a What fraction of the Year 7 students are girls?
b If there are 120 Year 7 students, how many boys and
girls are there?
13 To paint one classroom, 2
1
3
litres of paint are required.
How many litres of paint are required to paint fi ve
identical classrooms?
14 A scone recipe requires1
3
4
cups of self-raising fl our
and 3
4
of a cup of cream. James is catering for a large
group and needs to quadruple the recipe. How much
self-raising fl our and how much cream will he need?
15 Julie has fi nished an injury-plagued netball season
during which she was able to play only 2
3
of the matches.
The season consisted of 21 matches. How many games
did Julie miss as a result of injury?
6D
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Number and Algebra 291
16 Not all of the following fraction equations are correct. Copy them into your workbook, then
place a tick beside those that are correct and a cross beside those that are wrong. Provide the
correct solution for those you marked as incorrect.
a
1
3
1
4
1
7
+ = b
1
3
1
4
1
12
+ = c
1
3
1
4
2
7
× =
d
1
3
1
4
1
12
× = e
1
3
1
4
1
12
− = f
1
3
1
4
0
1
− =
−
17 Circle the correct alternative for the following statement and justify your answer. Using an
example, explain why the other alternatives are incorrect.
When multiplying a proper fraction by another proper fraction the answer is...
A a whole number B a mixed numeral
C an improper fraction D a proper fraction
18 Write two fractions that:
a multiply to 3
5
b multiply to 3
4
c multiply to 1
7
WORKING
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Enrichment: Who are we?
19 a Using the clues provided, work out which two fractions are being discussed.
• We are two proper fractions.
• Altogether we consist of four different digits.
• When added together our answer will still be a proper fraction.
• When multiplied together you could carry out some cancelling.
• The result of our product, when simplified, contains no new digits from our original four.
• Three of our digits are prime numbers and the fourth digit is a cube number.
b Design your own similar question and develop a set of appropriate clues. Ask a classmate to
answer your question.
c Design the ultimate challenging ‘Who are we?’ question. Make sure that there is only one
possible answer.
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Chapter 6 Computation with decimals 292 and fractions
Multiplying and dividing decimals by 10, 100, 1000 etc.
In this section, we will be multiplying decimals by
numbers such as 10, 100, 100 etc. We will be using
the phrase ‘powers of 10’ to represent numbers, such
as those shown opposite.
Note that 10 to the power of 3 is 1000, which contains three zeros.
This pattern continues, so 10 to the power of 6 is written as 1 followed by 6 zeros.
let’s start: Does the decimal point really ‘move’?
Consider the number 2.58.
Working from left to right:
• The digit 2 is in the units column.
• The decimal point sits between the units and the
tenths, as it always does.
• The digit 5 is in the tenths column.
• The digit 8 is in the hundredths
column.
Hundreds tens units Decimal point tenths Hundredths thousandths
2 • 5 8
Now, imagine that you buy 10 items for $2.58 each. The cost is $25.80.
• Did the digits change?
• Did the decimal point move? If so, which way and how many places?
• Or was it that the digits moved and the decimal point stayed still?
• If so, which way did the digits move? By how many places?
6E
= =
= = × =
= = × × =
10 to the power of 1 10 10
10 to the power of 2 10 10 10 100
10 to the power of 3 10 10 10 10 1000
1
2
3
■■ Every number contains a decimal point but it is usually not shown in integers.
For example: 345 is 345.0 and 2500 is 2500.0.
■■ Extra zeros can be added in the columns to the right of the decimal point without changing the
value of the decimal.
For example: 12.5 = 12.50 = 12.500 = 12.5000 etc.
■■ When a decimal is multiplied by 10 (which is 10 to the power of 1), the decimal point stays still
and the digits all move 1 place to the left.
However, it is easier to ‘visualise the decimal point moving’ 1 place to the right.
For example: 23 ↓. 758 × 10 = 237.58
operation Visualisation Example
Multiplying a decimal by 1000
(10 to the power of 3)
Decimal point moves 3 places to
the right.
23.758 × 1000 = 23758
Multiplying a decimal by 100
(10 to the power of 2)
Decimal point moves 2 places to
the right.
23.758 × 100 = 2375.8
Multiplying a decimal by 10
(10 to the power of 1)
Decimal point moves 1 place to
the right.
23.758 × 10 = 237.58
Dividing a decimal by 10
(10 to the power of 1)
Decimal point moves 1 place to
the left.
23.758 ÷ 10 = 2.3758
Key ideas
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Number and Algebra 293
Example 13 Multiplying by powers of 10
Evaluate:
a 36.532 × 100 b 4.31 × 10 000
Solution Explana tion
a 36.532 × 100 = 3653.2 100 has two zeros, therefore the decimal point appears to
move 2 places to the right. 36.532
b 4.31 × 10 000 = 43 100 Decimal point appears to move 4 places to the right and
additional zeros are inserted as necessary. 4.3100
Example 14 Dividing by powers of 10
Evaluate:
a 268.15 ÷ 10 b 7.82 ÷ 1000
Solution Explana tion
a 268.15 ÷ 10 = 26.815 10 has one zero, therefore the decimal point appears to
move 1 place to the left. 268.15
b 7.82 ÷ 1000 = 0.00782 Decimal point appears to move 3 places to the left and
additional zeros are inserted as necessary. .00782
Example 15 Working with ‘the missing’ decimal point
Evaluate:
a 567 × 10000 b 23 ÷ 1000
Solution Explana tion
a 567 × 10 000 = 5 670 000 If no decimal point is shown in the question, it must be
at the very end of the number. Four additional zeros must
be inserted to move the invisible decimal point 4 places
to the right.
5670000.
b 23 ÷ 1000 = 0.023 Decimal point appears to move 3 places to the left. 0.023
Dividing a decimal by 100
(10 to the power of 2)
Decimal point moves 2 places to
the left.
23.758 ÷ 100 = 0.23758
Dividing a decimal by 1000
(10 to the power of 3)
Decimal point moves 3 places to
the left.
23.758 ÷ 1000 = 0.023758
Key ideas
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Chapter 6 Computation with decimals 294 and fractions
Example 16 Evaluating using order of operations
Calculate this expression, using the order of operations:
426 ÷ 100 + 10(0.43 × 10 − 1.6)
SolutioN ExplANAtioN
426 ÷ 100 + 10(0.43 × 10 − 1.6)
= 4.26 + 10(4.3 − 1.6)
= 4.26 + 10 × 2.7
= 4.26 + 27
= 31.26
First, we must calculate the brackets.
The division by 100 can also be done in the fi rst step.
10(4.3 − 2.6) means 10 × (4.3 − 2.6).
Exercise 6E
1 Fill in the correct number of zeros in the multiplier to make the following product statements
correct. The fi rst one has been done for you.
a 56.321 × 1 0 0 = 5632.1
b 27.9234 × 1 = 27 923.4
c 0.03572 × 1 = 3.572
d 3200 × 1 = 320 000 000
2 Fill in the correct number of zeros in the divisor to make the following division statements
correct. The fi rst one has been done for you.
a 2345.1 ÷ 1 0 0 0 = 2.3451
b 7238.4 ÷ 1 = 72.384
c 0.00367 ÷ 1 = 0.000367
d 890 ÷ 1 = 0.0089
3 a How many places and in what direction must the decimal point in the number move if the
following operations occur?
i × 100 ii ÷ 10 iii × 1 000 000 iv ÷ 1
v ÷ 1000 vi × 1000 vii × 10 viii ÷ 10 000 000
b If all of the operations above had taken place on a number, one after the other, what would
be the fi nal position of the decimal place relative to its starting position?
4 Calculate:
a 4.87 × 10 b 35.283 × 10 c 422.27 × 10
d 14.304 × 100 e 5.69923 × 1000 f 1.25963 × 100
g 12.7 × 1000 h 154.23 × 1000 i 0.34 × 10 000
j 213.2 × 10 k 867.1 × 100 000 l 0.00516 × 100 000 000
Example 13
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Number and Algebra 295
5 Calculate:
a 42.7 ÷ 10 b 353.1 ÷ 10 c 24.422 ÷ 10
d 5689.3 ÷ 100 e 12 135.18 ÷ 1000 f 93 261.1 ÷ 10 000
g 2.9 ÷ 100 h 13.62 ÷ 10 000 i 0.54 ÷ 1000
j 36.7 ÷ 100 k 0.02 ÷ 10 000 l 1000.04 ÷ 100 000
6 Calculate:
a 22.913 × 100 b 0.03167 × 1000 c 4.9 ÷ 10
d 22.2 ÷ 100 e 6348.9 × 10 000 f 1.0032 ÷ 1000
7 Calculate:
a 156 × 100 b 43 × 1000 c 2251 ÷ 10
d 16 ÷ 1000 e 2134 × 100 f 2134 ÷ 100
g 7 ÷ 1000 h 99 × 100 000 i 34 ÷ 10 000
8 Calculate the following, using the order of operations.
a 1.56 × 100 + 24 ÷ 10 b 16 ÷ 100 + 32 ÷ 10
c 3 + 10(24 ÷ 100 + 8) d 10(6.734 × 100 + 32)
e 35.4 + 4.2 × 10 − 63.4 ÷ 10 f 4.7 − 24 ÷ 10 + 0.52 × 10
g 14 ÷ 100 + 1897 ÷ 1000 h 78.1 − 10(64 ÷ 100 + 5)
9 A service station charges $1.47 per litre of petrol. How much will it cost Tanisha to fill her car
with 100 litres of petrol?
10 A large bee farm produces 1200 litres of honey per day.
a If there are 1000 millilitres in 1 litre, how many millilitres of honey can the farm’s bees
produce in one day?
b The farm’s honey is sold in 100 millilitre jars. How many jars of honey can the farm’s bees
fill in one day?
11 Wendy is on a mobile phone plan that charges
her 3 cents per text message. On average, Wendy
sends 10 text messages per day. What will it cost
Wendy for 100 days of sending text messages
at this rate? Give your answer in cents and then
convert your answer to dollars.
12 Darren wishes to purchase 10 000 shares at $2.12 per share. Given that there is also an
additional $200 brokerage fee, how much will it cost Darren to purchase the shares?
Example 14
Example 15
Example 16
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Chapter 6 Computation with decimals 296 and fractions
13 The weight of a matchstick is 0.00015 kg. Find the weight of 10 000 boxes of matches, with
each box containing 100 matches. The weight of one empty match box is 0.0075 kg.
14 Complete the table below, listing at least one possible combination of operations that would
produce the stated answer from the given starting number. Justify your answers to a friend.
Starting number Answer possible two-step operations
12.357 1235.7 × 1000, ÷ 10
34.0045 0.0340045
0.003601 360.1
bac.dfg ba.cdfg ÷ 100, × 10
d.swkk dswkk
fwy f.wy
15 The number 12 345.6789 undergoes a series of multiplication and division operations by
different powers of 10. The fi rst four operations are: ÷ 1000, × 100, × 10 000 and ÷ 10. What is
the fi fth and fi nal operation if the fi nal number is 1.23456789?
Enrichment: Standard form
16 Extremely large numbers and extremely small numbers are often written in a more practical
way, known as standard form or scientifi c notation.
For example, the distance from the Earth to the Sun is 150 000 000 kilometres! The distance of
150 million kilometres can be written in standard form as 1.5 × 108 kilometres.
On a calculator, 150 000 000 can be represented as 1.5E8.
1.5 × 108 and 1.5E8 represent the same large number and indicate that the decimal place needs
to be moved 8 places to the right.
1.5E8 = 1.5 × 108 = 1.5 × 100 000 000
= 150000000 .
a Represent these numbers in standard form.
i 50 000 000 000 000 ii 42 000 000 iii 12 300 000 000 000 000
b Use a calculator to evaluate the following.
i 40 000 000 000 × 500 000 000 ii 9 000 000 × 120 000 000 000 000
c The distance from the Earth to the Sun is stated above as 150 million kilometres. The more
precise fi gure is 149 597 892 kilometres. Research how astronomers can calculate the
distance so accurately. Hint: It is linked to the speed of light.
d Carry out further research on very large numbers. Create a list of 10 very large numbers
(e.g. distance from Earth to Pluto, the number of grains in 1 kg of sand, the number of stars
in the galaxy, the number of memory bytes in a terabyte…). Rank your 10 large numbers in
ascending order.
e How are very small numbers, such as 0.000000000035, represented in standard form?
f Represent the following numbers in standard form.
i 0.000001 ii 0.0000000009 iii 0.000000000007653
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Number and Algebra 297
Multiplying by a decimal
There are countless real-life applications that involve the
multiplication of decimal numbers. For example, fi nding the area
of a block of land that is 34.5 m long and 5.2 m wide, or pricing a
4.5-hour job at a rate of $21.75 per hour. In general, the procedure
for multiplying decimal numbers is the same as multiplying whole
numbers. There is, however, one extra fi nal step, which involves
placing the decimal point in the correct position in the answer.
let’s start: Multiplication musings
Consider the following questions within your group.
• What happens when you multiply by a number that is
less than 1?
• Consider the product 15 × 0.75. Will the answer be more or
less than 15? Why?
• Estimate an answer to 15 × 0.75.
• What is the total number of decimal places in the numbers 15 and 0.75?
• Calculate 15 × 0.75. How many decimal places are there in the answer?
6F
Finding the area of a block of land usually
involves multiplying decimals.
■■ When multiplying decimals, start by ignoring any decimal points and perform the
multiplication as you would normally. On arriving at your answer, you must now place the
decimal point in the correct position.
■■ The correct position of the decimal point in the answer is found by following the rule that the
total number of decimal places in the question must equal the number of decimal places in the
answer.
For example: 5.34 × 1.2 3 decimal places in the question
534
12
1068
5340
6408
×
■■ It is always worthwhile estimating your answer. This allows you to check that your decimal
point is in the correct place and that your answer makes sense.
■■ When multiplying by multiples of 10, initially ignore the zeros in the multiplier and any
decimal points and perform routine multiplication. On arriving at your answer, position
your decimal point, remembering to move your decimal point according to the rules of
multiplying by powers of 10.
Key ideas
5.34 × 1.2 = 6.408 3 decimal places in the answer
decimal points
ignored here
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Chapter 6 Computation with decimals 298 and fractions
Example 17 Multiplying decimals
Calculate:
a 12.31 × 7 b 3.63 × 6.9
Solution Explana tion
a 1231
7
8617
×
12.31 × 7 = 86.17
Perform multiplication, ignoring decimal point.
There are 2 decimal places in the question, so there will
be 2 decimal places in the answer.
Estimation is less than 100 (≈ 12 × 7 = 84).
b 363
69
3267
21780
25047
×
3.63 × 6.9 = 25.047
Ignore both decimal points.
Perform routine multiplication.
Total of 3 decimal places in the question, so there must
be 3 decimal places in the answer.
Estimation is less than 28 (≈ 4 × 7 = 28).
Example 18 Multiplying decimals by multiples of 10
Calculate:
a 2.65 × 40 000 b 0.032 × 600
Solution Explana tion
a 2.65 × 40 000 = 106 000
265
4
1060
×
 10.60 × 10 000 = 106000.
Ignore the decimal point and zeros.
Multiply 265 × 4.
Position the decimal place in your answer.
There are 2 decimal places in the question, so must have
2 decimal places in the answer.
Move the decimal point 4 places to the right.
b 0.032 × 600 = 19.2
32
6
192
×
 0.192 × 100 = 19.2
Ignore the decimal point and zeros.
Multiply 32 × 6.
Position decimal place in the answer.
Shift decimal place 2 places to the right because there
are two zeros in the question.
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Number and Algebra 299
1 Work out the total number of decimal places in each of the following product statements.
a 4 × 6.3 b 3.52 × 76 c 42 × 5.123
d 8.71 × 11.2 e 5.283 × 6.02 f 2.7 × 10.3
g 4.87 × 3241.21 h 0.003 × 3 i 0.00103 × 0.0045
2 Insert the decimal point into each of the following answers so that the multiplication is true.
a 6.4 × 3 = 192 b 6.4 × 0.3 = 192 c 0.64 × 0.3 = 192
3 Why is it worthwhile to estimate an answer to a multiplication question involving decimals?
4 a What is the difference between a decimal point and a decimal place?
b How many decimal points and how many decimal places are in the number 423.1567?
5 Copy and complete the rule for multiplying decimal numbers (see the Key Ideas in the section).
The total number of decimal places ________________________ must equal the number of
_______________________ in the answer.
6 Calculate:
a 5.21 × 4 b 3.8 × 7 c 22.93 × 8
d 14 × 7.2 e 3 × 72.82 f 1.293 × 12
g 3.4 × 6.8 h 5.4 × 2.3 i 0.34 × 16
j 43.21 × 7.2 k 0.023 × 0.042 l 18.61 × 0.071
7 Calculate:
a 2.52 × 40 b 6.9 × 70 c 31.75 × 800
d 1.4 × 7000 e 3000 × 4.8 f 7.291 × 50 000
g 0.0034 × 200 h 0.0053 × 70 000 i 3.004 × 30
8 Calculate and then round your answer to the nearest dollar. Check your answers using a calculator.
a 5 × $6.30 b 3 × $7.55 c 4 × $18.70
d $1.45 × 12 e $30.25 × 4.8 f 7.2 × $5200
g 34.2 × $2.60 h 0.063 × $70.00 i 0.085 × $212.50
9 Anita requires 4.21 m of material for each dress she is making.
She is planning to make a total of seven dresses. How much
material does she need?
10 The net weight of a can of spaghetti is 0.445 kg. Find the net
weight of eight cans of spaghetti.
11 Jimbo ran 5.35 km each day for the month of March. How many
kilometres did he run for the month?
Example 17
Example 18
Exercise 6F
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Chapter 6 Computation with decimals 300 and fractions
12 Bernard is making a cubby house for his children. He needs 32 lengths of timber, each
2.1 metres long.
a What is the total length of timber needed to build the cubby house?
b What is the cost of the timber if the price is $2.95 per metre?
13 A lawyer charges $125.00 per hour to assist her client. How much does the lawyer charge the
client if she works on the job for 12.25 hours?
14 According to its manufacturer, a particular car can travel 14.2 km on 1 litre of petrol.
a How far could the car travel on 52 litres of petrol?
b The car has 23.4 litres of fuel in the tank and must complete a journey of 310 km. Will it
make the journey without refuelling?
c If the car does make the journey, how much petrol is left in the tank at the end of the trip? If
the car doesn’t make the journey, how many extra litres of fuel is needed?
15 Write down two numbers, each with 2 decimal places, that when multiplied by 1.83 will give an
answer between 0.4 and 0.5.
16 Write down one number with 4 decimal places that when multiplied by 345.62 will give an
answer between 1 and 2.
17 a If 68 × 57 = 3876, what is the answer to 6.8 × 5.7? Why?
b If 23 × 32 = 736, what is the answer to 2.3 × 32? Why?
c If 250 × 300 = 75 000, what is the answer to 2.5 × 0.3? Why?
d What is 7 × 6? What is the answer to 0.7 × 0.6? Why?
Enrichment: Creating a simple cash register
18 Using a spreadsheet program, such as Excel, design a user-friendly
cash register interface. You must be able to enter up to 10 different
items into your spreadsheet. You will need a quantity column and a cost
per item column.
Using appropriate formulae, the total cost of the bill should be
displayed, and there should then be room to enter the amount of money
paid and, if necessary, what change should be given.
When your spreadsheet is set up, enter the following items.
4 chocolate bars @$1.85 each toothpaste @$4.95
3 loaves of bread @$3.19 each 2 kg sausages @$5.99 per kg
newspaper @$1.40 tomato sauce @$3.20
2 × 2 litres of milk @$3.70 each 2 packets of Tim Tams@$3.55 each
washing powder @$8.95 5 × 1.25 litres of soft drink @$0.99 each
Money paid = $80.00
If your program is working correctly, the amount of change given should be $13.10.
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Number and Algebra 301
Dividing fractions
Remember that division used to be referred to as ‘how many’.
Thinking of division as ‘how many’ helps us to understand dividing fractions.
For example, to fi nd
1
2
1
4
÷ , think of 1
2
how many 1
4
s, or how many 1
4
s are in a 1
2
?
Consider this strip of paper that is divided into four equal sections.
In our example of 1
2
1
4
÷ , we have only 1
2
a strip, so we will
shade in half the strip.
By thinking of the ÷ sign as ‘how many’, the question is asking how many quarters are in half the strip.
From our diagram, we can see that the answer is 2. Therefore, 1
2
1
4
÷ = 2.
In a game of football, when it is half-time, you have played two quarters. This is another way of
confi rming that 1
2
1
4
÷ = 2.
let’s start: ‘Divvy up’ the lolly bag
To ‘divvy up’ means to divide up or divide out
or share equally.
Consider a lolly bag containing 24 lollies.
In pairs, students answer the following questions.
• How many lollies would each person get if
you ‘divvy up’ the lollies between three
people?
• If you got
1
3
of the lollies in the bag, how
many did you get?
Can you see that ‘divvying up’ by 3 is the same
as getting 1
3
? Therefore, ÷ 3 is the same as × 1
3
.
• How many lollies would each person get if you ‘divvy
up’ the lollies between eight people?
• If you got 1
8
of the lollies in the bag, how many did you get?
Can you see that ‘divvying up’ by 8 is the same as getting 1
8
? Therefore, ÷ 8 is the same as × 1
8
.
• What do you think is the same as dividing by n?
• What do you think is the same as dividing by a
b
?
6G
How many ways can these 24 lollies be divided?
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Chapter 6 Computation with decimals 302 and fractions
■■ To find the reciprocal of a fraction, you must invert the fraction. This is done by swapping the
numerator and the denominator. ‘Inverting’ is sometimes known as turning the fraction upside
down or flipping the fraction.
–– The reciprocal of a
b
b
a
is ..
For example: The reciprocal of 3
5
is 5
3
.
■■ Dividing by a number is the same as multiplying by its reciprocal.
For example: 15 ÷ 3 = 5 and 15
1
3
× = 5..
–– Dividing by 2 is the same as multiplying by 1
2
.
■■ When asked to divide by a fraction, instead choose to multiply by the fraction’s reciprocal.
Therefore, to divide by a
b
we multiply by b
a
.
■■ When dividing, mixed numbers must be changed to improper fractions.
Key ideas
Example 19 Finding reciprocals
State the reciprocal of the following.
a 2
3
b 5 c 1
3
7
Solution Explana tion
a Reciprocal of 2
3
is 3
2
. The numerator and denominator are swapped.
b Reciprocal of 5 is 1
5
. Think of 5 as 5
1
and then invert.
c Reciprocal of 1
3
7
is 7
10
. Convert 1
3
7
to an improper fraction; i.e. 10
7
,
and then invert.
Example 20 Dividing a fraction by a whole number
Find:
a 5
8
÷ 3 b 2
3
11
÷ 5
Solution Explana tion
a 5
8
3
5
8
1
3
5
24
÷ = ×
=
Change the ÷ sign to a × sign and invert the 3.
Multiply the numerators and denominators.
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Number and Algebra 303
b 2
3
11
5
25
11
5
1
25
11
1
5
5
11
5
1
÷ = ÷
= ×
=
Convert the mixed numeral to an improper fraction.
Write 5 as an improper fraction.
Change the ÷ sign to a × sign and invert the divisor.
Simplify by cancelling.
Multiply numerators and denominators.
Example 21 Dividing a whole number by a fraction
Find:
a 6
1
3
÷ b 24
3
4
÷
Solution Explana tion
a 6
1
3
6
1
3
1
18
1
18
÷ = ×
= =
Instead of ÷ 1
3
, change to × 3
1
.
Simplify.
b 24
3
4
24
1
4
3
32
8
1 ÷ = ×
=
Instead of ÷ 3
4
, change to × 4
3
.
Cancel and simplify.
Example 22 Dividing fractions by fractions
Find:
a 3
5
3
8
÷ b 2
2
5
1
3
5
÷
Solution Explana tion
a 3
5
3
8
3
5
8
3
8
5
1
3
5
÷ = ×
= =
Change the ÷ sign to a × sign and invert the divisor.
(Note: The divisor is the second fraction.)
Cancel and simplify.
b 2
2
5
1
3
5
12
5
8
5
12
5
5
8
3
2
1
1
2
3
1
1
2
÷ = ÷
= ×
= =
Convert mixed numerals to improper fractions.
Change the ÷ sign to a × sign and invert the divisor.
Cancel, multiply and simplify.
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Chapter 6 Computation with decimals 304 and fractions
1 Which of the following is the correct fi rst step for fi nding 3
5
4
7
÷ ?
A
3
5
7
4
× B
5
3
4
7
× C
5
3
7
4
×
2 Write the correct fi rst step for each of these division questions. (Do not go on and fi nd the
fi nal answer.)
a 5
11
3
5
÷ b 1
3
1
5
÷ c
7
10
12
17
÷ d
8
3
÷ 3
3 When dividing mixed numerals, the fi rst step is to convert to improper fractions and the second
step is to multiply by the reciprocal of the divisor. Write the correct fi rst and second steps
for each of the following mixed numeral division questions. (Do not go on and fi nd the fi nal
answer.)
a 2
1
2
1
1
3
÷ b 24 3
1
5
÷ c 4
3
11
5
1
4
÷ d
8
3
11
3
7
÷
4 Make each sentence correct, by inserting the word more or less in the gap.
a 10 ÷ 2 gives an answer that is ________ than 10.
b 10
1
2
÷ gives an answer that is ________ than 10.
c 3
4
2
3
÷ gives an answer that is ________ than 3
4
.
d 3
4
3
2
× gives an answer that is ________ than 3
4
.
e 5
7
8
5
÷ gives an answer that is ________ than 5
7
.
f 5
7
5
8
× gives an answer that is ________ than 5
7
.
5 State the reciprocal of each of the following.
a
5
7
b
3
5
c
2
9
d
1
8
e 2
1
3
f 4
3
5
g 1
5
6
h 8
2
3
i 12 j 101 k
1
9 l 1
6 Find:
a
3
4
÷ 2 b
5
11
÷ 3 c
8
5
÷ 4 d
15
7
÷ 3
e 2
1
4
÷ 3 f 5
1
3
÷ 4 g 12
4
5
÷ 8 h 1
13
14
÷ 9
Example 19
Example 20
Exercise 6G
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Number and Algebra 305
7 Find:
a 5
1
4
÷ b 7
1
3
÷ c 10
1
10
÷ d 24
1
5
÷
e 12
2
5
÷ f 15
3
8
÷ g 14
7
2
÷ h 10
3
2
÷
8 Find:
a
2
7
2
5
÷ b
1
5
1
4
÷ c
3
7
6
11
÷ d
2
3
8
9
÷
e 2
1
4
1
1
3
÷ f 4
1
5
3
3
10
÷ g 12
1
2
3
3
4
÷ h 9
3
7
12
4
7
÷
9 Find:
a
3
8
÷ 5 b 22
11
15
÷ c 2
2
5
1
3
4
÷ d
3
4
9
4
÷
e 7
1
4
÷ f 2
6
15
÷ 9 g 7
2
3
1
1
6
÷ h
3
5
2
7
÷
10 Repeat Question 9, using a calculator. Don’t forget to estimate your answer first, so that you
pick up any calculator keystroke errors.
11 If 2
1
4
leftover pizzas are to be shared between three friends, what fraction of pizza will each
friend receive?
12 A property developer plans to subdivide 7
1
2
acres of land into blocks of at least
3
5
of an
acre. Through some of the land runs a creek, where a protected species of frog lives. How
many of the blocks can the developer sell if two blocks must be reserved for the creek and
its surroundings?
13 Miriam cuts a 10-millimetre sisal rope into four equal pieces. If the rope is 3
3
5
metres long
before it is cut, how long is each piece?
14 A carpenter takes
3
4
of an hour to make a chair. How
many chairs can he make in 6 hours?
15 Justin is a keen runner and regularly runs at a
pace of 3
1
2
minutes per kilometre. Justin
finished a Sunday morning run in 77 minutes.
How far did he run?
Example 21
Example 22
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Chapter 6 Computation with decimals 306 and fractions
16 Pair up the equivalent expressions and state the simplifi ed answer.
1
2
of 8 12 ÷ 4 10
1
2
× 10 ÷ 2
3
1
2
÷ 12
1
4
× 1
2
1
8
÷ 3 × 2
17 Find:
a
3
8
4
5
2
3
× ÷ b
3
8
4
5
2
3
÷ ÷ c
3
8
4
5
2
3
÷ × d
3
8
4
5
2
3
× ×
18 a A car travels 180 kilometres in 1
1
2
hours. How far will it travel in 2 hours if it travels at a
constant speed?
b A different car took 2
1
4
hours to travel 180 kilometres. How far did it travel in 2 hours, if it
maintained a constant speed?
Enrichment: You provide the question
19 Listed below are six different answers.
You are required to make up six questions that will result in the following six answers.
All questions must involve a division sign. Your questions should increase in order of diffi culty by
adding extra operation signs and extra fractions.
a Answer 1:
3
5 b Answer 2: 2
1
3
c Answer 3:
7
1
d Answer 4: 0 e Answer 5: 1
100
f Answer 6: 4
4
5
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Number and Algebra 307
Dividing decimals
Similar to multiplication of decimal numbers, there are countless real-life applications that involve the
division of decimal numbers. However, unlike multiplying decimal numbers, where we basically ignore
the decimal points until the very end of the question, with division we try to manipulate the question in
such a way as to prevent dividing by a decimal number.
Terminology reminders:
Example: 24 ÷ 4 = 6 or
24
4
= 6 or 4)24
6
24 is known as the dividend (the amount you have or the number being divided), 4 is known as the
divisor (the number doing the dividing) and 6 is known as the quotient (or the answer).
let’s start: Division decisions
Consider the following questions within your group.
• What happens when you divide by a number that is less than 1?
• Consider the answer of 10 ÷ 0.2. Will the answer be more or less than 10? Why?
• Estimate an answer to 10 ÷ 0.2.
• Calculate the answer of 100 ÷ 2. How does this compare to the answer of 10 ÷ 0.2?
• Can you think of an easier way to calculate 21.464 ÷ 0.02?
6H
■■ Division of decimal numbers by whole numbers
– Complete as you would normally with any other division question.
– The decimal point in the quotient (answer) goes directly above the decimal point in the
dividend.
For example: 60.524 ÷ 4
4)60 5 24
15 1 31
2 1
.
.
■■ Division of decimal numbers by other decimals
– Change the divisor into a whole number.
– Whatever change is made to the divisor must also be made to the dividend.
For example: 24.562 ÷ 0.02
24.562 ÷ 0.02 = 2456.2 ÷ 2
– When dividing by multiples of 10, initially ignore the zeros in the divisor and perform
routine division. On arriving at your answer, you must then re-position your decimal
point according to the rules of dividing by powers of 10. For each zero in the question
that you ignored initially, the decimal point must move 1 place to the left.
Key ideas
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Chapter 6 Computation with decimals 308 and fractions
Example 23 Dividing decimals by whole numbers
Calculate:
a 42.837 ÷ 3 b 0.0234 ÷ 4
Solution Explana tion
a 14.279
3)4 2 8 3 7
14 2 7 9
1 2 2
.
.
Carry out division, remembering that the decimal point
in the answer is placed directly above the decimal point
in the dividend.
b 0.00585
4)0 02 3 4 0
0 00 5 8 5
2 3 2
.
.
Remember to place zeros in the answer every time the
divisor ‘doesn’t go’.
Again, align the decimal place in the answer directly
above the decimal place in the question.
An additional zero is required at the end of the dividend
to terminate the decimal answer.
Basic arithmetic calculators
automatically treat fractions
as division operations and
convert them to decimals, but
mathematical calculators and
spreadsheets can be set to
work with fractions.
Example 24 Dividing decimals by decimals
Calculate:
a 62.316 ÷ 0.03 b 0.03152 ÷ 0.002
Solution Explana tion
a 62.316 ÷ 0.03
= 6231.6 ÷ 3 = 2077.2
3)62 3 1 6
20 7 7 2
2 2
.
.
Need to divide by a whole number.
62.316 ÷ 0.03
Move each decimal point 2 places to the right.
Carry out the division question 6231.6 ÷ 3.
b 0.03152 ÷ 0.002
= 31.52 ÷ 2 = 15.76
2)3 1 5 2
15 7 6
1 1 1
.
.
Multiply divisor and dividend by 1000.
0.03152 ÷ 0.002
Move each decimal point 3 places to the right.
Carry out the division question 31.52 ÷ 2.
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Number and Algebra 309
Example 25 Dividing decimals by multiples of 10
Calculate 67.04 ÷ 8000.
SolutioN ExplANAtioN
8)67 04
08 38
.
.
8.38 ÷ 1000 = 0.00838
67.04 ÷ 8000 = 0.00838
Ignore the three zeros in the 8000.
Divide 67.04 by 8.
Now divide by 1000, resulting in moving the decimal
point 3 places to the left.
Example 26 Evaluating using order of operations
Calculate using the order of operations:
3.8 − 1.6 × 0.45 + 5 ÷ 0.4
SolutioN ExplANAtioN
3.8 − 1.6 × 0.45 + 5 ÷ 0.4
= 3.8 − 0.72 + 12.5
= 3.08 + 12.5
= 15.58
First carry out × and ÷, working from left to right.
Then carry out + and −, working from left to right.
Exercise 6H
1 For the question 36.52 ÷ 0.4 = 91.3, which of the following options uses the correct
terminology?
A 36.52 is the divisor, 0.4 is the dividend and 91.3 is the quotient.
B 36.52 is the dividend, 0.4 is the divisor and 91.3 is the quotient.
C 36.52 is the quotient, 0.4 is the dividend and 91.3 is the divisor.
D 36.52 is the divisor, 0.4 is the quotient and 91.3 is the dividend.
2 Explain where you place the decimal point in the quotient (i.e. answer), when dividing a decimal by
a whole number.
3 Calculate:
a 1200 ÷ 20 b 120 ÷ 2 c 12 ÷ 0.2 d 1.2 ÷ 0.02
e Explain why these questions all give the same answer.
4 For each of the following pairs of numbers, move the decimal points the same number of places
so that the second number becomes a whole number.
a 3.2456, 0.3 b 120.432, 0.12
c 0.00345, 0.0001 d 1234.12, 0.004
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Chapter 6 Computation with decimals 310 and fractions
5 Calculate:
a 8.4 ÷ 2 b 30.5 ÷ 5 c 64.02 ÷ 3
d 2.822 ÷ 4 e 4.713 ÷ 3 f 2.156 ÷ 7
g 38.786 ÷ 11 h 1491.6 ÷ 12 i 0.0144 ÷ 6
j 234.21 ÷ 2 k 3.417 ÷ 5 l 0.01025 ÷ 4
6 Calculate:
a 6.14 ÷ 0.2 b 23.25 ÷ 0.3 c 2.144 ÷ 0.08
d 5.1 ÷ 0.6 e 0.3996 ÷ 0.009 f 45.171 ÷ 0.07
g 0.0032 ÷ 0.04 h 0.04034 ÷ 0.8 i 10.78 ÷ 0.011
j 4.003 ÷ 0.005 k 0.948 ÷ 1.2 l 432.2 ÷ 0.0002
7 Design three decimal division questions for your partner. Make sure you calculate the answer to
each question. Swap questions with your partner. Go to work solving your partner’s questions.
Pass your answers back for your partner to correct. Discuss any mistakes made by either person.
8 Calculate:
a 236.14 ÷ 200 b 413.35 ÷ 50 c 3.71244 ÷ 300
d 0.846 ÷ 200 e 482.435 ÷ 5000 f 0.0313 ÷ 40
9 Calculate the following, rounding your answers to 2 decimal places. Check your answers using
a calculator.
a 35.5 kg ÷ 3 b $213.25 ÷ 7 c 182.6 m ÷ 0.6 m
d 287 g ÷ 1.2 e 482.523 L ÷ 0.5 f $5235.50 ÷ 9
10 Calculate the following, using the order of operations. Check your answers using a calculator.
a 3.68 ÷ 2 + 5.7 ÷ 0.3 b 6(3.7 × 2.8 + 5.2)
c 17.83 − 1.2(8.1 − 2.35) d 9.81 ÷ 0.9 + 75.9 ÷ 10
e (56.7 − 2.4) ÷ (0.85 ÷ 2 + 0.375) f 34.5 × 2.3 + 15.8 ÷ (0.96 − 0.76)
11 Find the missing digits in these division questions.
a
3) 2 6 7
0
.
. b
3)1
0 6 4
. 2
.
c
) .
.
10 7
2 5 d
) .
.
15
2 1 4
2 9
12 Charlie paid $12.72 to fi ll his ride-on lawnmower with 8 L of fuel. What was the price per litre
of the fuel that he purchased?
13 Dibden is a picture framer and has recently purchased 214.6 m of timber. The average-sized
picture frame requires 90 cm (0.9 m) of timber. How many average picture frames could Dibden
make with his new timber?
14 A water bottle can hold 600 mL of water. How many water bottles can be fi lled from a large
drink container that can hold 16 L?
Example 23
Example 24
Example 25
Example 26
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Number and Algebra 311
15 Six friends go out for dinner. At the end of the evening, the restaurant’s bill is $398.10.
a As the bill is split equally among the six friends, how much does each person pay?
b Given that the friends are happy with the food and service, they decide to round the
amount they each pay to $70. What is the waiter’s tip?
16 Clara purchases 1.2 kg of apples for $3.90. Her friend Sophia buys 900 g of bananas for $2.79
at the same shop. Find the cost per kilogram of each fruit. Which type of fruit is the best value
in terms of price per kilogram?
17 A police radar gun measures a car to be 231.5 m away.
At 0.6 seconds later, the radar gun measures the same car
to be 216.8 m away.
a Determine the speed of the car in metres per second (m/s).
b Multiply your answer to part a by 3.6 to convert your answer to
km/h.
c The car is travelling along an 80 km/h stretch of road.
Is the car speeding?
18 Given that 24.53 × 1.97 = 48.3241, write down the value of each of
the following questions, without using a calculator.
a 48.3241 ÷ 1.97 b 48.3241 ÷ 2.453 c 4832.41 ÷ 1.97
d 483.241 ÷ 245.3 e 0.483241 ÷ 0.197 f 483 241 ÷ 2453
Enrichment: What number am I?
19 I am thinking of a number. Given the following clues for each, find the number.
a When I add 4.5 and then multiply by 6, the answer is 30.
b When I divide it by 3 and then add 2.9, the answer is 3.
c When I multiply it by 100 and then add 9, the answer is 10.
d When I multiply it by 5 and then add a half, the answer is 6.
e When I subtract 0.8, then divide by 0.2 and then divide by 0.1, the answer is 200.
f Make up three of your own number puzzles to share with the class.
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Chapter 6 Computation with decimals 312 and fractions
Computation with negative fractions
The English mathematician John Wallis (1616–1703) invented a number line that displayed numbers
extending in both the positive and negative directions.
So, just as we can have negative whole numbers, we can also have negative fractions. In fact, each
positive fraction has an opposite (negative) fraction. Two examples are highlighted on the number line below.
–3 –2 –2 –1 0 1 2 3 2
3
2 2
3
1
2
– 1
2
let’s start: Where do you end up?
–5 –4 –3 –2 –1 0 1 2 3 4 5
You are given a starting point and a set of instructions to follow. You must determine where the fi nishing
point is. The fi rst set of instructions reviews the addition and subtraction of integers. The other two sets
involve the addition and subtraction of positive and negative fractions.
• Starting point is +1. Add 3, subtract 5, add −2, subtract −4, subtract 3.
Finishing point =
• Starting point is 0. Subtract
3
5
, add
1
5
, add − 4
5
, subtract
2
5
, subtract − 3
5
.
Finishing point =
• Starting point is
1
2
, subtract
3
4
, add − 1
3
, subtract − 1
2
, subtract
1
12
, add
1
6
.
Finishing point =
6I EXTENSION
■■ Operations with positive fractions also apply to negative fractions.
■■ The arithmetic rules we observed for integers (Chapter 1) also apply to fractions.
■■ Subtracting a larger positive fraction from a smaller positive fraction will result in a negative
fraction.
For example:
1
5
2
3
3
15
10
15
7
15
− = − = −
■■ Adding a negative fraction is equivalent to subtracting its opposite.
For example:
1
2
1
3
1
2
1
3
+  −

     

    =
−

+
   


   

= − 1
2
1
3
■■ Subtracting a negative fraction is equivalent to adding its opposite.
For example:
1
2
1
3
1
2
1
3
−  −
 
  


   

= +  +
   


   

= + 1
2
1
3
■■ The product or quotient of two fractions of the same sign (positive or negative) is a positive
fraction.
– Product: 1
3
2
5
2
15
× = or − × − 
   


   

= 1
3
2
5
2
15
– Quotient:
2
15
1
3
2
5
÷ = or − ÷ − 
   


   

= 2
15
1
3
2
5
Key ideas
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Number and Algebra 313
Example 27 Adding and subtracting negative fractions
Evaluate:
a
2
7
5
7
+  −

     

   

b
2
3
4
3
−  −
 
  


   

c
1
5
1
4
+  −

     

   

d − − − 
   


   

7
3
3
2
3
SolutioN ExplANAtioN
a
2
7
5
7
2
7
5
7
3
7
+  −

     

   

= −
= −
Adding − 5
7
is equivalent to subtracting
5
7
.
b
2
3
4
3
2
3
4
3
6
3
2
−  −
 
  


   

= +
= =
Subtracting − 4
3
is equivalent to adding
4
3
.
c
1
5
1
4
1
5
1
4
4
20
5
20
1
20
+  −

     

   

= −
= −
= −
Adding − 1
4
is equivalent to subtracting
1
4
.
The LCM of 5 and 4 is 20.
Write equivalent fractions with LCD of 20.
Subtract the numerators.
d − −  −
   


   

= − +
= − +
=
7
3
3
2
3
7
3
3
2
3
7
3
11
3
4
3
Subtracting −3
2
3
is equivalent to adding 3
2
3
.
Convert mixed numeral to improper fraction.
Denominators are the same, therefore add numerators
−7 + 11 = 4.
■■ The product or quotient of two fractions of the opposite sign (positive and negative) is a
negative fraction.
– Product:
1
2
1
4
1
8
×  −

     

   

= − or − × = − 1
2
1
4
1
8
– Quotient:
1
8
1
2
1
4
÷  −

     

   

= − or − ÷ = − 1
8
1
2
1
4
Key ideas
ISBN: 9781107626973
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Chapter 6 Computation with decimals 314 and fractions
Example 28 Multiplying with negative fractions
Evaluate:
a
2
3
4
5
×  −

     

   

b − × − 
   


   

6
5
3
4
SolutioN ExplANAtioN
a
2
3
4
5
8
15
×  −

     

   

= −
The two fractions are of opposite sign, so the answer
is a negative.
b − ×  −
   


   

= ×
= × =
6
5
3
4
6
5
3
4
3
5
3
2
9
10
The two fractions are of the same sign, so the answer
is a positive.
Cancel where possible, then multiply numerators
and multiply denominators.
1 Draw a number line from −4 to 4 and on it mark the following negative and positive fractions.
a − 1
4
b
1
4
c 1
1
2
d −2
1
2
e −3
4
5
f − 7
3
Example 29 Dividing with negative fractions
Evaluate:
a − ÷ − 
   


   

2
5
3
4
b −1 ÷ 1
3
3
SolutioN ExplANAtioN
a − ÷ − 
   


   

= − ×  −

     

   

= ×
=
2
5
3
4
2
5
4
3
2
5
4
3
8
15
The reciprocal of  −
   


   

3
4
is  −
   


   

4
3
.
The two fractions are of the same sign, so the answer
is a positive.
The answer is in simplest form.
b − ÷ = − ×
= −
1 1
3
3
4
3
1
3
4
9
The reciprocal of 3 is
1
3
.
The two numbers are of opposite sign, so the answer
is a negative.
Exercise 6I E X T E N S I O N
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Number and Algebra 315
2 Complete these sentences.
a Adding  −
   


   

1
4 is equivalent to subtracting ______.
b Adding
1
3
is equivalent to subtracting ______.
c Subtracting − 
   


   

3
5 is equivalent to adding ______.
d Subtracting
2
7
is equivalent to adding ______.
3 Do not evaluate the following expressions. Simply state whether the answer will be positive
or negative.
a − ×  −
   


   

3
5
1
3
b −5 × 1
5
9
11
c 5
3
3
5
÷  −

     

   

d − ÷ − 
   


   

2 1
7
8
1
3
4 Evaluate:
a − + 6
7
2
7
b − + 3
5
4
5
c − − 5
9
2
9
d − − 11
3
5
3
e 1
3
2
3
+ −


 
 

   

f + −    
 

 
  1
5
3
5
g −  −   
 

 
  1
4
5
4
h −  −   
 
 
  3
11
4
11
5 Evaluate:
a + −    
 

 
  1
4
1
3
b + −    
 

 
  3
7
4
5
c −  −   
 

 
  1
2
3
5
d 2
9
2
3
− −

 
 


   

e − − −  
   
  3
2
5
4
f − − −  
   
  5
8
3
4
g − − −  
   
  7
5
1
1
4
h − − − 
   


   

8
3
2
2
5
6 Evaluate:
a × −    
 

 
  3
5
4
7
b − × 2
5
8
11
c − × −  
   
  1
3
4
5
d − × −  
   
  5
9
3
2
e − × 3
9
4
7
f × −    
 

 
  2
6
3
8
g × −    
 

 
  –1
1
2
2
7
h − × 3
8
3
1
5
7 Evaluate:
a − ÷ 5
7
3
4
b − ÷ 1
4
5
9
c − ÷ −  
   
  2
3
5
4
d − ÷ −  
   
  4
9
1
3
e − ÷ 4
7
2 f − ÷ 3
5
4 g −1 ÷ (− ) 1
2 ( 2) h – 5 –
1
3
2
2
9
÷ 
 
  


   

Example 27a,b
Example 27c,d
Example 28
Example 29
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Chapter 6 Computation with decimals 316 and fractions
8 Evaluate, using a calculator. Estimate your answer fi rst.
a
13
5
17
5
−  −
 

 
b − + − 
 

 
7
13
4
2
3
c − × − 
 

 
24
33
15
40
d −2 ÷
3
7
5
4
5
9 Arrange these fractions from smallest to largest.
− − − − − 3
4
,
1
2
,
5
3
,
3
4
, 11
2
,
1
16
,
1
5
, 3
1
10
10 Toolapool has an average maximum temperature of 13
1
2
°C and an average minimum
temperature of −3
1
4
°C. The average temperature range is calculated by subtracting the
average minimum temperature from the average maximum temperature. What is the average
temperature range for Toolapool?
11 Xaio aims to get 8 hours sleep per week night. On Monday night he slept for 6
1
3
hours, on
Tuesday night 7
1
2
hours, on Wednesday night 5
3
4
hours and on Thursday night 8
1
4
hours.
a State the difference between the amount of sleep Xaio achieved each night and his goal of
8 hours. Give a negative answer if the amount of sleep is less than 8 hours.
b After four nights, how much is Xaio ahead or behind in terms of his sleep goal?
c If Xaio is to meet his weekly goal exactly, how much sleep must he get on Friday night?
12 Maria’s mother wants to make eight curtains that
each require 2
1
5
metres of material in a standard
width, but has only 16
1
4
metres. She asks Maria to
buy more material. How much more material must
Maria buy?
WORKING
MATHEMATICALLY
U F
R PS
C
WORKING
MATHEMATICALLY
U F
R PS
C
6I
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Number and Algebra 317
13 Place an inequality sign (< or >) between the following fraction pairs to make a true statement.
a − − 1
3
1
2
b −3 − 1
5
2
3
7
c − 1
4
1
2
d − 3
5
1
11
e 2 −
1
5
4
3
5
f 0 −
1
100
g 4
9
5
9
h − − 4
9
5
9
14 Do not evaluate the following expressions. Simply state whether the answer will be positive
or negative.
a − ×  −
   


   

×  −

     

   

2
7
1
7
3
11
b − × − 
   


   

4 1
5
9
11
2
c − ÷  −
   


   

× ×  −
   


   

5
6
2
7
1
3
4
9
d − 
   


   

÷  −

     

   

3
5
4
1
5
3 3 15 Using a calculator, evaluate the expressions given in Question 14.
16 If a > 0, b > 0 and a < b, place an inequality sign between the following fraction pairs to
make a true statement.
a a
b
b
a
b a
b
a
b
− c − − a
b
b
a
d − − b
a
a
b
Enrichment: positive and negative averages
17 a Calculate the average (also known as the mean) of the following sets of numbers, by adding
and then dividing by the number of numbers.
i 1
1
2
3
4
5
4
2
1
2
, , ,
ii − − 2
3
5
6
7
6
1
3
1
1
3
, ,
, ,
iii −2 − − 1
5
3
5
0
1
5
1
3
5
, ,
, ,
iv −7 − − − 1
3
2
1
2
5 1
6
3
3
10
, ,
,
b List a set of fi ve different fractions that have an average of 0.
c List a set of fi ve different fractions that have an average of − 3
4
.
WORK ING
MATHEMATICALLY
U F
R PS
C
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Chapter 6 Computation with decimals 318 and fractions
Best Buy
The concept of a ‘best buy’ relates to purchasing a product that is the best value for money. To
determine the ‘best buy’ you need to compare the prices of similar products for the same weight.
STRAWBERRY JAM jar 375 g
$ 3.95
$ 10.53 per kg
STRAWBERRY JAM jar 250 g
$ 2.95
$ 11.80 per kg
Converting units
a Convert the following to a price per kg.
i 2 kg of apples for $3.40 ii 5 kg of sugar for $6.00
iii 1.5 kg of cereal for $4.50 iv 500 g of butter for $3.25
b Convert the following to a price per 100 g.
i 300 g of grapes for $2.10 ii 1 kg of cheese for $9.60
iii 700 g of yoghurt for $7.49 iv 160 g of dip for $3.20
Finding ‘best buys’
a By converting to a price per kg, determine which is the best buy.
i 2 kg of sauce A for $5.20 or 1 kg of sauce B for $2.90
ii 4 kg of pumpkin A for $3.20 or 3 kg of pumpkin B for $2.70
iii 500 g of honey A for $5.15 or 2 kg of honey B for $19.90
iv 300 g of milk A for $0.88 or 1.5 kg of milk B for $4.00
investigation
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Number and Algebra 319
b By converting to a price per 100 g, determine which is the best buy.
i 500 g of paper A for $3.26 or 200 g of paper B for $1.25
ii 250 g of salami A for $4.50 or 150 g of salami B for $3.10
iii 720 g of powder A for $3.29 or 350 g of powder B for $1.90
iv 1.1 kg of shampoo A for $12.36 or 570 g of shampoo B for $6.85
Problem solving
a Star Washing Liquid is priced at $3.85 for 600 g, whereas Best Wash Liquid is priced at $5.20 for
1 kg. Find the difference in the price per 100 g, correct to the nearest cent.
b Budget apples cost $6.20 per 5 kg bag. How much would a 500 g bag of Sunny apples have to be if
it was the same price per 100 g?
c 1.5 kg of cheddar cheese costs $11.55, and 800 g of feta cheese costs $7.25. Sally works out the
best value cheese, then buys $5 worth of it. How much and what type of cheese did Sally buy?
Investigate
Go to a local supermarket and choose a type of product of which there are many brands to choose from.
a Record the following information for each brand of the same type of product.
i price ii weight iii brand name
b Calculate the price of each brand per:
i kg ii 100 g
c Compare the prices of each brand. Comment on the following.
i Which brand is the ‘best buy’?
ii The differences between the cheapest and most expensive brands.
iii The reasons why some brands might be more expensive.
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Chapter 6 Computation with decimals 320 and fractions
Decimal vs percentage
Draw two horizontal lines on a long sheet of graph paper that is 100 small squares wide. On line 1,
make a number line from 0 to 1, labelling every tenth and marking hundredths. On line 2, mark
percentages from 0% to 100%, labelling every 10% and marking every percentage.
0
Example of the number lines
0.1 0.2
0% 10% 20%
0.3
30%
Answer the following questions, using your number lines to help you.
a Which is larger, 73% or 0.74?
b Which is smaller, 26% or 0.3?
c List in ascending order: 45%, 0.72, 49%, 37%, 0.58, 0.7, 51%, 0.64, 60%, 0.5.
d List in descending order: 37%, 0.03, 82%, 0.37, 0.8, 77%, 0.23, 38%, 2%, 0.4.
e List in ascending order: 78%, 0.683, 77.5%, 79.9%, 0.78452, 0.76, 0.784, 69.9%, 0.6885.
f Write three decimal numbers between 0.47 and 0.57.
g In decimal form, list all the whole tenths that are greater than 0.34 and less than 0.78.
h In decimal form, list all the decimal hundredths that are greater than 0.32 and less than 0.41.
i Write down three pairs of decimals so that each pair has a difference of 0.02. Now write each of
your number pairs as percentages. What is the difference between these percentage values?
j Play some decimal/percentage games in pairs.
• Person A gives a decimal (between 0 and 1) and person B states the percentage that is equivalent
to two-tenths less than person A’s decimal.
• Person A states a percentage and person B gives the decimal that is equivalent to 5% more than
person A’s percentage.
• Use your decimal and percentage number lines to make up your own games.
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Number and Algebra 321
1 According to legend, this sank to the bottom of the Atlantic Ocean.
To fi nd the answer:
• Work out the problems.
• Locate each answer in the table.
• Place the letter by the answer in the blank next to the questions.
• Match the numbers and letters in the puzzle below.
1. ______ The sum of 21.36 and 14.4 minus 8.2 A = 0.2
C = 0.875
E = 0.4
F = 0.81
I = 14.26
L = 17
N = 0.9909
O = 27.56
S = 6
T = 7
2. ______ Digit in the hundredths place in 347.879
3. ______ The square of 0.9
4. ______ 5.1 divided by 0.3
5. ______ The decimal equivalent of
7
8
6. ______ The sum of 0.0415, 0.415 and 0.0041, less 0.062,
to 1 decimal place
7. ______ 3.15 × 0.05, to 1 decimal place
8. ______ The area of a rectangle with length 6.2 cm and
width 2.3 cm
9. ______ The difference between 9 and 8.0091
10. ______ The number of decimal places when 0.6235 is
multiplied by 6.23
5 1 9 2 8 9 6 9 2 1 3 7 2 4 7 9 2 8 10
2 Consider the ladder in the diagram. The heights of each rung on
the ladder are separated by an equal amount. Determine the heights
for each rung of the ladder.
3 Find the digits represented by the letters in these decimal problems.
a A. 2B
9.C5
11. 12
+
b 2A. 43
9. B4
C7. 8D
−
c 3.A × B.4 = 8.16
d
) .
.
A 2 2 1
0 757
B
0.54 m high
1.26 m high
puzzles and challenges
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322
Chapter summary
Chapter 6 Computation with decimals and fractions
Addition
+
9.807
26.350
36.157
Align decimal points.
Multiplication
2.78 × 34 = 94.52
2.78 × 3.4 = 9.452
0.278 × 3.4 = 0.9452
0.278 × 0.34 = 0.09452
278
1112
8340
9452
Number of decimal places in
the question equals number of
decimal places in the answer.
× 34
Multiplication by
powers of 10
2.76 × 10 000
= 27 600.0
Division by powers of 10
2.76 ÷ 10 000
= 0.000276
Subtraction
–
216.94
31.53
185.41
1
1
Division
8.547 ÷ 0.03
= 854.7 ÷ 3
2 8 4 . 9
8 5 4 . 7
2 1 2 3
Decimal point moves left.
Decimal point moves right.
1
1
Decimals
Adding fractions
4
5
2
3 +
12
15
10
15 = +
22
15
7
15 = = 1
Dividing fractions
1
6
25
6
10
9 =
=
÷ Reciprocal
10
9
9
10
of is .
25
6
9
10 =
4
3
4
3
= 15
4
1
9 ÷
2
5
2
3
1
Subtracting fractions
=
1
4
3 2
3
– 1 Or
5
4
2
= 7
12
1
= (2 – 1) +
2
3 – 1
1
4
3 2
3 – 1
= 13
4
5
3 –
= 39
12
20
12 –
7
12
= 19 1
12 =
= 39 – 20
12
5
4
2
3 –  


= 1 +
15
12
8
12 –  

 
Multiplying fractions
5
10
20
50 = ×
2
5 =
5
10
20
50
of
1
1
2
5
16
5
9
4 = ×
×
36
5 =
1
5
3 1
4 ×
4
1
2
1
5 = 7
Percentage of a quantity
=
25% of $40
= $10
1
4 × 40
=
= 60 minutes
= 1 hour
120
100
× 50
120% of
50 minutes
2
Fractions
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Number and Algebra 323
Multiple-choice questions
1 The next number in the pattern 0.023, 0.025, 0.027, 0.029 is:
A 0.0003 B 0.030 C 0.0031 D 0.031 E 0.033
2 0.05 is equivalent to:
A 5
10
B 5
100
C 5
1000
D 5
500
E 5
3 The smallest number out of 0.012, 10.2, 0.102, 0.0012 and 1.02 is:
A 0.012 B 0.102 C 0.0012 D 1.02 E 10.2
4 0.36 ÷ 1000 is equal to:
A 3.6 B 360 C 0.036 D 0.0036 E 0.00036
5 6.2 × 0.2 is equal to:
A 1.24 B 12.4 C 0.124 D 124 E 0.0124
6 What is the answer to 0.08 × 0.6?
A 0.48 B 4.8 C 0.0048 D 0.048 E 48
7 When rounded to 1 decimal place, 84.553 becomes:
A 80 B 84 C 84.5 D 84.6 E 84.55
8 As a decimal, 23
90
is equal to:
A 0.2 B
.
0.25 C 0.26 D
. .
0.28 E
.
0.256
9 7 + 0.7 + 0.07 + 0.007, to 2 decimal places, is:
A 7.78 B 7.77 C 7 D 7.7 E 7.777
10 5.624 means:
A 5.62444… B 6.6242424… C 5.624624624…
D 5.6246464… E 5.62456245624…
Short-answer questions
1 Arrange each group in descending order, from largest to smallest.
a 0.4, 0.04, 0.44 b 2.16, 2.016, 2.026 c 0.932, 0.98, 0.895
2 Write each fraction as a decimal.
a 81
10
b 81
100
c 801
100
d 801
1000
3 What is the place value of the digit 3 in the following numbers?
a 12.835 b 6.1237 c 13.5104
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Chapter 6 Computation with decimals 324 and fractions
4 State whether each of the following is true or false.
a 8.34 < 8.28 b 4.668 > 4.67 c 8.2 > 8.182
d 3 08
308
100
.  e 62
100
 6.20 f 7
10
70
100
=
5 Evaluate:
a 2841 ÷ 3 b 2841 ÷ 0.3 c 2841 ÷ 0.03
6 Round each of the following to the specifi ed number of decimal places (which is given in brackets).
a 423.46 (1) b 15.8892 (2) c 7.25432 (1)
d 69.999531 (3) e 2
3
4
(1) f
2
3
(2)
g 5
11
(3) h 1
81
(44) (Hint: Look for the short method!)
7 Evaluate:
a 13.85 − 4.32 b 19.12 − 14.983
c 27.6 + 15.75 d 204.70837 + 35.7902
e 472.427 − 388.93 f 210.8 − (26.3 − 20.72)
8 State whether each of the following is true or false.
a 10.34 ÷ 100 = 0.1034 b 3.125 × 0.1 = 31.25
c 115.23 ÷ 10 = 1.1523 × 1000 d 115.23 has 3 decimal places
e 24.673 = 24.7 when rounded to 1 decimal place
9 State the number of decimal places in each answer to these calculations.
a A number with 2 decimal places × a number with 3 decimal places
b A number with 1 decimal place × a number with 4 decimal places
c A number with 2 decimal places × a number with 2 decimal places
d A number with 2 decimal places × a number with 0 decimal places
10 Solve each of the following, using the order of operations.
a 1.37 × 100 b 0.79 × 1000 c 225.1 ÷ 10
d 96.208 ÷ 1000 e 75.68 + 6.276 × 100 − 63.24 ÷ 10
f 3.56 × 100 + 45 ÷ 10 g 100 × (56.34 × 100 + 0.893)
11 Calculate the following.
a 2.4 × 8 b 9 × 7.11 c 2.3 × 8.4
d 3.8 ÷ 4 e 12.16 ÷ 8 f 3 ÷ 0.5
g 4 ÷ 0.25 h 1.2 ÷ 0.4 i 3.42 ÷ 1.1
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Number and Algebra 325
12 Copy and complete this table, stating fractions both with the denominator 100 and in their
simplest form.
Decimal Fraction Percentage
0.45
?
100
7
10
=
32%
0.06
79
100
1.05
?
100
7
20
=
65%
?
1000
1
8
=
Extended-response questions
1 Find the answer in these practical situations.
a Jessica is paid $125.70 for 10 hours of work and Jaczinda is paid $79.86 for 6 hours of work.
Who receives the higher rate of pay per hour, and by how much?
b Petrol is sold for 144.9 cents per litre. Jacob buys 30 L of petrol for his car. Find the total
price he pays, to the nearest 5 cents.
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Chapter 6 Computation with decimals 326 and fractions
c The Green family are preparing to go to the Great Barrier Reef for a holiday. For each of the four
family members, they purchase a goggles and snorkel set at $37.39 each, fi ns at $18.99 each and
rash tops at $58.48 each. How much change is there from $500?
d For her school, a physical education teacher buys 5 each of basketballs, rugby union and soccer
balls. The total bill is $711.65. If the rugby balls cost $38.50 each and the basketballs cost $55.49
each, what is the price of a soccer ball?
2 A car can use 25% less fuel per kilometre when travelling at 90 km/h than it would when travelling at
110 km/h. Janelle’s car uses 7.8 litres of fuel per 100 km when travelling at 110 km/h, and fuel costs
145.6 cents per litre.
a How much money could Janelle save on a 1000-km trip from Sydney to Brisbane if she travels at a
constant speed of 90 km/h instead of 110 km/h?
b During a 24-hour period, 2000 cars travel the 1000-km trip between Sydney and Brisbane. How
much money could be saved if 30% of these cars travel at 90 km/h instead of 110 km/h?
3 Siobhan is on a 6-week holiday in the United Kingdom, and is using her mobile phone to keep in
contact with her friends and family in Australia. The mobile phone charge for voice calls is $0.40
‘fl agfall’ and $0.65 per 45 seconds; SMS messages are $0.38 each.
During her holiday, Siobhan makes 27 voice calls and sends 165 SMS messages to Australia. If her
mobile phone bill is $832.30, determine the average length of Siobhan’s voice calls.
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Semester review 1 327
Chapter 1: Computation with positive integers
Multiple-choice questions
1 Using numerals, thirty-fi ve thousand, two hundred and six is:
A 350 260 B 35 260 C 35 000 206 D 3526 E 35 206
2 The place value of 8 in 2 581 093 is:
A 8 thousand B 80 thousand C 8 hundred D 8 tens E 8 ones
3 The remainder when 23 650 is divided by 4 is:
A 0 B 4 C 1 D 2 E 3
4 18 − 3 × 4 + 5 simplifi es to:
A 65 B 135 C 11 D 1 E 20
5 800 ÷ 5 × 4 is the same as:
A 160 × 4 B 800 ÷ 20 C 800 ÷ 4 × 5 D 40 E 4 × 5 ÷ 800
Short-answer questions
1 Write the number seventy-four in:
a Babylonian numerals
b Roman numerals
c Egyptian numerals
2 Write the numeral for:
a 6 × 10 000 + 7 × 1000 + 8 × 100 + 4 × 10 + 9 × 1
b 7 × 100 000 + 8 × 100 + 5 × 10
3 Calculate:
a 96 481 + 2760 + 82 b 10 963 − 4096 c 147 × 3
d 980 × 200 e 4932 ÷ 3 f 9177 ÷ 12
4 State whether each of the following is true or false.
a 18 < 20 − 2 × 3 b 9 × 6 > 45 c 23 = 40 ÷ 2 + 3
5 How much more than 17 × 18 is 18 × 19?
6 Calculate:
a 7 × 6 − 4 × 3 b 8 × 8 − 16 ÷ 2 c 12 × (6 − 2)
d 16 × [14 − (6 − 2)] e 24 ÷ 6 × 4 f 56 − (7 − 5) × 7
7 State whether each of the following is true or false.
a 4 × 25 × 0 = 1000 b 0 ÷ 10 = 0 c 8 ÷ 0 = 0
d 8 × 7 = 7 × 8 e 20 ÷ 4 = 20 ÷ 2 ÷ 2 f 8 + 5 + 4 = 8 + 9
8 Insert brackets to make 18 × 7 + 3 = 18 × 7 + 18 × 3 true.
9 How many times can 15 be subtracted from 135 before an answer of zero occurs?
Semester review 1
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Semester review 1
328
10 Write 3 859 643 correct to the nearest:
a 10 b thousand c million
Extended-response question
1 Tom works as a labourer, earning $25 an hour on weekdays and $60 an hour on weekends.
a During a particular week, Tom works from 7 a.m. to 2 p.m. Monday to Thursday. How many
hours does he work that week?
b How much does Tom earn for this work?
c If Tom works 5 hours on Saturday in the same week, what is his total income for the week?
d How many more hours on a Friday must Tom work to earn the same amount as working
5 hours on a Saturday?
Chapter 2: Angle relationships
Multiple-choice questions
1 Which statement is correct?
A Line m is perpendicular to line l.
B Line m bisects line l.
C Line m is parallel to line l.
D Line m is shorter than line l.
E Line m is longer than line l.
2 An angle of 181° is classifi ed as:
A acute B refl ex C straight D obtuse E sharp
3 Which two angles represent alternate angles?
A a° and e°
B d ° and f °
C a° and f °
D g° and b°
E c° and f °
4 Which of the following shows a pair of supplementary angles?
A B C
D E
l
m
d° c°
e° f °
h° g°
a° b°
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Semester review 1 329
5 The value of x + y is:
A 70
B 220
C 35
D 140
E 110
Short-answer questions
1 a Name two pairs of parallel lines.
b Name a pair of perpendicular lines.
c List any three lines that are concurrent. At what point
do they cross?
d Name two points that are collinear with point C.
e Name the point at which line BE and line FD intersect.
2 Measure these angles.
a b
c
3 What is the complement of 65°?
4 What is the supplement of 102°?
5 Find the value of a in each of the following angles.
a
40° a°
b
40° a°
c
a°
120° 100°
x° y°
110°
A
F
B C
E D
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Semester review 1
330
d
25°
a°
e
a° 62°
f a°
56°
6 Find the value of each angle formed when these two parallel lines are
crossed by the transversal, as shown.
7 Explain why AB is not parallel to CD.
8 Use your ruler and a pair of compasses to draw a 60° angle. Bisect it to form two 30° angles.
Extended-response question
1 Consider the diagram shown.
a Find the value of:
i x ii y iii z
b What is the value of x + y + z ?
Chapter 3: Computation with positive and negative integers
Multiple-choice questions
1 Which of the following statements is incorrect?
A −2 > −4 B 0 < 5 C 0 < −10 D −9 < −8 E −5 < 3
2 12 + (−9) – (−3) is the same as:
A 12 + 9 + 3 B 12 – 9 + 3 C 12 – 9 – 3 D 12 – 12 E 12
3 If a = −3, the value of −2a2 is:
A 36 B −36 C 18 D −18 E 12
4 The coordinates of the point that is 3 units below (3, 1) is:
A (0, 1) B (0, −2) C (0, −1) D (3, 4) E (3, −2)
5 12 × (−4 + (−8) ÷ 2) equals:
A −96 B 72 C −72 D 60 E 96
c°
d°
f ° e°
g°
b°
a° 80°
A
C D
65° B
125°
x° z°
y°
151°
56°
125° 140°
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Semester review 1 331
Short-answer questions
1 For each of the following, insert =, > or <.
a −3 3 b −10 ÷ 2 5 c −20 × (−1)
−
−
40
2
2 Calculate:
a −5 + (−8) b 12 – 96 c −12 – 96
d −4 – 8 – 9 e −12 + 96 f −7 – (−7)
3 Find:
a −6 × 4 b −9 × 8 × (−1) c (−12)2
d −9 −7
3
× ( )
e −150 ÷ (−2 – 3) f −10 + 7 × (−3)
4 State whether the answer to each of the following is positive or negative.
a −3 × (−3) × (−3) b −109 × 142 × (−83) c −2 × (−1 – (−3))
5 Copy and complete.
a + 9 = −6 b × (−3) = −6 × (−4) c 16 × = −64
6 If a = 6 and b = −4, fi nd the value of:
a −a + b b a – b c 2(b – a)
d −ab2 e a2 + b2 f 24 ÷ (ab)
Extended-response question
1 Refer to the given Cartesian plane when answering
these questions.
a Name any point that lies in the fi rst quadrant.
b Name any point(s) with a y value of zero. Where does
each point lie?
c Which point has coordinates (−1, −2)?
d Find the distance between points:
i A and B ii D and E
e What shape is formed by joining the points IDAG ?
f What is the area of IDAG ?
g ABXD are the vertices of a square. What are the coordinates of X ?
h Decode: (2, 2), (2, −3), (0, 2), (−1, 2), (2, 2), (2, −3)
Chapter 4: Understanding fractions, decimals and percentages
Multiple-choice questions
1 Which of the following is equivalent to
12
7
?
A
24
7
B 1
5
7
C 1
5
12
D
112
17
E
7
12
2
1
2
+
1
3
is equal to:
A
2
5
B
2
6
C
5
6
D
1
5
E
7
6
2
3
1
-2
-3
-3 -2 -1-1 1 2 3
H
G
F
E
A
D
I C
B
O 4 5
y
x
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Semester review 1
332
3
350
450
in simplest form is:
A
35
45
B
4
5
C
3
4
D
3 5
4 5
.
.
E
7
9
4 What fraction of $2 is 40 cents?
A
1
20
B
20
1
C
5
1
D
1
5
E
1
40
5 2
1
2
÷
3
4
is the same as:
A
5
2
×
4
3
B
5
2
×
3
4
C
2
5
÷
3
4
D
2
5
×
4
3
E
3
2
×
3
4
Short-answer questions
1 Arrange
1
2
,
1
3
,
2
5
and
3
10
in ascending order.
2 Express 5
2
3
as an improper fraction.
3 Find each of the following.
a
2
3
+
1
4
b 4 − 1
1
3
c 2
1
2
+ 3
3
4
d
2
5
×
1
2
e
2
3
÷
1
6
f 1
1
5
×
5
12
4 Write 15% as a simple fraction.
5 Find 25% of $480.
6 Find 12
1
2
% of $480.
7 State whether each of the following is true or false.
a 25% of x = x ÷ 4 b 10% of w =
w
10
c 20% of 50 = 50% of 20 d 1% of x = 100x
8 Which is larger,
2
3
or 67%?
Extended-response question
1 Caleb’s cold and fl u prescription states: ‘Take two pills three times a day with food.’
The bottle contains 54 pills.
a How many pills does Caleb take each day?
b What fraction of the bottle remains after Day 1?
c How many days will it take for the pills to run out?
d If Caleb takes his fi rst dose Friday night before going to bed, on what day will he take his
last dose?
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Semester review 1 333
Chapter 5: Probability
Multiple-choice questions
1 What is the probability of rolling a 6 with a fair 6-sided die?
A 6% B 16% C 16
2
3
% D 17% E 60%
2 Sophie has some jelly beans. Six are red and four are green. She eats a red one, then randomly
chooses another jelly bean. The probability that it is red is:
A 3
5
B 2
5
C 4
9
D 5
9
E 1
2
3 The letters of the alphabet are written on cards and placed in a hat. One letter is chosen randomly.
The probability that it is a vowel is closest to:
A 0.05 B 0.1 C 0.2 D 0.3 E 0.26
4 There are 30 students in a class, of which 17 are boys. A student is chosen at random. The
probability of choosing a girl is:
A 17
30
B 13
17
C 13
30
D 17
13
E 1
13
5
Right
handed
Left
handed
Boys 25 7
Girls 20 8
A survey of some Year 7 students is conducted, asking them whether they are left handed or right
handed. If a student is chosen at random, the chance of choosing a left-handed girl is:
A 1
3
B 2
15
C 2
7
D 5
7
E 8
15
Short-answer questions
1 A spinner is designed with different numbers in each sector. From the spinners A to D shown below:
A
2
4 3
5
1
B
2
3 1
1
C
2
3
1
D
2
1
4 3
5
6
2 1
a Which spinner has the lowest probability of landing on the number 1 in a single spin?
b Which spinner has a 50% probability of landing on the number 1 in a single spin?
c List the spinners in order, from the most likely to land on the number 1 to the least likely.
2 One card is randomly selected from a standard deck of 52 playing cards. Find the probability that
the selected card is:
a red b black c a heart
d an ace e a king f a red 7
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Semester review 1
334
3 The students attending an after school music programme are asked,
‘Do you play the piano?’ and ‘Do you play the violin?’. Their responses
are shown in the Venn diagram. One of the music students is chosen at
random. Find the probability that:
a The student plays neither the piano nor the violin.
b The student plays the piano and the violin.
c The student plays the piano but not the violin.
d The student plays the violin.
e The student plays the piano or the violin.
4 Arrange these events from least likely to most likely.
A: tossing ‘heads’ with a fair coin
B: randomly choosing a king from a standard deck of playing cards
C: rolling a 6 with a fair 6-sided die
D: randomly choosing a red card or a spade from a standard deck of playing cards
E: rolling a number greater than 1 on a fair 6-sided die
Extended-response question
1 A standard deck of playing cards includes 13 cards for each suit: hearts, diamonds, clubs and
spades. Each suit has an ace, king, queen, jack, 2, 3, 4, 5, 6, 7, 8, 9 and 10. One card is drawn at
random from the deck.
Find the following probabilities.
a P(heart) b P(club) c P(diamond or spade)
d P(ace of hearts) e P(number less than 4 and not an ace)
f P(king) g P(ace or heart) h P(queen or club)
Chapter 6: Computation with decimals and fractions
Multiple-choice questions
1 80 +
6
10
+
7
1000
is the same as:
A 8067 B 867 C 80.67 D 80.067 E 80.607
2 Select the incorrect statement.
A 0.707 > 0.7 B 0.770 = 77
100
C 0.07 × 0.7 = 0.49
D 0.7 ×
1
10
= 0.07 E 0.7 × 10 = 7
3 The best estimate for 23.4 × 0.96 is:
A 234 B 230 C 0.234 D 23 E 20
4
3
8
is the same as:
A 0.375 B 3.8 C 0.38 D 2.6
i E 38%
5 6.8 ÷ 0.04 is the same as:
A 68 ÷ 4 B 680 ÷ 4 C 17 D
4
68
E 7 ÷ 0.05
8 2 15
5
piano violin
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Semester review 1 335
Short-answer questions
1 Write each of the following as a decimal.
a two-tenths b
13
100
c
17
10
2 In the decimal 136.094:
a What is the value of the digit 6?
b What is the value of the digit 4?
c What is the decimal, correct to the nearest tenth?
3 Round 18.398741 correct to:
a the nearest whole b 1 decimal place c 2 decimal places
4 Evaluate:
a 15 − 10.93 b 19.7 + 240.6 + 9.03 c 20 − 0.99
d 0.6 × 0.4 e (0.3)2 f
12
0 2 .
5 Find:
a 1.24 − 0.407 b 1.2 + 0.6 × 3 c 1.8 × 0.2 ÷ 0.01
6 If 369 × 123 = 45 387, write down the value of:
a 3.69 × 1.23 b 0.369 × 0.123 c 45.387 ÷ 36.9
7 Find:
a 36.49 × 1000 b 1.8 ÷ 100 c 19.43 × 200
8 For each of the following, circle the larger of each pair.
a
4
5
, 0.79 b 1.1, 11% c
2
3
, 0.6
9 State if each of the following is true or false.
a 0.5 = 50% b 0 15
5
20
. = c 38% = 0.19
d 126% = 1.26 e
4
5
= 0.08 f 1
3
4
= 1.75
Extended-response question
1 The cost of petrol is 146.5 cents per litre.
a Find the cost of 55 L of petrol, correct to the nearest cent.
b Mahir pays cash for his 55 L of petrol. What is the amount that he pays, correct to the nearest
5 cents?
c If the price of petrol is rounded to the nearest cent before the cost is calculated, how much
would 55 L of petrol cost now?
d By how much is Mahir better off if the rounding occurs at the end rather than the beginning?
e Is the result the same if the price drops to 146.2 cents per litre?
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336 Chapter 7 Time
Chapter Time 7
What you will learn
Units of time
Working with time
Using time zones
7A
7B
7C
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luxor temple obelisk
Thousands of years before the use of clocks and even
the introduction of the Gregorian calendar, sundials
were used to tell the time during the day. Egyptian
obelisks dating back to 3500 BC were some of the
earliest sundials used.
The two 3300-year-old twin obelisks, once
marking the entrance of the Luxor temple in Egypt,
are still standing today. One of them, however, was
gifted to France and in 1836 was placed at the centre
of Place de la Concorde in Paris, where it still stands.
nSW Syllabus
for the australian
Curriculum
Strand: Measurement and Geometry
Substrand: tiME
Outcome
A student performs calculations of time
that involve mixed units, and interprets
time zones.
(MA4–13MG)
337
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338 Chapter 7 Time
1 How many:
a hours in one day? b seconds in one minute?
c minutes in one hour? d days in one week?
e months in one year? f days in one year (not a leap year)?
2 What day is it:
a 3 days after Tuesday? b 6 days before Sunday?
c 3 weeks after Wednesday? d 10 minutes after 11:55 p.m. Saturday?
3 Give the time, using a.m. or p.m., that matches these descriptions.
a 2 hours after 3 p.m. b 1 hour before 2:45 a.m.
c 6 hours before 10:37 a.m. d 4 hours after 4:49 p.m.
e 1
1
2
hours after 2:30 p.m. f 3
1
2
hours before 7:15 p.m.
g 2 hours before 12:36 p.m. h 5 hours after 9:14 a.m.
4 Convert the following to the units shown in brackets.
a 60 seconds (minutes) b 120 minutes (hours)
c 49 days (weeks) d 6 hours (minutes)
5 Melissa watched two movies on the weekend. One lasted 1 hour 36 minutes and the other
lasted 2 hours 19 minutes.
a What was the total time Melissa spent watching movies, in hours and minutes?
b What was the total time in minutes?
6 Write the following times as they would be displayed on a digital clock; e.g. 8:15.
a 3 o’clock b half past 2 c a quarter to 6
pre-test
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Measurement and Geometry 339
units of time
Time in minutes and seconds is based on the number
60. Other units of time, including the day and year, are
defi ned by the rate at which the Earth spins on its axis
and the time that the Earth takes to orbit the Sun.
The origin of the units seconds and minutes dates
back to the ancient Babylonians, who used a base 60
number system. The 24-hour day dates back to the
ancient Egyptians, who described the day as 12 hours
of day and 12 hours of night. Today, we use a.m. (ante
meridiem, which is Latin for ‘before noon’) and p.m.
(post meridiem, which is Latin for ‘after noon’) to
represent the hours before and after noon (midday).
During the rule of Julius Caesar, the ancient Romans
introduced the solar calendar, which recognised that the
Earth takes about 365
1
4
days to orbit the Sun. This gave
rise to the leap year, which includes one extra day (in February) every 4 years.
The calendar we use today is called the Gregorian calendar. It was formally introduced by Pope
Gregory XIII in 1582 and includes the 12 months, each with 30 or 31 days, except for February, which
has 28 days or 29 days in a leap year.
number 1 2 3 4 5 6 7 8 9 10 11 12
Month Jan Feb Mar April May June July Aug Sep Oct Nov Dec
Days 31 28/29 31 30 31 30 31 31 30 31 30 31
Time after Christ (ad) is now often referred to as the Common Era (ce) and the time before Christ (bc) is
also referred to as Before the Common Era (bce).
let’s start: Knowledge of time
Do you know the answers to these questions about time and the calendar?
• When is the next leap year?
• Why do we have a leap year?
• Which months have 31 days?
• Why are there different times in different countries or parts of a country?
• What do bc (or bce) and ad (or ce) mean on time scales?
7A
The Earth takes 1 year to orbit the Sun.
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340 Chapter 7 Time
■■ The standard unit of time is the second (s).
■■ Units of time include:
– 1 minute (min) = 60 seconds (s)
– 1 hour (h) = 60 minutes (min)
– 1 day = 24 hours (h)
– 1 week = 7 days
– 1 year = 12 months
■■ a.m. or p.m. is used to describe the 12 hours before and after noon (midday).
■■ 24-hour time shows the number of hours and minutes after midnight.
– 0330 is 3:30 a.m.
– 1530 is 3:30 p.m.
■■ DMS conversion: Most scientifi c and graphics calculators have a DMS (Degrees, Minutes and
Seconds) button or function that converts time in fraction or decimal form to hours, minutes and
seconds.
– For example: 2.26 hours → 2° 15′ 36″, meaning 2 hours, 15 minutes and 36 seconds.
×24
÷24
day hour
×60
÷60
minute
×60
÷60
second
Key ideas
Example 1 Converting units of time
Convert these times to the units shown in brackets.
a 3 days (minutes) b 30 months (years)
Solution Explanation
a 3 days = 3 × 24 h
= 3 × 24 × 60 min
= 4320 min
1 day = 24 hours
1 hour = 60 minutes
b 30 months = 30 ÷ 12 years
= 2
1
2
years
There are 12 months in 1 year.
Example 2 using 24-hour time
Write these times using the system given in brackets.
a 4:30 p.m. (24-hour time) b 1945 hours (a.m./p.m.)
Solution Explanation
a 4:30 p.m. = 1200 + 0430
= 1630 hours
Since the time is p.m., add 12 hours to 0430 hours.
b 1945 hours = 7:45 p.m. Since the time is after 1200 hours, subtract 12 hours.
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Measurement and Geometry 341
Example 3 Converting to hours, minutes and seconds
Convert the following to hours, minutes and seconds.
a Convert mentally 4
1
3
hours. b Use a calculator to convert 6.42 hours.
Solution Explanation
a 4
1
3
= 4 hours, 20 minutes 1
3
of an hour is 20 minutes because
1
3
of 60 = 20.
b 4.42 → 4°25′12″ = 4 hours,
25 minutes and 12 seconds
Use the DMS bottom on your calculator. Ensure your
calculator is in Degree mode.
1 Which months of the year contain:
a 28 or 29 days? b 30 days? c 31 days?
2 From options a to F, match up the time units with the most appropriate description.
a single heartbeat a 1 hour
b 40 hours of work B 1 minute
c duration of a university lecture C 1 day
d bank term deposit D 1 week
e 200-m run E 1 year
f fl ight from Australia to the UK F 1 second
3 State whether you would multiply by 60 (M) or divide by 60 (D) when converting:
a hours to minutes b seconds to minutes
c minutes to hours d minutes to seconds
4 Find the number of:
a seconds in 2 minutes b minutes in 180 seconds
c hours in 120 minutes d minutes in 4 hours
e hours in 3 days f days in 48 hours
g weeks in 35 days h days in 40 weeks
Exercise 7A
WORKING
MATHEMATICALLY
U F
R PS
C
5 Convert these times to the units shown in brackets.
a 3 h (min) b 10.5 min (s) c 240 s (min)
d 90 min (h) e 6 days (h) f 72 h (days)
g 1 week (h) h 1 day (min) i 14 400 s (h)
j 20 160 min (weeks) k 2 weeks (min) l 24 h (s)
m 3.5 h (min) n 0.25 min (s) o 36 h (days)
p 270 min (h) q 75 s (min) r 7200 s (h)
Example 1
WORKING
MATHEMATICALLY
U F
R PS
C
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342 Chapter 7 Time
6 Write these times, using the system shown in brackets.
a 1:30 p.m. (24-hour) b 8:15 p.m. (24-hour) c 10:23 a.m. (24-hour)
d 11:59 p.m. (24-hour) e 0630 hours (a.m./p.m.) f 1300 hours (a.m./p.m.)
g 1429 hours (a.m./p.m.) h 1938 hours (a.m./p.m.) i 2351 hours (a.m./p.m.)
7 Write each of these digital clock displays as a number of hours expressed as a decimal;
e.g. 4:30 is 4.5 hours.
a 1:30 b 4:45 c 7:15 d 3:20
8 Write these times in hours and minutes.
a 2
1
2
hours b 4
1
4
hours c 1
1
3
hours
d 6.5 hours e 3.75 hours f 9.25 hours
9 Round these times to the nearest hour.
a 1:32 p.m. b 5:28 a.m. c 1219 hours d 1749 hours
10 Use the DMS button/function on your calculator to convert the following to hours, minutes
and seconds.
a 7.12 hours b 2.28 hours c 3.05 hours d 8.93 hours
Example 2
Example 3a
Example 3b
WORKING
MATHEMATICALLY
U F
R PS
C
11 Marion reads the following times on an airport display panel. Re-write the times using a.m.
or p.m.
a 0630 b 1425 c 1927
12 When there are 365 days in a year, how many weeks are there in a year? Round your answer to
2 decimal places.
13 Assuming there are 365 days in a year and my birthday falls on a Wednesday this year, on what day
will my birthday fall in 2 years’ time?
WORKING
MATHEMATICALLY
U F
R PS
C
7A
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Measurement and Geometry 343
14 Explain why:
a 4.2 hours is 4 hours and 12 minutes
b 2 hours and 10 minutes is 2.16
.
hours
15 a To convert from hours to seconds, what single number do you multiply by?
b To convert from days to minutes, what single number do you multiply by?
c To convert from seconds to hours, what single number do you divide by?
d To convert from minutes to days, what single number do you divide by?
16 Without the use of the DMS function on your calculator, but allowing the use of your calculator’s
basic functions, convert the following to hours, minutes and seconds.
a 2.4 hours b 7.18 hours c 9.92 hours
WORKING
MATHEMATICALLY
U F
R PS
C
Enrichment
17 There is a rich history associated with the way civilisations
have recorded time and with the development of today’s
calendar. Use the internet as a research tool and write
5–10 points summarising this topic.
You may want to include topics relating to:
■■ the introduction of the Gregorian calendar
■■ the number of days in the year and in a leap year
■■ the lunar calendar
■■ some ancient methods of recording time
■■ the Roman infl uence on today’s calendar
■■ the use of sundials.
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344 Chapter 7 Time
Working with time
It is a common activity to make a calculation involving
time. For example, working out the duration of a sporting
event or show, fi nding a train’s time of arrival, using a
timetable, or estimating time durations for trade quotes
or cooking. These calculations may involve operations
such as addition or subtraction, and the use of the
different ways that time can be displayed.
let’s start: Your mental strategy
Mental strategies are helpful in working out the sums
and differences associated with time. Try to work out
the answer to these simple problems and then describe
the mental strategy you used to your group or class.
• A football match begins at 1:45 p.m. and fi nishes at
4:10 p.m. What is the duration of the match?
• A train leaves the city station at 8:40 a.m. and
arrives in town 2
1
2
hours later. At what time does
the train reach town?
• The construction of the Great Pyramid of Giza
began in 2560 bc. How old does that make the
pyramid now?
7B
■■ Mental strategies should be used to solve simple problems involving time.
– The total time to build two models, which took 45 minutes and 55 minutes each, is
55 + 5 + 40 = 1 hour 40 minutes.
– The time duration of a taxi ride beginning at 2:50 p.m. and ending at 3:35 p.m. is
35 + 10 = 45 minutes or 60 – 15 = 45 minutes.
■■ The bc and ad timeline is similar to a number line in that it extends in both directions, with 0
representing the birth of Christ. The time duration from 500 bc to 2000 ad is therefore
500 + 2000 = 2500 years.
■■ When solving problems to do with time, it may be helpful to use the same type of units.
2000 1500 1000 500 0 500 1000 1500 2000
BC AD
Key ideas
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Measurement and Geometry 345
Example 4 Calculating time intervals
Mentally calculate the time interval between these pairs of times.
a 4:35 p.m. to 9:10 p.m. b 5 h 20 min 20 s to 7 h 40 min 10 s
Solution Explanation
a Time interval = 4 h + 25 min + 10 min
= 4 h 35 min
There are 4 hours from 5 p.m. to 9 p.m., another
25 minutes before 5 p.m., and then 10 minutes after
9 p.m.
b Time interval = 2 h + 20 min - 10 s
= 2 h 19 min 50 s
2 hours and 20 minutes after 5 h 20 min and 20 s is
10 seconds too many, so subtract 10 seconds. A
calculator method might include entering 7°40′10″ -
5°20′20″, using the DMS button or °, ′ and ″ symbols.
Example 5 Using timetables
Use this train timetable for Bathurst to Penrith to answer these questions.
Station a.m. p.m.
Bathurst 7:11 2:41
Lithgow 8:15 3:45
Bell 8:32 4:02
Mount Victoria 8:42 4:13
Katoomba 8:57 4:29
Springwood 9:29 5:01
Penrith 9:54 5:30
a How long does it take to travel from:
i Bathurst to Lithgow in the morning?
ii Lithgow to Penrith in the morning?
iii Bathurst to Penrith in the afternoon?
b Luke travels from Lithgow to Bell in the morning and then from Bell to Katoomba in the afternoon.
What is Luke’s total travel time?
Solution Explanation
a i 1 h 4 min
ii 1 h + 45 min – 6 min = 1 h 39 min
iii 3 h – 11 min = 2h 49 min
8:15 is 1 hour plus 4 minutes after 7:11.
1 hour and 45 minutes takes 8:15 to 10:00, so subtract
6 minutes to get 9:54.
3 hours after 2:41 is 5:41, so subtract 11 minutes.
b 17 min + 27 min = 44 min 8:15 to 8:32 is 17 minutes, and 4:02 to 4:29 is 27 minutes.
This gives a total of 44 minutes.
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346 Chapter 7 Time
1 State whether each of the following is true or false.
a There are 60 seconds in 1 hour.
b 12 noon is between morning and afternoon.
c There are 35 minutes between 9:35 a.m. and 10:10 a.m.
d There are 17 minutes between 2:43 p.m. and 3:10 p.m.
e The total of 39 minutes and 21 minutes is 1 hour.
2 What is the time difference between these times?
a 12 noon and 6:30 p.m. b 12 midnight and 10:45 a.m.
c 12 midnight and 4:20 p.m. d 11 a.m. and 3:30 p.m.
3 Add these time durations to give a total time.
a 1 h 30 min and 2 h 30 min b 4 h 30 min and 1 h 30 min
c 2 h 15 min and 1 h 15 min d 6 h 15 min and 2 h 30 min
e 3 h 45 min and 1 h 30 min f 4 h 45 min and 2 h 45 min
Exercise 7B
WORKING
MATHEMATICALLY
U F
R PS
C
4 Mentally calculate the time interval between these pairs of times.
a 2:40 a.m. to 4:45 a.m. b 4:20 p.m. to 6:30 p.m.
c 1:50 p.m. to 5:55 p.m. d 12:07 p.m. to 2:18 p.m.
e 6:40 a.m. to 8:30 a.m. f 1:30 a.m. to 5:10 a.m.
g 10:35 p.m. to 11:22 p.m. h 3:25 a.m. to 6:19 a.m.
i 6:18 a.m. to 9:04 a.m. j 7:51 p.m. to 11:37 p.m.
5 Write the time for these descriptions.
a 4 hours after 2:30 p.m. b 10 hours before 7 p.m.
c 3
1
2
hours before 10 p.m. d 7
1
2
hours after 9 a.m.
e 6
1
4
hours after 11:15 a.m. f 1
3
4
hours before 1:25 p.m.
6 Calculate the time interval between these pairs of times. You may wish to use the DMS or °, ′, ″
buttons on your calculator.
a 2 h 10 min 20 s to 4 h 20 min 30 s b 5 h 30 min 15 s to 8 h 45 min 21 s
c 9 h 46 min 13 s to 10 h 50 min 27 s d 1 h 30 min 10 s to 2 h 25 min 5 s
e 6 h 43 min 28 s to 8 h 37 min 21 s f 4 h 51 min 42 s to 10 h 36 min 10 s
7 For each of the following, add the time durations to fi nd the total time. Give your answers in hours,
minutes and seconds.
a 2
1
4
hours and 3
1
2
hours b 5
1
3
hours and 2
2
3
hours
c 6.2 hours and 2.9 hours d 0.3 hours and 4.2 hours
e 2 h 40 min 10 s and 1 h 10 min 18 s f 10 h 50 min 18 s and 2 h 30 min 12 s
Example 4a
Example 4b
WORKING
MATHEMATICALLY
U F
R PS
C
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Measurement and Geometry 347
8 A ferry takes Selina from Cabarita to Circular Quay
in 23 minutes and 28 seconds. The return trip takes
19 minutes and 13 seconds. What is Selina’s total
travel time?
9 Use this train timetable for Fairfi eld to Redfern to answer these questions.
Station a.m. p.m.
Fairfi eld 7:32 2:43
Granville 7:44 2:56
Auburn 7:48 2:59
Ashfi eld 8:01 3:12
Redfern 8:11 3:23
a How long does it take to travel from:
i Fairfi eld to Auburn in the morning?
ii Granville to Redfern in the morning?
iii Auburn to Redfern in the afternoon?
iv Fairfi eld to Redfern in the afternoon?
b Does it take longer to travel from Fairfi eld to Redfern
in the morning or afternoon?
c Jeremiah travels from Fairfi eld to Auburn in the
morning and then from Auburn to Redfern in the
afternoon. What is Jeremiah’s total travel time?
10 A scientist argues that dinosaurs died out
52 million years ago, whereas another says they
died out 108 million years ago. What is the
difference in their time estimates?
Example 5
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348 Chapter 7 Time
11 Three essays are marked by a teacher. The fi rst takes 4 minutes and 32 seconds to mark,
the second takes 7 minutes and 19 seconds, and the third takes 5 minutes and 37 seconds.
What is the total time taken to complete marking the essays?
12 Adrian arrives at school at 8:09 a.m. and leaves at 3:37 p.m. How many hours and minutes is
Adrian at school?
13 On a fl ight to Europe, Janelle spends 8 hours and 36 minutes on a fl ight from Melbourne to Kuala
Lumpur, Malaysia, 2 hours and 20 minutes at the airport at Kuala Lumpur, and then
12 hours and 19 minutes on a fl ight to Geneva, Switzerland. What is Janelle’s total travel time?
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14 To convert the speed 10 metres per second (m/s) to kilometres per hour (km/h), you must
multiply by 3600 and divide by 1000 to give a factor of 3.6. Explain why.
15 Give a reason why airports and other workplaces might use a 24-hour display rather than
use a.m. or p.m.
16 Calculate a rent of $400 per week as a yearly amount. Assume 365 days in a year and show
your working.
WORKING
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Enrichment: time challenges
17 A doctor earns $180 000 working 40 weeks per year, 5 days per week, 10 hours per day. What does
the doctor earn in each of these time periods?
a per day b per hour
c per minute d per second (in cents)
18 What is the angle between the hour and minute hands on an analogue clock at these times?
a 6:15 a.m. b 4:55 p.m. c 5:47 a.m.
19 Rex takes 3 hours to paint a standard-sized bedroom, whereas his mate Wilbur takes 4 hours
to paint a room of the same size. How long will it take to paint a standard-sized room if they
work together?
6
4
7 5
3
8
9
10 2
11 12 1
7B
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Measurement and Geometry 349
using time zones
During the 19th century, as railways and telecommunications
developed, it became increasingly important to deal with the
difference in local times. Standard time zones were introduced
around the world, most of which are one-hourly deviations
from standard time, which is taken to be the time in Greenwich,
England (United Kingdom). Standard time or Greenwich Mean
Time (GMT) is now called Coordinated Universal Time (UTC).
In Australia, we use three main time zones: the Western,
Central and Eastern Standard Time zones, which alter for
daylight saving in some States and Territories.
let’s start: Time zone discussion
In groups or as a class, discuss what you know about Australian
and international time zones. You may wish to include:
• time zones and time differences within Australia
• UTC (or GMT)
• daylight saving
• the timing of telecasts of sporting events around the world.
7C
■■ The Earth is divided into 24 time zones (one for each hour).
– Twenty-four 15° lines of longitude divide the Earth into its time zones. Time zones
also depend on a country’s borders and its proximity to other countries. (See map on
pages 350–351 for details.)
– Time is based on the time in a place called Greenwich, United Kingdom, and this is
called Coordinated Universal Time (UTC) or Greenwich Mean Time (GMT).
– Places east of Greenwich are ahead in time.
– Places west of Greenwich are behind in time.
■■ Australia has three time zones:
– Eastern Standard Time (EST), which is UTC plus 10 hours.
– Central Standard Time (CST), which is UTC plus 9.5 hours.
– Western Standard Time (WST), which is UTC plus 8 hours.
Key ideas
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350 Chapter 7 Time
Sun
1:00
Sun
2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00
ANGOLA
ARGENTINA
BRAZIL
BOLIVIA
CANADA
CHILE
COLUMBIA
FRANCE
GERMANY
GREENLAND
ICELAND
IRELAND
ITALY
ALGERIA LIBYA
MEXICO
PERU
PORTUGAL SPAIN
SWEDEN
CHAD
MALI NIGER
NAMIBIA
MAURITANIA
MOROCCO
UNITED
KINGDOM
NORWAY
NIGERIA
DEM. REP.
OF THE CONGO
VENEZUELA
CUBA
UNITED STATES
ALASKA
GREECE
POLAND
o
1
3 1
6 4 0
6
4
3½
6 5
8
7
1
5
4
3
9
J Auckland
K Edinburgh
L Greenwich
M Johannesburg
N London
P New York
Q Vancouver
R Washington, DC
K
L
N
P
R
Q
World cities key
11 10 9 8 7 6 5 4 3 2 1 0 1
11 10 9 8 7 6 5 4 3 2 1 0 1
9½
8
20:00 21:00 22:00
9 10
Western
Australia
South
Australia
Northern
Territory
Queensland
New South Wales
Tasmania
Eastern
Standard
Time
Central
Standard
Time
Western
Standard
Time
A Adelaide
B Alice Springs
C Brisbane
D Cairns
E Canberra, ACT
F Darwin
G Hobart
H Melbourne
I Perth
Australian cities key
AUSTRALIA
A
G
H
I
B
F
D
C
Victoria E
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Measurement and Geometry 351
Sun
24:00
Sun
20:00
Sun
13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00 1:00
ANGOLA
CHINA
EGYPT
ETHIOPIA
FRANCE
GERMANY
INDIA
SRI LANKA
PAKISTAN
IRAQ
ITALY
JAPAN
KAZAKHSTAN
LIBYA
MALAYSIA
NEW ZEALAND
SOUTH
AFRICA
SWEDEN
THAILAND
TURKEY
ZIMBABWE
CHAD SUDAN
NIGER
NAMIBIA
TANZANIA
SAUDI
ARABIA
UNITED
KINGDOM
NORWAY
FINLAND
IRAN
RUSSIA
MONGOLIA
MADAGASCAR
ZAMBIA
NIGERIA
DEM. REP.
OF THE CONGO
AFGHANISTAN
NEPAL
BURMA
PHILIPPINES
INDONESIA
AUSTRALIA
GREECE
UKRAINE
POLAND
ROMANIA
SYRIA
1 2
3
3
4
4
4
3½
4½ 5
5
6
7
5¾
5½
5½
6½
8
8
9
9
10
10
11 12
9½
11½
12¾
5
3
9
J
M
1 2 3 4 5 6 7 8 9 10
11 12 12 11
1 2 3 4 5 6 7 8 9 10 11 12 12 11
Daylight saving
No daylight saving
WA
NT
QLD
SA
NSW
ACT
VIC
TAS
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352 Chapter 7 Time
Example 7 Using time zones
Coordinated Universal Time (UTC) is based on the time in Greenwich, United Kingdom. Use the world
time zone map (on pages 350–351) to answer the following.
a When it is 2 p.m. UTC, find the time in these places.
i France ii China iii Queensland iv Alaska
b When it is 9:35 a.m. in New South Wales, Australia, find the time in these places.
i Alice Springs ii Perth iii London iv central Greenland
Solution Explanation
a i 2 p.m. + 1 hour = 3 p.m. Use the time zone map to see that France is to the east of
Greenwich and is in a zone that is 1 hour ahead.
ii 2 p.m. + 8 hours = 10 p.m. From the time zone map, China is 8 hours ahead of
Greenwich.
iii 2 p.m. + 10 hours = 12 midnight Queensland uses Eastern Standard Time, which is 10 hours
ahead of Greenwich.
iv 2 p.m. - 9 hours = 5 a.m. Alaska is to the west of Greenwich, in a time zone that is
9 hours behind.
Example 6 Working with Australian time zones
Use the Australian standard time zones map (on pages 350–351) to help with these questions. When it is
8:30 a.m. in New South Wales, what time is it in each of the following?
a Queensland b Northern Territory c Western Australia
Solution Explanation
a 8:30 a.m. Using standard time, NSW and Qld are in the same time zone.
b 8:00 a.m. NT is UTC + 9
1
2
hours, whereas NSW is UTC + 10 hours.
So NT is 1
2
hour behind.
c 6:30 a.m. WA is UTC + 8 hours and so is 2 hours behind NSW.
■■ Daylight saving is the practice of moving clocks forward, usually by one hour, to create more
daylight in the evening and less daylight in the morning. In some parts of Australia, daylight saving
runs from the first Sunday in October to the first Sunday in April.
■■ The International Date Line separates one calendar day from the next. So crossing the date line
from west to east subtracts one day.
Key ideas
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Measurement and Geometry 353
1 Which Australian States and Territories use daylight saving time?
2 a How many hours in front of Coordinated Universal Time (UTC) are these countries and
Australian States?
i Victoria, Australia ii South Australia iii Western Australia
iv Thailand v China vi Egypt
b How many hours behind Coordinated Universal Time (UTC) are the following countries?
i Iceland ii eastern Brazil
iii Columbia iv Peru
3 When it is 10 a.m. Monday in New Zealand, what day of the week is it in the USA?
Exercise 7C
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4 Use the Australian standard time zones map (on pages 350–351) to help answer the following.
When it is 10 a.m. in New South Wales, what time is it in these States and Territories?
a Victoria b South Australia
c Tasmania d Northern Territory
e Western Australia f The Australian Capital Territory (ACT)
5 Use the Australian standard time zones map (on pages 350–351) to help answer the following. When
it is 4:30 p.m. in Western Australia, what time is it in the following States?
a South Australia b New South Wales c Tasmania
6 Use the time zone map (on pages 350–351) to fi nd the time in the following places, when it
is 10 a.m. UTC.
a Spain b Turkey c Tasmania d Darwin
e Argentina f Peru g Alaska h Portugal
7 Use the time zone map (on pages 350–351) to fi nd the time in these places, when it is
3:30 p.m. in New South Wales.
a United Kingdom b Libya c Sweden
d Perth e Japan f central Greenland
g Alice Springs h New Zealand
Example 6
Example 7a
Example 7b
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b i 9:35 a.m. –
1
2
hour = 9:05 a.m. Alice Springs uses Central Standard Time, which is
1
2
hour
behind Eastern Standard Time.
ii 9:35 a.m. - 2 hours = 7:35 a.m. Perth uses Western Standard Time, which is 2 hours behind
Eastern Standard Time.
iii 9:35 a.m. - 10 hours = 11:35 p.m.
(the day before)
UTC (time in Greenwich, United Kingdom) is 10 hours
behind EST.
iv 9:35 a.m. - 13 hours = 8:35 p.m.
(the day before)
Central Greenland is 3 hours behind UTC in Greenwich,
so is 13 hours behind EST.
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354 Chapter 7 Time
13 Use the daylight saving time zone map (on pages 350–351) to help answer the following. During
daylight saving time, when it is 9:30 a.m. in Sydney, what time is it in the following States?
a Queensland b Victoria c South Australia d Western Australia
14 During daylight saving time, Alice drives from Kingscliff in New South Wales to the Gold
Coast in Queensland. How will she need to adjust her wristwatch when she crosses the border?
15 Explain why Eastern Standard Time in Australia is 11 hours ahead of the United Kingdom for a
proportion of the year.
16 Monty departs on a 20-hour fl ight from Brisbane to London, United Kingdom, at 5 p.m. on
20 April. Give the time and date of his arrival in London (ignoring UK daylight saving time).
17 Elsa departs on an 11-hour fl ight from Johannesburg, South Africa, to Perth at 6:30 a.m. on
25 October. Give the time and date of her arrival in Perth. (Note: South Africa does not use
daylight saving time.)
WORKING
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Enrichment: time anomalies
18 There are a number of interesting anomalies associated with time zones. You may wish to use the
internet to help explore these topics.
a Usually, States and Territories to the east are ahead of those in the west. During daylight saving
time, however, this is not true for all States in Australia. Can you fi nd these States and explain
why?
b Broken Hill is in New South Wales but does not use the New South Wales time zone. Explore.
c Does Lord Howe Island (part of New South Wales) use the same time as New South Wales all year
round? Discuss.
d Are there any other time zone anomalies in Australia or overseas that you can describe?
8 What is the time difference between these pairs of places?
a United Kingdom and Kazakhstan b South Australia and New Zealand
c Queensland and Egypt d Peru and Angola (in Africa)
e Mexico and Germany
9 Rick in Wollongong, NSW wants to watch a soccer match that
is being televised at 2 p.m. in England (United Kingdom).
What time will he need to switch on his television in Wollongong?
(Use Eastern Standard Time.)
10 At the London Olympics a rowing race is scheduled to
begin at 11:35 a.m. What time will this be in Broome,
Western Australia?
11 A 2-hour football match starts at 2:30 p.m. Eastern Standard Time
(EST) in Newcastle, NSW. What time will it be in the
United Kingdom when the match fi nishes?
12 If the date is 29 March and it is 3 p.m. in Perth, what is the time and date in these places?
a Italy b Alaska c Chile
WORKING
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7C
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Measurement and Geometry 355
1 Teaghan takes 7 hours to fl y from Dubbo, New South Wales to Esperance, Western Australia. She
departs at 7 a.m. What is the time in Esperance when she arrives? (Use Australian standard time.)
2 Albert can dig a post hole in 6 minutes and Sue can dig a post hole in 8 minutes. How long will
it take for them to dig one hole if they work together?
3 When it is a Tuesday on 25 October in a particular year, what day will it be on 25 October in the
following year, if it is not a leap year?
4 The average time for fi ve snails to complete a race is 2 min 30 s. Four of the snails’ race times are
2 min 20 s, 3 min, 2 min 10 s and 1 min 50 s. What is the fi fth snail’s race time?
5 At a speedway, the yellow car completes a lap every 30 seconds and the blue car completes a lap
every 50 seconds. If the cars both start at the same place, how long will it take for the blue car to
‘lap’ the yellow car?
puzzles and challenges
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356 Chapter 7 Time
Chapter summary
Finding duration
4:35 to 6:44 = 2 h + 9 min
6:19 to 8:07 = 2 h − 12 min
= 1 h 48 min
2 h 26 min 12s + 3 h 24 min + 56 s
= 5 h + 50 min + 68 s
= 5 h 51 min 8 s
Mental strategies
Addition
Conversion
day hour minute second
÷ 24 ÷ 60 ÷ 60
× 24 × 60 × 60
DMS conversion
1.47 hours →
1 h 28′ 12″
Time zones
Australian time zones
UTC (Coordinated Universal
Time) is the time in
Greenwich, England
(United Kingdom).
- Countries to the east
are ahead
- Countries to the west
are behind
State/
Territory
WA
NT
SA
Tas.
Vic.
NSW
ACT
Qld
+ 8
+ 9
+ 9
+ 10
+ 10
+ 10
+ 10
+ 10
+ 8
+ 9
+ 10
+ 11
+ 11
+ 11
+ 11
+ 10
Standard
Daylight
saving
1
21
2
1
2
1
2
9:26 a.m. is 0926
5:03 p.m. is 1703
24-hour time
Time
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Measurement and Geometry 357
Multiple-choice questions
1 The number of minutes in 3 hours is:
A 180 B 60 C 90 D 300 E 200
2 How many years are there in 42 months?
A 2 B 2.5 C 2 D 3.5 E 4
3 When written using 24-hour time, 4:26 p.m. is:
A 0626 B 1226 C 0426 D 1426 E 1626
4 Converting 2
2
3
hours to hours and minutes gives:
A 2 h 67 min B 2 h 35 min C 2 h 23 min
D 2 h 40 min E 2 h 30 min
5 Converting 2.64 hours to hours, minutes and seconds gives:
A 2 h 40 min 12 s B 2 h 38 min 24 s C 3 h 4 min 0 s
D 2 h 30 min 10 s E 2 h 60 min 4 s
6 The time taken to make and assemble two chairs is 3 hours 40 minutes and 15 seconds and
2 hours 38 minutes and 51 seconds. Hence, the total build time is:
A 5 h 58 min 6 s B 6 h 20 min 6 s C 6 h 19 min 6 s
D 6 h 19 min 66 s E 6 h 18 min 6 s
7 The time interval from 3:36 a.m. to 4:27 a.m. is:
A 51 min B 49 min C 41 min D 39 min E 61 min
8 How many hours is Western Australia behind New South Wales during Australian standard time?
A 5 B 4 C 3 D 2 E 1.5
9 If it is 12 noon during daylight saving time in South Australia, what time is it in Queensland?
A 2 p.m. B 2:30 p.m. C 1 p.m. D 12:30 p.m. E 11:30 a.m.
10 When it is 4 a.m. UTC, the time in Sydney is:
A 1:30 p.m. B 1 p.m. C 2 p.m. D 3 p.m. E 3 a.m.
Short-answer questions
1 Convert these times to the units shown in brackets.
a 1
1
2
h (min) b 120 s (min) c 48 h (days)
d 3 weeks (days) e 1 day (min) f 1800 s (h)
2 Re-write these times, using the system shown in brackets.
a 4 a.m. (24-hour time) b 3:30 p.m. (24-hour time)
c 7:19 p.m. (24-hour time) d 0635 (a.m./p.m.)
e 1251 (a.m./p.m.) f 2328 (a.m./p.m.)
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358 Chapter 7 Time
3 Re-write these times, using hours and minutes.
a 3
1
2
hours b 4
1
3
hours c 6.25 hours d 1.75 hours
4 Use the DMS button/function on your calculator to convert the following to hours, minutes
and seconds.
a 3.6 hours b 6.92 hours c 11.44 hours
5 Margaret is catching a train leaving at 1330 in
London and arriving at 1503 in York. What
will be Margaret’s travel time?
6 Calculate the time interval between these pairs of times. Give your answer in hours, minutes
and seconds.
a 7:43 a.m. to 1:36 p.m. b 2 h 30 min 10 s to 6 h 36 min 5 s
c 5 h 52 min 6 s to 7 h 51 min 7 s d 0931 to 1309
7 Use this train timetable for Telarah to Newcastle to answer the following questions.
Station a.m. p.m.
Telarah 7:30 2:52
Metford 7:42 3:04
Sandgate 7:55 3:16
Hamilton 8:10 3:30
Newcastle 8:16 3:36
a How long does it take to travel from:
i Telarah to Sandgate in the morning?
ii Metford to Newcastle in the morning?
iii Sandgate to Newcastle in the afternoon?
iv Telarah to Newcastle in the afternoon?
b Does it take longer to travel from Telarah to Newcastle in the morning or afternoon?
c Ashdi travels from Telarah to Sandgate in the morning, then from Sandgate to Newcastle in the
afternoon. What is Ashdi’s total travel time?
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Measurement and Geometry 359
8 Use the Australian time zone maps (on pages 350–351) to help answer these questions.
a During Australian standard time it is 7:45 a.m. in South Australia. What time is it in:
i New South Wales? ii Western Australia?
b During Australian daylight saving time it is 4:36 p.m. in New South Wales. What time is it in:
i Western Australia? ii Queensland?
9 An AFL match telecast begins at 2:10 p.m. Eastern Standard Time. At what time will someone in the
Northern Territory need to switch on the television if they want to watch the game?
Extended-response question
1 Use the International time zone maps (on pages 350–351) to answer these questions.
a When it is 11 a.m. UTC, state the time in:
i Sydney ii Ethiopia iii Pakistan
b When it is 3:30 p.m. in New South Wales, state the time in:
i Zimbabwe ii China iii Bolivia
c When it is 6 a.m. Tuesday in New South Wales, state the day of the week in:
i India ii Canada
d Chris flies from Sydney, leaving at 8 a.m., and travels for 7 hours, arriving in Kuala
Lumpur, Malaysia. What is the time in Kuala Lumpur when he arrives?
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Chapter 8 Algebraic 360 techniques 1
Chapter
Algebraic
techniques 1
8
What you will learn
Introduction to formal algebra
Substituting positive numbers into algebraic expressions
Equivalent algebraic expressions
Like terms
Multiplying, dividing and mixed operations
Expanding brackets
Applying algebra EXTENSION
Substitution involving negative numbers and mixed operations
Number patterns EXTENSION
Spatial patterns EXTENSION
Tables and rules EXTENSION
The Cartesian plane and graphs EXTENSION
8A
8B
8C
8D
8E
8F
8G
8H
8I
8J
8K
8L
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number and algebra Designing robots
nSW Syllabus
for the australian
Curriculum
Strand: number and algebra
Substrand: alGEBRaiC TECHniQuES
Outcome
A student generalises number properties
to operate with algebraic expressions.
(MA4–8NA)
361
Algebra provides a way to describe everyday activities
using mathematics alone. By allowing letters like x or y
to stand for unknown numbers, different concepts and
relationships can be described easily.
Engineers apply their knowledge of algebra and
geometry to design buildings, roads, bridges, robots, cars,
satellites, planes, ships and hundreds of other structures
and devices that we take for granted in our world today.
To design a robot, engineers use algebraic rules to
express the relationship between the position of the
robot’s ‘elbow’ and the possible positions of a robot’s
‘hand’. Although they cannot think for themselves,
electronically programmed robots can perform tasks
cheaply, accurately and consistently, without ever getting
tired or sick or injured, or the need for sleep or food!
Robots can have multiple arms, reach much farther than a
human arm and can safely lift heavy, awkward objects.
Robots are used extensively in car manufacturing.
Using a combination of robots and humans, Holden’s
car manufacturing plant in Elizabeth, South Australia
fully assembles each car in 76 seconds!
Understanding and applying mathematics has made
car manufacturing safer and also extremely effi cient.
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362 Chapter 8 Algebraic techniques 1
1 If = 7, write the value of each of the following.
a + 4 b – 2 c 12 – d 3 ×
2 Write the value of × 4 if:
a = 2 b = 9 c = 10 d = 2.5
3 Write the answer to each of the following computations.
a 4 and 9 are added b 3 is multiplied by 7
c 12 is divided by 3 d 10 is halved
4 Write down the following, using numbers and the symbols +, ÷, × and –.
a 6 is tripled b 10 is halved
c 12 is added to 3 d 9 is subtracted from 10
5 For each of the tables, describe the rule relating the input and output numbers. For example:
Output = 2 × input.
a
Input 1 2 3 5 9
Output 3 6 9 15 27
b Input 1 2 3 4 5
Output 6 7 8 9 10
c Input 1 5 7 10 21
Output 7 11 13 16 27
d Input 3 4 5 6 7
Output 5 7 9 11 13
6 If the value of x is 8, fi nd the value of:
a x + 3 b x – 2 c x × 5 d x ÷ 4
7 Find the value of each of the following.
a 4 × 3 + 8 b 4 × (3 + 8) c 4 × 3 + 2 × 5 d 4 × (3 + 2) × 5
8 Find the value of each of the following.
a 50 – (3 × 7 + 9) b 24 ÷ 2 – 6 c 24 ÷ 6 – 2 d 24 ÷ (6 – 2)
9 If = 5, write the value of each of the following.
a – 4 b × 2 – 1 c ÷ 5 + 2 d × 7 + 10
e × f × ÷ g 3 × – 15 h 2
pre-test
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number and algebra 363
introduction to formal algebra
A pronumeral is a letter that can represent any number. The
choice of letter used is not signifi cant mathematically, but can
be used as an aide to memory. For instance, h might stand for
someone’s height and w might stand for someone’s weight.
The table shows the salary Petra earns for various hours of
work if she is paid $12 an hour.
number of hours Salary earned ($)
1 12 × 1 = 12
2 12 × 2 = 24
3 12 × 3 = 36
n 12 × n = 12n
Rather than writing 12 × n, we write 12n because multiplying
a pronumeral by a number is common and this notation saves
space. We can also write 18 ÷ n as 18
n
.
let’s start: Pronumeral stories
Ahmed has a jar with b biscuits in it that he is taking to a birthday party. He eats 3 biscuits and then
shares the rest equally among 8 friends. Each friend receives b − 3
8
biscuits. This is a short story for
the expression b − 3
8
.
• Try to create another story for b − 3
8
, and share it with others in the class.
• Can you construct a story for 2t + 12? What about 4(k + 6)?
8A
Using pronumerals, we can work out a total
salary for any number of hours of work.
■■ x + y + 3 is an example of an algebraic expression.
■■ x and y are pronumerals, which are letters that stand for numbers.
■■ In the example x + y + 3, x and y could represent any numbers, so they could be called variables.
■■ a × b is written as ab and a ÷ b is written as a
b
.
■■ A term consists of numbers and pronumerals combined with multiplication or division.
For example, 5 is a term, x is a term, 9a is a term, abc is a term, 4
3
xyz is a term.
■■ A term that does not contain any pronumerals is called a constant term. All numbers by
themselves are constant terms.
■■ An (algebraic) expression consists of numbers and pronumerals combined with any
mathematical operations. For example, 3x + 2yz is an expression and 8 ÷ (3a – 2b) + 41
is also an expression. Any term is also an expression.
■■ A coeffi cient is the number in front of a pronumeral. For example, the coeffi cient of y in the
expression 8x + 2y + z is 2. If there is no number in front, then the coeffi cient is 1, since 1z
and z are equal.
Key ideas
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Chapter 8 Algebraic 364 techniques 1
Example 1 The terminology of algebra
a List the individual terms in the expression 3a + b + 13c.
b State the coefficient of each pronumeral in the expression 3a + b + 13c.
c Give an example of an expression with exactly two terms, one of which is a constant term.
Solution Explanation
a There are three terms: 3a, b and 13c. Each part of an expression is a term.
Terms get added (or subtracted) to make an expression.
b The coefficient of a is 3, the coefficient
of b is 1 and the coefficient of c is 13.
The coefficient is the number in front of a pronumeral.
For b the coefficient is 1 because b is the same as 1 × b.
c 27a + 19
(There are many other expressions.)
This expression has two terms, 27a and 19, and 19 is
a constant term because it is a number without any
pronumerals.
Example 2 Writing expressions from word descriptions
Write an expression for each of the following.
a 5 more than k b 3 less than m c the sum of a and b
d double the value of x e the product of c and d
Solution Explanation
a k + 5 5 must be added to k to get 5 more than k.
b m – 3 3 is subtracted from m.
c a + b a and b are added to obtain their sum.
d 2 × x or just 2x x is multiplied by 2. The multiplication sign is optional.
e c × d or just cd c and d are multiplied to obtain their product.
Example 3 Expressions involving more than one operation
Write an expression for each of the following without using the × or ÷ symbols.
a p is halved, then 4 is added
b the sum of x and y is taken and then divided by 7
c the sum of x and one-seventh of y
d 5 is subtracted from k and the result is tripled
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number and algebra 365
SoluTion ExplanaTion
a p
2
+ 4
p is divided by 2, then 4 is added.
b x y
x y ( + ) ÷ = +
7
7
x and y are added. This whole expression is divided
by 7. By writing the result as a fraction, the brackets
are no longer needed.
c x
y +
7
or x + y 1
7
x is added to one-seventh of y, which is
y
7
.
d (k – 5) × 3 = 3(k – 5) 5 subtracted from k gives the expression k – 5. Brackets
must be used to multiply the whole expression by 3.
1 The expression 4x + 3y + 24z + 7 has four terms.
a List the terms. b What is the constant term?
c What is the coeffi cient of x? d Which letter has a coeffi cient of 24?
2 Match each of the word descriptions on the left with the correct mathematical expression on
the right.
a the sum of x and 4 a x - 4
b 4 less than x B
x
4
c the product of 4 and x C 4 - x
d one-quarter of x D 4x
e the result from subtracting x from 4 E
4
x
f 4 divided by x F x + 4
Example 1
Example 2
Exercise 8A
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3 For each of the following expressions, state:
i the number of terms; and
ii the coeffi cient of n.
a 17n + 24 b 31 – 27a + 15n
c 15nw + 21n + 15 d 15a – 32b +
4
3
xy + 2n
e n + 51 f 5bn – 12 +
d
5
+ 12n
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Chapter 8 Algebraic 366 techniques 1
4 Write an expression for each of the following without using the × or ÷ symbols.
a 1 more than x b the sum of k and 5
c double the value of u d 4 lots of y
e half of p f one-third of q
g 12 less than r h the product of n and 9
i t is subtracted from 10 j y is divided by 8
5 Write an expression for each of the following without using the × or ÷ symbols.
a 5 is added to x, then the result is doubled.
b a is tripled, then 4 is added.
c k is multiplied by 8, then 3 is subtracted.
d 3 is subtracted from k, then the result is multiplied by 8.
e The sum of x and y is multiplied by 6.
f x is multiplied by 7 and the result is halved.
g p is halved and then 2 is added.
h The product of x and y is subtracted from 12.
6 Describe each of these expressions in words.
a 7x b a + b c (x + 4) × 2 d 5 – 3a
Example 3
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7 Nicholas buys 10 lolly bags from a supermarket.
a If there are 7 lollies in each bag, how many lollies does he buy in total?
b If there are n lollies in each bag, how many lollies does he buy in total? Hint: Write an
expression involving n.
8 Mikayla is paid $x per hour at her job. Write an expression for each of the following.
a How much does Mikayla earn if she works 8 hours?
b If Mikayla gets a pay rise of $3 per hour, what is her new hourly wage?
c If Mikayla works for 8 hours at the increased hourly rate, how much does she earn?
9 Recall that there are 100 centimetres in 1 metre and 1000 metres in 1 kilometre. Write expressions for
each of the following.
a How many metres are there in x km?
b How many centimetres are there in x metres?
c How many centimetres are there in x km?
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Number and Algebra 367
10 A group of people go out to a restaurant, and the total amount they must pay is $A. They
decide to split the bill equally. Write expressions to answer the following questions.
a If there are 4 people in the group,
how much do they each pay?
b If there are n people in the group,
how much do they each pay?
c One of the n people has a voucher
that reduces the total bill by $20.
How much does each person pay
now?
11 There are many different ways of describing the expression
a + b
4
in words. One way is ‘The sum of
a and b is divided by 4.’ What is another way?
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12 If x is a whole number between 10 and 99, classify each of these statements as true or false.
a x must be smaller than 2 × x.
b x must be smaller than x + 2.
c x – 3 must be greater than 10.
d 4 × x must be an even number.
e 3 × x must be an odd number.
13 If b is an even number greater than 3, classify each of these statements as true or false.
a b + 1 must be even.
b b + 2 could be odd.
c 5 + b could be greater than 10.
d 5b must be greater than b.
14 If c is a number between 10 and 99, sort the following in ascending order (i.e. smallest to largest).
3c, 2c, c – 4, c ÷ 2, 3c + 5, 4c – 2, c + 1, c × c.
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Enrichment: Many words compressed
15 One advantage of writing expressions in symbols rather than words is that it takes up less space.
For instance, ‘twice the value of the sum of x and 5’ uses eight words and can be written as
2(x + 5).
Give an example of a worded expression that uses more than 10 words and then write it as a
mathematical expression.
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Chapter 8 Algebraic 368 techniques 1
Substituting positive numbers into
algebraic expressions
Substitution involves replacing pronumerals (like x and y) with numbers and obtaining a single number as
a result. For example, we can evaluate 4 + x when x is 11, to get 15.
let’s start: Sum to 10
The pronumerals x and y could stand for any
number.
• What numbers could x and y stand for if
you know that x + y must equal 10? Try to
list as many pairs as possible.
• If x + y must equal 10, what values could
3x + y equal? Find the largest and smallest
values.
8B
■■ To evaluate an expression or to substitute values means to replace each pronumeral in an
expression with a number to obtain a fi nal value. For example, if x = 3 and y = 8, then x + 2y
evaluated gives 3 + 2 × 8 =19.
■■ A term like 4a means 4 × a. When substituting a number we must include the multiplication sign,
since two numbers written as 42 is very different from the product 4 × 2.
■■ Replace all the pronumerals with numbers, then evaluate using the normal order of operations
seen in Chapter 1:
– brackets
– multiplication and division from left to right
– addition and subtraction from left to right.
For example: (4 3) 2 20 4 2 7 2 20 4 2
14 5 2
9 2
11
+ × − ÷ + = × − ÷ +
= − +
= +
=
Key ideas
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Number and Algebra 369
Example 4 Substituting a pronumeral
Given that t = 5, evaluate:
a t + 7 b 8t c
10
4
t
+ − t
Solution Explanation
a t + = +
=
7 5 7
12
Replace t with 5 and then evaluate the expression,
which now contains no pronumerals.
b 8 8
8 5
40
t = t
= ×
=
× Insert × where it was previously implied, then
substitute in 5. If the multiplication sign is not
included, we might get a completely incorrect answer
of 85.
c 10
4
10
5
4 5
2 4 5
1
t
+ − t = + −
= + −
=
Replace all occurrences of t with 5 before evaluating.
Note that the division (10 ÷ 5) is calculated before the
addition and subtraction.
Example 5 Substituting multiple pronumerals
Substitute x = 4 and y = 7 to evaluate these expressions.
a 5x + y + 8
b 80 – (2xy + y)
Solution Explanation
a 5 8 5 8
5 4 7 8
20 7 8
35
x + y + = × x + y +
= × + +
= + +
=
Insert the implied multiplication sign between 5 and x
before substituting the values for x and y.
b 80 2 80 2
80 2 4 7 7
80 56
− + = − × × +
= − × × +
= −
( ) ( )
( )
(
xy y x y y
+
= −
=
7
80 63
17
)
Insert the multiplication signs, and remember the order
in which to evaluate.
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Chapter 8 Algebraic 370 techniques 1
Example 6 Substituting with powers and roots
If p = 4 and t = 5, fi nd the value of:
a 3p2 b t 2 + p3 c p2 + 32
SoluTion ExplanaTion
a 3 3
3 4 4
48
p2 = × p × p
= × ×
=
Note that 3p2 means 3 × p × p, not (3 × p)2.
b t2 p3 52 43
5 5 4 4 4
25 64
89
+ = +
= × + × ×
= +
=
t is replaced with 5, and p is replaced with 4.
Remember that 43 means 4 × 4 × 4.
c p2 32 42 32
25
5
+ = +
=
=
Recall that the square root of 25 must be 5 because
5 × 5 = 25.
Exercise 8B
1 Use the correct order of operations to evaluate the following.
a 4 + 2 × 5 b 7 – 3 × 2 c 3 × 6 – 2 × 4 d (7 – 3) × 2
2 What number would you get if you replaced b with 5 in the expression 12 + b?
3 What number is obtained when x = 3 is substituted into the expression 5 × x ?
4 What is the result of evaluating 10 – u if u is 7?
5 Calculate the value of 12 + b if:
a b = 5 b b = 8 c b = 60 d b = 0
6 If x = 5, evaluate each of the following. Set out your solution in a manner similar to that
shown in Example 4.
a x + 3 b x × 2 c 14 – x
d 2x + 4 e 3x + 2 – x f 13 – 2x
g 2(x + 2) + x h 30 – (4x + 1) i
20
3
x
+
j (x )
x
+ 5 ×
10 k
x + 7
4
l 10 − x
x
Example 4a
Example 4b,c
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Number and Algebra 371
m 7x + 3(x – 1) n 40 – 3x – x o x + x( x + 1)
p + +
x
x x
30
2 ( 3) q 100 – 4(3 + 4x) r 6 3 8
2
( x )
x
−
+
7 Substitute a = 2 and b = 3 into each of these expressions and evaluate.
a 2a + 4 b 3a – 2 c a + b
d 3a + b e 5a – 2b f 7ab + b
g ab – 4 + b h 2 × (3a + 2b) i 100 – (10a + 10b)
j 12 6
a b
+ k
ab
b
3
+ l 100
a + b
8 Evaluate the expression 5x + 2y when:
a x = 3 and y = 6 b x = 4 and y = 1 c x = 7 and y = 3
d x = 0 and y = 4 e x = 2 and y = 0 f x = 10 and y = 10
9 Copy and complete each of these tables.
a n 1 2 3 4 5 6
n + 4 5 8
b x 1 2 3 4 5 6
12 – x 9
c b 1 2 3 4 5 6
2(b – 1)
d q 1 2 3 4 5 6
10q – q
10 Evaluate each of the following, given that a = 9, b = 3 and c = 5.
a 3c 2 b 5b 2 c a 2 – 3 3 d 2b 2 +
a
3
– 2c
e a + 3ab f b2 + 42 g 24
2
6
3
+ b
h (2c)2 – a2
11 A number is substituted for b in the expression 7 + b and gives the result 12. What is the
value of b?
12 A number is substituted for x in the expression 3x – 1. If the result is a two-digit number, what
value might x have? Try to describe all the possible answers.
13 Copy and complete the table.
x 5 9 12
x + 6 11 7
4x 20 24 28
Example 5
Example 6
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Chapter 8 Algebraic 372 techniques 1
14 Assume x and y are two numbers, where xy = 24.
a What values could x and y equal if they are whole numbers? Try to list as many as possible.
b What values could x and y equal if they can be decimals, fractions or whole numbers?
15 Dugald substitutes different whole numbers into the expression 5 × (a + a). He notices that the
result always ends in the digit 0. Try a few values and explain why this pattern occurs.
Enrichment: Missing numbers
16 a Copy and complete the following table, in which x and y are whole numbers.
x 5 10 7
y 3 4 5
x + y 9 14 7
x – y 2 3 8
xy 40 10 0
b If x and y are two numbers where x + y and x × y are equal, what values might x and y have?
Try to fi nd at least three (they do not have to be whole numbers).
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8B
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number and algebra 373
Equivalent algebraic expressions
In algebra, as when using words, there are often many ways to express the same thing. For example, we
can write ‘the sum of x and 4’ as x + 4 or 4 + x, or even x + 1 + 1 + 1 + 1.
No matter what number x is, x + 4 and 4 + x will always be equal. We say that the expressions
x + 4 and 4 + x are equivalent because of this.
By substituting different numbers for the pronumerals it is possible to see whether two expressions are
equivalent. Consider the four expressions in this table.
3a + 5 2a + 6 7a + 5 – 4a a + a + 6
a = 0 5 6 5 6
a = 1 8 8 8 8
a = 2 11 10 11 10
a = 3 14 12 14 12
a = 4 17 14 17 14
From this table it becomes apparent that 3a + 5 and 7a + 5 – 4a are equivalent, and that 2a + 6 and
a + a + 6 are equivalent.
let’s start: Equivalent expressions
Consider the expression 2a + 4.
• Write as many different expressions as possible that are equivalent to 2a + 4.
• How many equivalent expressions are there?
• Try to give a logical explanation for why 2a + 4 is equivalent to 4 + a × 2.
8C
■■ Two expressions are called equivalent when they are equal, regardless of what numbers are
substituted for the pronumerals.
■ ■ ■ For example, 5x + 2 is equivalent to 2 + 5x and to 1 + 5x + 1 and to x + 4x + 2.
Key ideas
This collection of pronumerals and numbers can be arranged into many different equivalent expressions.
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Chapter 8 Algebraic 374 techniques 1
Example 7 Equivalent expressions
Which two of these expressions are equivalent: 3x + 4, 8 – x, 2x + 4 + x ?
SoluTion ExplanaTion
3x + 4 and 2x + 4 + x are equivalent. By drawing a table of values, we can see straight away
that 3x + 4 and 8 – x are not equivalent, since they differ
for x = 2.
3x + 4 and 2x + 4 + x are equal for all values, so they are
equivalent.
x = 1 x = 2 x = 3
3x + 4 7 10 13
8 – x 7 6 5
2x + 4 + x 7 10 13
Exercise 8C
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1 a Copy the following table into your workbook and complete. C
x = 0 x = 1 x = 2 x = 3
2x + 2
(x + 1) × 2
b Fill in the gap: 2x + 2 and (x + 1) × 2 are __________ expressions.
2 a Copy the following table into your workbook and complete.
x = 0 x = 1 x = 2 x = 3
5x + 3
6x + 3
b Are 5x + 3 and 6x + 3 equivalent expressions?
3 Show that 6x + 5 and 4x + 5 + 2x are equivalent by completing the table.
6x + 5 4x + 5 + 2x
x = 1
x = 2
x = 3
x = 4
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Number and Algebra 375
4 For each of the following, choose a pair of equivalent expressions.
a 4x, 2x + 4, x + 4 + x
b 5a, 4a + a, 3 + a
c 2k + 2, 3 + 2k, 2(k + 1)
d b + b, 3b, 4b – 2b
5 Match up the equivalent expressions below.
a 3x + 2x A 6 – 3x
b 4 – 3x + 2 B 2x + 4x + x
c 2x + 5 + x C 5x
d x + x – 5 + x D 4 – x
e 7x E 3x + 5
f 4 – 3x + 2x F 3x – 5
6 Write two different expressions that are equivalent
to 4x + 2.
7 The rectangle shown opposite has a perimeter given
by b + l + b + l. Write an equivalent expression
for the perimeter.
8 There are many expressions that are equivalent to 3a + 5b + 2a – b + 4a. Write an equivalent
expression with as few terms as possible.
9 The expressions a + b and b + a are equivalent and only contain two terms. How many
expressions are equivalent to a + b + c and contain only three terms? Hint: Rearrange the
pronumerals.
10 Prove that no two of these four expressions are equivalent: 4 + x, 4x, x – 4, x ÷ 4.
11 Generalise each of the following patterns in numbers to give two equivalent expressions. The
first one has been done for you.
a Observation: 3 + 5 = 5 + 3 and 2 + 7 = 7 + 2 and 4 + 11 = 11 + 4.
Generalised: The two expressions x + y and y + x are equivalent.
b Observation: 2 × 5 = 5 × 2 and 11 × 5 = 5 × 11 and 3 × 12 = 12 × 3.
c Observation: 4 × (10 + 3) = 4 × 10 + 4 × 3 and 8 × (100 + 5) = 8 × 100 + 8 × 5.
d Observation: 100 – (4 + 6) = 100 – 4 – 6 and 70 – (10 + 5) = 70 – 10 – 5.
e Observation: 20 – (4 – 2) = 20 – 4 + 2 and 15 – (10 – 3) = 15 – 10 + 3.
f Observation: 100 ÷ 5 ÷ 10 = 100 ÷ (5 × 10) and 30 ÷ 2 ÷ 3 = 30 ÷ (2 × 3).
Example 7
b


b
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Chapter 8 Algebraic 376 techniques 1
12 a Show that the expression 4 × (a + 2) is equivalent to 8 + 4a using a table of values for a
between 1 and 4.
b Write an expression using brackets that is equivalent to 10 + 5a.
c Write an expression without brackets that is equivalent to 6 × (4 + a).
Enrichment: Thinking about equivalence
13 3a + 5b is an expression containing two terms. List two expressions containing three terms that
are equivalent to 3a + 5b.
14 Three expressions are given: expression A, expression B and expression C.
a If expressions A and B are equivalent, and expressions B and C are equivalent, does this
mean that expressions A and C are equivalent? Try to prove your answer.
b If expressions A and B are not equivalent, and expressions B and C are not equivalent, does
this mean that expressions A and C are not equivalent? Try to prove your answer.
Each shape above is made from three identically-sized tiles of length l and breadth b. Which of the shapes
have the same perimeter?
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number and algebra 377
like terms
Whenever we have terms with exactly the same pronumerals, they are called ‘like terms’ and can be
collected and combined. For example, 3x + 5x can be simplifi ed to 8x. If the two terms do not have
exactly the same pronumerals, they must be kept separate; for example, 3x + 5y cannot be simplifi ed – it
must be left as it is.
let’s start: Simplifying expressions
• Try to fi nd a simpler expression that is equivalent to
1a + 2b + 3a + 4b + 5a + 6b + … + 19a + 20b
• What is the longest possible expression that is equivalent to 10a + 20b + 30c?
Assume that all coeffi cients must be whole numbers greater than zero.
• Compare your expressions to see who has the longest one.
8D
■■ Like terms are terms containing exactly the same pronumerals, although not necessarily in
the same order.
– 5ab and 3ab are like terms.
– 4a and 7b are not like terms.
– 2acb and 4bac are like terms.
■■ Like terms can be combined within an expression to create a simpler expression that is
equivalent. For example, 5ab + 3ab can be simplifi ed to 8ab.
■■ If two terms are not like terms (such as 4x and 5y), they can still be added to get an
expression like 4x + 5y, but this expression cannot be simplifi ed further.
Key ideas
Example 8 identifying like terms
Which of the following pairs are like terms?
a 3x and 2x b 3a and 3b c 2ab and 5ba
d 4k and k e 2a and 4ab f 7ab and 9aba
SoluTion ExplanaTion
a 3x and 2x are like terms. The pronumerals are the same.
b 3a and 3b are not like terms. The pronumerals are different.
c 2ab and 5ba are like terms. The pronumerals are the same, even though they are
written in a different order (one a and one b).
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Chapter 8 Algebraic 378 techniques 1
d 4k and k are like terms. The pronumerals are the same.
e 2a and 4ab are not like terms. The pronumerals are not exactly the same (the first
term contains only a and the second term has a and b).
f 7ab and 9aba are not like terms. The pronumerals are not exactly the same (the first
term contains one a and one b, but the second term
contains two a terms and one b).
Example 9 Simplifying using like terms
Simplify the following by collecting like terms.
a 7b + 2 + 3b
b 12d – 4d + d
c 5 + 12a + 4b – 2 – 3a
d 13a + 8b + 2a – 5b – 4a
e 12uv + 7v – 3vu + 3v
Solution Explanation
a 7b + 2 + 3b = 10b + 2 7b and 3b are like terms, so they are combined. They
cannot be combined with 2 because it contains no
pronumerals.
b 12d – 4d + d = 9d All the terms here are like terms. Remember that d
means 1d when combining them.
c 5 + 12a + 4b - 2 - 3a
= 12a - 3a + 4b + 5 - 2
= 9a + 4b + 3
12a and 3a are like terms. We subtract 3a because it
has a minus sign in front of it. We can also combine the
5 and the 2 because they are like terms.
d 13a + 8b + 2a - 5b - 4a
= 13a + 2a - 4a + 8b - 5b
= 11a + 3b
Combine like terms, remembering to subtract any term
that has a minus sign in front of it.
e 12uv + 7v - 3vu + 3v
= 12uv + 3vu + 7v + 3v
= 9uv + 10v
Combine like terms. Remember that 12uv and 3vu are
like terms (i.e. they have the same pronumerals).
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number and algebra 379
1 For each of the following terms, state all the pronumerals that occur in it.
a 4xy b 3abc c 2k d pq
2 Copy the following sentences into your workbook and fi ll in the gaps to make the sentences
true. More than one answer might be possible.
a 3x and 5x are ____________ terms.
b 4x and 3y are not ____________ ____________.
c 4xy and 4yx are like ____________.
d 4a and ____________ are like terms.
e x + x + 7 and 2x + 7 are ____________ expressions.
f 3x + 2x + 4 can be written in an equivalent way as ____________.
3 Classify the following pairs as like terms (L) or not like terms (N).
a 7a and 4b b 3a and 10a c 18x and 32x
d 4a and 4b e 7 and 10b f x and 4x
g 5x and 5 h 12ab and 4ab i 7cd and 12cd
j 3abc and 12abc k 3ab and 2ba l 4cd and 3dce
4 Simplify the following by collecting like terms.
a a + a b 3x + 2x c 4b + 3b d 12d – 4d
e 15u – 3u f 14ab – 2ab g 8ab + 3ab h 4xy – 3xy
5 Simplify the following by collecting like terms.
a 2a + a + 4b + b b 5a + 2a + b + 8b c 3x – 2x + 2y + 4y
d 4a + 2 + 3a e 7 + 2b + 5b f 3k – 2 + 3k
g 7f + 4 – 2f + 8 h 4a – 4 + 5b + b i 3x + 7x + 3y – 4x + y
j 10a + 3 + 4b – 2a k 4 + 10h – 3h l 10x + 4x + 31y – y
m 10 + 7y – 3x + 5x + 2y n 11a + 4 – 3a + 9 o 3b + 4b + c + 5b – c
p 7ab + 4 + 2ab q 9xy + 2x – 3xy + 3x r 2cd + 5dc – 3d + 2c
s 5uv + 12v + 4uv – 5v t 7pq + 2p + 4qp – q u 7ab + 32 – ab + 4
6 Ravi and Marissa each work for n hours per week. Ravi earns $27 per hour and Marissa earns
$31 per hour.
a Write an expression for the amount Ravi earns in one week.
b Write an expression for the amount Marissa earns in one week.
c Write a simplifi ed expression for the total amount Ravi and Marissa earn in one week.
Example 8
Example 9
Exercise 8D
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Chapter 8 Algebraic 380 techniques 1
7 The length of the line segment shown could be expressed as a + a + 3 + a + 1.
a a 3 a 1
a Write the length in the simplest form.
b What is the length of the segment if a is equal to 5?
8 Let x represent the number of marbles in a
standard-sized bag. Xavier bought
4 bags and Cameron bought 7 bags. Write
simplifi ed expressions for:
a the number of marbles Xavier has
b the number of marbles Cameron has
c the total number of marbles that Xavier
and Cameron have
d the number of extra marbles that
Cameron has compared to Xavier
9 Simplify the following by collecting like
terms.
a 3xy + 4xy + 5xy b 4ab + 5 + 2ab c 5ab + 3ba + 2ab
d 10xy – 4yx + 3 e 10 – 3xy + 8xy + 4 f 3cde + 5ecd + 2ced
g 4 + x + 4xy + 2xy + 5x h 12ab + 7 – 3ab + 2 i 3xy – 2y + 4yx
10 a Test, using a table of values, that 3x + 2x is equivalent to 5x.
b Prove that 3x + 2y is not equivalent to 5xy.
11 a Test that 5x + 4 – 2x is equivalent to 3x + 4.
b Prove that 5x + 4 – 2x is not equivalent to 7x + 4.
c Prove that 5x + 4 – 2x is not equivalent to 7x – 4.
Enrichment: How many rearrangements?
12 The expression a + 3b + 2a is equivalent to 3a + 3b.
a List two other expressions with three terms that are equivalent to 3a + 3b.
b How many expressions, consisting of exactly three terms added together, are equivalent to
3a + 3b? All coeffi cients must be whole numbers greater than 0.
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number and algebra 381
Multiplying, dividing and mixed operations
To multiply a number by a pronumeral, we have already seen we can write them next to each other. For
example, 7a means 7 × a, and 5abc means 5 × a × b × c. The order in which numbers or pronumerals are
multiplied is unimportant, so 5 × a × b × c = a × 5 × c × b = c × a × 5 × b. When writing a product
without × signs, the numbers are written fi rst.
We write
7
3
xy
xz
as shorthand for (7xy) ÷ (3xz).
We can simplify fractions like
10
15
by dividing by common factors, such as
10
15
5 2
5 3
2
3
= ×
×
= .
Similarly, common variables can be cancelled in a division like 7
3
xy
xz
, giving
7
3
7
3
xy
xz
y
z
= .
let’s start: Rearranging terms
5abc is equivalent to 5bac because the order of multiplication does
not matter. In what other ways could 5abc be written?
8E
5 × a × b × c = ?
■■ a × b is written ab.
■■ a ÷ b is written
a
b
.
■■ a × a is written a2.
■■ Because of the commutative property of multiplication (e.g. 2 × 7 = 7 × 2), the order in
which values are multiplied is not important. So 3 × a and a × 3 are equivalent.
■■ Because of the associative property of multiplication (e.g. 3 × (5 × 2) and (3 × 5) × 2 are
equal), brackets are not required when only multiplication is used. So 3 × (a × b) and
(3 × a) × b are both written as 3ab.
■■ Numbers should be written fi rst in a term and pronumerals are generally written in
alphabetical order. For example, b × 2 × a is written as 2ab.
■■ When dividing, any common factor in the numerator and denominator can be cancelled.
For example:
2 1
1 1
4
2
a b 2
bc
a
c
=
Key ideas
Example 10 Simplifying expressions with multiplication
a Write 4 × a × b × c without multiplication signs.
b Simplify 4a × 2b × 3c, giving your fi nal answer without multiplication signs.
c Simplify 3w × 4w.
SoluTion ExplanaTion
a 4 × a × b × c = 4abc When pronumerals are written next to each other they
are being multiplied.
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Chapter 8 Algebraic 382 techniques 1
b 4a × 2b × 3c = 4 × a × 2 × b × 3 × c
= 4 × 2 × 3 × a × b × c
= 24abc
c 3w × 4w = 3 × w × 4 × w
= 3 × 4 × w × w
= 12w2
First, insert the missing multiplication signs.
Now we can rearrange to bring the numbers to the
front.
4 × 2 × 3 = 24 and a × b × c = abc, giving the fi nal
answer.
First, insert the missing multiplication signs.
Rearrange to bring numbers to the front.
3 × 4 = 12 and w × w is written as w2.
Example 11 Simplifying expressions with division
a Write (3x + 1) ÷ 5 without a division sign.
b Simplify the expression
8
12
ab
b
.
SoluTion ExplanaTion
a (3x + 1) ÷ 5 = 3 1
5
x + The brackets are no longer required as it becomes
clear that all of 3x + 1 is being divided by 5.
b
8
12
8
12
ab
b
a b
b
= × ×
×
= × × ×
× ×
2 4
3 4
a b
b
= 2
3
a
Insert multiplication signs to help spot common
factors.
8 and 12 have a common factor of 4.
Cancel out the common factors of 4 and b.
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1 Chen claims that 7 × d is equivalent to d × 7. C
a If d = 3, fi nd the values of 7 × d and d × 7. b If d = 5, fi nd the values of 7 × d and d × 7.
c If d = 8, fi nd the values of 7 × d and d × 7. d Is Chen correct in his claim?
2 Classify each of the following statements as true or false.
a 4 × n can be written as 4n. b n × 3 can be written as 3n.
c 4 × b can be written as b + 4. d a × b can be written as ab.
e a × 5 can be written as 50a.
3 a Simplify the fraction
12
18
. (Note: This is the same as 2 6
3 6
×
×
.)
b Simplify the fraction
2000
3000
. (Note: This is the same as 2 1000
3 1000
×
×
.)
c Simplify 2
3
a
a
. (Note: This is the same as
2
3
×
×
a
a
.)
Exercise 8E
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Number and Algebra 383
4 Match up these expressions with the correct way to write them.
a 2 × u A 3u
b 7 × u B 5
u
c 5 ÷ u C 2u
d u × 3 D
u
5
e u ÷ 5 E 7u
5 Write each of these expressions without any multiplication signs.
a 2 × x b 5 × p c 8 × a × b
d 3 × 2 × a e 5 × 2 × a × b f 2 × b × 5
g x × 7 × z × 4 h 2 × a × 3 × b × 6 × c i 7 × 3 × a × 2 × b
j a × 2 × b × 7 × 3 × c k 9 × a × 3 × b × d × 2 l 7 × a × 12 × b × c
6 Simplify these expressions.
a 3a × 12 b 7d × 9 c 2 × 4e
d 3 × 5a e 4a × 3b f 7e × 9g
g 8a × bc h 4d × 7af i a × 3b × 4c
j 2a × 4b × c k 4d × 3e × 5fg l 2cb × 3a × 4d
7 Simplify these expressions.
a w × w b a × a c 3d × d
d 2k × k e p × 7p f q × 3q
g 6x × 2x h 3z × 5z i 9r × 4r
8 Simplify these expressions.
a x ÷ 5 b z ÷ 2 c a ÷ 12
d b ÷ 5 e 2 ÷ x f 5 ÷ d
g x ÷ y h a ÷ b i (4x + 1) ÷ 5
j (2x + y) ÷ 5 k (2 + x) ÷ (1 + y) l (x – 5) ÷ (3 + b)
m 2x + y ÷ 5 n 2 + x ÷ 1 + y o x – 5 ÷ 3 + b
p 3 × 2b - 2b q 3 × (2b - 2b) r 6b + 15b ÷ 3
s (6b + 15b) ÷ 3 t (c - 2c) × 4 u c - 2c × 4
9 Simplify the following expressions by dividing by any common factors. Remember that
a
a
1
= .
a
2
5
x
x
b
5
9
a
a
c
9
4
ab
b
d
2
5
ab
a
e
2
4
x
f
9
12
x
g
10
15
a
a
h
30
40
y
y
i
4
2
a
j
21
7
x
x
k
4
2
xy
x
l
9
3
x
xy
Example 10a
Example 10b
Example 10c
Example 11a
Example 11b
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Chapter 8 Algebraic 384 techniques 1
10 Write a simplifi ed expression for the area of the following rectangles. Recall that for
rectangles, Area = length × breadth.
a k
3
b
x
6 c 3x
4y
11 The weight of a single muesli bar is x grams.
a What is the weight of 4 bars? Write an expression.
b If Jamila buys n bars, what is the total weight of
her purchase?
c Jamila’s cousin Roland buys twice as many bars
as Jamila. What is the total weight of Roland’s
purchase?
12 We can factorise a term like 15ab by writing it as
3 × 5 × a × b. Numbers are written in prime factor
form and pronumerals are given with multiplication signs. Factorise the following.
a 6ab b 21xy c 4efg d 33q2r
13 Five friends go to a restaurant. They split the bill evenly, so each spends the same amount.
a If the total cost is $100, how much do they each spend?
b If the total cost is $C, how much do they each spend? Write an expression.
14 The expression 3 × 2p is the same as the expression 2 2 2
1 2 3
p p p
( ) ( ) ( )
+ + .
a What is a simpler expression for 2p + 2p + 2p? (Hint: Combine like terms.)
b 3 × 2p is shorthand for 3 × 2 × p. How does this relate to your answer in part a?
15 The area of the rectangle shown is 3a. The length and breadth of this rectangle are now doubled.
a Draw the new rectangle, showing its dimensions.
b Write a simplifi ed expression for the area of the new rectangle.
c Divide the area of the new rectangle by the area of the old rectangle. What
do you notice?
d What happens to the area of the original rectangle if you triple both the
length and the breadth?
Enrichment: Managing powers
16 The expression a × a can be written as a 2 and the expression a × a × a can be written as a 3.
a What is 3a 2b 2 when written in full with multiplication signs?
b Write 7 × x × x × y × y × y without any multiplication signs.
c Simplify 2a × 3b × 4c × 5a × b × 10c × a.
d Simplify 4a 2 × 3ab 2 × 2c 2.
3
a
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number and algebra 385
Expanding brackets
We have already seen that there are different ways of writing two equivalent expressions. For example,
4a + 2a is equivalent to 2 × 3a, even though they look different.
Note that 3(7 + a) = 3 × (7 + a), which is equivalent to 3 lots of 7 + a.
So, 3(7 + a) = 7 + a + 7 + a + 7 + a
= 21 + 3a
It is sometimes useful to have an expression that is written with brackets, like 3 × (7 + a), and
sometimes it is useful to have an expression that is written without brackets, like 21 + 3a.
let’s start: Total area
What is the total area of the rectangle shown at right? Try to write two
expressions, only one of which includes brackets.
8F
3
7 a
■■ Expanding (or eliminating) brackets involves writing an equivalent expression without
brackets. This can be done by writing the bracketed portion a number of times or by
multiplying each term.
2
2 2
(a b) a b a b
a b
+ = + + +
= +
or 2 2 2
2 2
(a b) a b
a b
+ = × + ×
= +
■■ To eliminate brackets, you can use the distributive law, which states that:
a(b + c) = ab + ac and a(b – c) = ab – ac
■■ The distributive law is used in arithmetic.
■■ The process of removing brackets using the distributive law is called expansion.
■■ When expanding, every term inside the brackets must be multiplied by the term outside the
brackets.
Key ideas
For example: 5 × 27 = 5(20 + 7)
= 5 × 20 + 5 × 7
= 100 + 35
= 135
Many of the simpler expressions in algebra can be
thought of in terms of the areas of rectangles.
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Chapter 8 Algebraic 386 techniques 1
Example 12 Expanding brackets by simplifying repeated terms
Repeat the expression that is inside the brackets and then collect like terms. The number outside the
brackets is the number of repeats.
a 2(a + k) b 3(2m + 5)
Solution Explanation
a 2(a + k) = a + k + a + k
= 2a + 2k
Two repeats of the expression a + k.
Simplify by collecting the like terms.
b 3(2m + 5) = 2m + 5 + 2m + 5 + 2m + 5
= 6m + 15
Three repeats of the expression 2m + 5.
Simplify by collecting the like terms.
Example 13 Expanding brackets using rectangle areas
Write two equivalent expressions for the area of each rectangle shown, only one of which includes
brackets.
a
2
5 x b
3
12
a
c 2
7
b
a
Solution Explanation
a Using brackets: 2(5 + x)
Without brackets: 10 + 2x
The whole rectangle has height 2 and breadth 5 + x.
The smaller rectangles have area 2 × 5 = 10 and
2 × x = 2x, so they are added.
b Using brackets: 12(a + 3)
Without brackets: 12a + 36
The dimensions of the whole rectangle are 12 and a + 3.
Note that, by convention, we do not write (a + 3)12.
The smaller rectangles have area
12 × a = 12a and 12 × 3 = 36.
c Using brackets: (a + 7)(b + 2)
Without brackets: ab + 2a + 7b + 14
The whole rectangle has height a + 7 and breadth b + 2.
Note that brackets are used to ensure we are multiplying
the entire height by the entire breadth.
The diagram can be split into four rectangles, with areas
ab, 2a, 7b and 14.
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number and algebra 387
Example 14 Expanding using the distributive law
Expand the following expressions.
a 5(x + 3) b 8(a – 4)
c 3(a + 2b) d 5a(3p – 7q)
SoluTion ExplanaTion
a 5(x + 3) = 5 × x + 5 × 3
= 5x + 15
Use the distributive law.
5(x + 3) = 5x + 5 × 3
Simplify the result.
b 8(a – 4) = 8 × a – 8 × 4
= 8a – 32
Use the distributive law with subtraction.
8(a - 4) = 8a - 8 × 4
Simplify the result.
c 3(a + 2b) = 3 × a + 3 × 2b
= 3a + 6b
Use the distributive law.
3(a + 2b) = 3a + 3 × 2b
Simplify the result, remembering that 3 × 2b = 6b.
d 5a (3p – 7q) = 5a × 3p – 5a × 7q
= 15ap – 35aq
Use the distributive law.
5a(3p - 7q) = 5a × 3p - 5a × 7q
Simplify the result, remembering that
5a × 3p = 15ap and 5a × 7q = 35aq.
Exercise 8F
1 The expression 3(a + 2) can be written as (a + 2) + (a + 2) + (a + 2).
a Simplify this expression by collecting like terms.
b Write 2(x + y) in full without brackets and simplify the result.
c Write 4(p + 1) in full without brackets and simplify the result.
d Write 3(4a + 2b) in full without brackets and simplify the result.
2 The area of the rectangle shown can be written as 4(x + 3).
a What is the area of the green rectangle?
b What is the area of the red rectangle?
c Write the total area as an expression, without using brackets.
3 Copy and complete the following computations, using the distributive law.
a 3 × 21 = 3 × (20 + 1) b 7 × 34 = 7 × (30 + 4) c 5 × 19 = 5 × (20 - 1)
= 3 × 20 + 3 × 1 = 7 × ___ + 7 × ___ = 5 × ___ -5 × ___
= ___ + ___ = ___ + ___ = ___ - ___
= ___ = ___ = ___
Example 12
x 3
4
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Chapter 8 Algebraic 388 techniques 1
4 a Copy and complete the following table. Remember to follow the rules for correct order
of operations.
4(x + 3) 4x + 12
x = 1 = 4(1 + 3)
= 4(4)
= 16
= 4(1) + 12
= 4 + 12
= 16
x = 2
x = 3
x = 4
b Fill in the gap: The expressions 4(x + 3) and 4x + 12 are _____________.
5 For the following rectangles, write two equivalent expressions for the area.
a 4
12
x b 8
9
z
c 3
3
a
b
6 Use the distributive law to expand the following.
a 6( y + 8) b 7(l + 4) c 8(s + 7) d 4(2 + a)
e 7(x + 5) f 3(6 + a) g 9(9 – x) h 5( j – 4)
i 8( y – 8) j 8(e – 7) k 6(e – 3) l 10(8 – y)
7 Use the distributive law to expand the following.
a 10(6g – 7) b 5(3e + 8) c 5(7w + 10) d 5(2u + 5)
e 7(8x – 2) f 3(9v – 4) g 7(q – 7) h 4(5c – v)
i 2(2u + 6) j 6(8l + 8) k 5(k – 10) l 9(o + 7)
8 Use the distributive law to expand the following.
a 6i (t – v) b 2d (v + m) c 5c (2w – t) d 6e (s + p)
e d (x + 9s) f 5a (2x + 3v) g 5j (r + 7p) h i (n + 4w)
i 8d (s – 3t ) j f (2u + v) k 7k (2v + 5y) l 4e (m + 10y)
9 Write an expression for each of the following and then expand it.
a A number, x, has 3 added to it and the result is multiplied by 5.
b A number, b, has 6 added to it and the result is doubled.
c A number, z, has 4 subtracted from it and the result is multiplied by 3.
d A number, y, is subtracted from 10 and the result is multiplied by 7.
Example 13
Example 14a,b
Example 14c
Example 14d
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Number and Algebra 389
10 In a school classroom there is one teacher as well as an unknown number of boys and girls.
a If the number of boys is b and the number of girls is g, write an expression for the total
number of people in the classroom, including the teacher.
b The teacher and all the students are each wearing two socks. Write two different expressions
for the total number of socks being worn, one with brackets and one without.
11 When expanded, 4(3x + 6y) gives 12x + 24y. Find two other expressions that expand to
12x + 24y.
12 The distance around a rectangle is given by the expression 2(l + b), where l is the length and
b is the breadth. What is an equivalent expression for this distance?
13 Use a diagram of a rectangle like that in Question 2 to prove that 5(x + 3) = 5x + 15.
14 Use a diagram of a rectangle to prove that (a + 2)(b + 3) = ab + 2b + 3a + 6.
15 When expanded, 5(2x + 4y) gives 10x + 20y.
a How many different ways can the missing numbers be filled with whole numbers for the
equivalence ( x + y) = 10x + 20y?
b How many different expressions expand to give 10x + 20y if fractions or decimals are
included?
Enrichment: Expanding sentences
16 Using words, people do a form of expansion. Consider these two statements.
Statement A: ‘John likes tennis and football.’
Statement B: ‘John likes tennis and John likes football.’
Statement B is an ‘expanded form’ of statement A, which is equivalent in its meaning but shows
more clearly that two facts are being communicated. Write an ‘expanded form’ of the following
sentences.
a Rosemary likes Maths and English.
b Priscilla eats fruit and vegetables.
c Bailey and Lucia like the opera.
d Frank and Igor play video games.
e Pyodir and Astrid like chocolate and
tennis. (Note: There are four facts being
communicated here.)
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Chapter 8 Algebraic 390 techniques 1
applying algebra
An algebraic expression can be used to describe problems
relating to many different areas, including costs, speeds
and sporting results. Much of modern science relies on the
application of algebraic rules and formulas. It is important
to be able to convert word descriptions of problems to
mathematical expressions in order to solve these problems
mathematically.
let’s start: Garden bed area
The garden shown at right has an area of 34 m2, but the length
and breadth are unknown.
• What are some possible values that b and l could equal?
• Try to fi nd the dimensions of the garden that make the fencing
around the outside as small as possible.
8G
In many sports, results and details can be
expressed using algebra.
Area = 34 m2
3 m
2 m
 = ?
b = ?
EXTENSION
■■ Many different situations can be modelled with algebraic expressions.
■■ To apply an expression, the pronumerals should be defi ned clearly. Then known values should be
substituted for the pronumerals.
Key ideas
Example 15 applying an expression
The perimeter of a rectangle is given by the expression 2l + 2b, where l is the length and b is
the breadth.
a Find the perimeter of a rectangle if l = 4 and b = 7.
b Find the perimeter of a rectangle with breadth 8 cm and height 3 cm.
SoluTion ExplanaTion
a Perimeter is given by
2l + 2b = 2(4) + 2(7)
= 8 + 14
= 22
To apply the rule, we substitute l = 4 and b = 7
into the expression.
Evaluate using the normal rules of arithmetic
(i.e. multiplication before addition).
b Perimeter is given by
2l + 2b = 2(8) + 2(3)
= 16 + 6
= 22 cm
Substitute l = 8 and b = 3 into the expression.
Evaluate using the normal rules of arithmetic,
remembering to include appropriate units (cm) in the
answer.
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number and algebra 391
Example 16 Constructing expressions from problem descriptions
Write expressions for each of the following.
a The total cost, in dollars, of 10 bottles, if each bottle costs $x.
b The total cost, in dollars, of hiring a plumber for n hours. The plumber charges a $30 call-out fee
plus $60 per hour.
c A plumber charges a $60 call-out fee plus $50 per hour. Use an expression to fi nd how much an
8-hour job would cost.
SoluTion ExplanaTion
a 10x Each of the 10 bottles costs $x, so the total cost is
10 × x = 10x.
b 30 + 60n For each hour, the plumber charges $60, so must pay
60 × n = 60n. The $30 call-out fee is added to the total
bill.
c Expression for cost: 60 + 50n
If n = 8, then cost is 60 + 50 × 8 = $460
Substitute n = 8 to fi nd the cost for an 8-hour job.
Cost will be $460.
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1 The area of a rectangle is given by the expression l × b, where l is its length and b is C
its breadth.
a Find the area if b = 5 and l = 7.
b Find the area if b = 2 and l = 10.
2 The perimeter of a square with breadth b is given by the expression 4b.
a Find the perimeter of a square with breadth 6 cm (i.e. b = 6).
b Find the perimeter of a square with breadth 10 m (i.e. b = 10).
3 Consider the equilateral triangle shown.
x x
x
a Write an expression that gives the perimeter of this triangle.
b Use your expression to fi nd the perimeter if x = 12.
Example 15a
Example 15b
Exercise 8G E X T E N S I O N
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Chapter 8 Algebraic 392 techniques 1
4 If pens cost $2 each, write an expression for the cost of n pens.
5 If pencils cost $x each, write an expression for the cost of:
a 10 pencils
b 3 packets of pencils, if each packet contains 5 pencils
c k pencils
6 A car travels at 60 km/h, so in n hours it has travelled
60n kilometres.
a How far does the car travel in 3 hours (i.e. n = 3)?
b How far does the car travel in 30 minutes?
c Write an expression for the total distance travelled
in n hours for a motorbike with speed 70 km/h.
7 A carpenter charges a $40 call-out fee and then $80 per
hour. This means the total cost for x hours of work is
$(40 + 80x).
a How much would it cost for a 2-hour job (i.e. x = 2)?
b How much would it cost for a job that takes 8 hours?
c The call-out fee is increased to $50. What is the new
expression for the total cost of x hours?
8 Match up the word problems with the expressions (a to E) below.
a The area of a rectangle with height 5 and breadth x. a 10 + 2x
b The perimeter of a rectangle with height 5 and breadth x. B 5x
c The total cost, in dollars, of hiring a DVD for x days if the price is $1 per day. C 5 + x
d The total cost, in dollars, of hiring a builder for 5 hours if the builder charges
a $10 call-out fee and then $x per hour.
D x
e The total cost, in dollars, of buying a $5 magazine and a book that costs $x. E 10 + 5x
9 A plumber charges a $50 call-out fee and $100 per hour.
a Copy and complete the table below.
no. of hours ( t ) 1 2 3 4 5
Total cost ($)
b Find the total cost, in dollars, if the plumber works
for t hours. Give an expression.
c Substitute t = 30 into your expression to fi nd how
much it will cost for the plumber to work 30 hours.
10 To hire a tennis court, you must pay a $5 booking fee plus $10 per hour.
a What is the cost of booking a court for 2 hours?
b What is the cost, in dollars, of booking a court for x hours? Write an expression.
c A tennis coach hires a court for 7 hours. Substitute x = 7 into your expression to fi nd the
total cost.
Example 16a
Example 16c
Example 16b
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Number and Algebra 393
11 In Australian Rules football a goal is worth 6 points and a ‘behind’ is worth 1 point. This
means the total score for a team is 6g + b, if g goals and b behinds are scored.
a What is the score for a team that has scored 5 goals and 3 behinds?
b What are the values of g and b for a team that has scored 8 goals and 5 behinds?
c If a team has a score of 20, this could be because g = 2 and b = 8. What are the other
possible values of g and b?
12 Adrian’s mobile phone costs 30 cents to make a connection, plus 60 cents per minute of talking.
This means that a t-minute call costs 30 + 60t cents.
a What is the cost of a 1-minute call?
b What is the cost of a 10-minute call? Give your answer in dollars.
c Write an expression for the cost of a t-minute call in dollars.
13 During a sale, a shop sells all CDs for $c each, books cost $b each and DVDs cost $d each.
Claudia buys 5 books, 2 CDs and 6 DVDs.
a What is the cost, in dollars, of Claudia’s order? Give your answer as an expression involving
b, c and d.
b Write an expression for the cost of Claudia’s order if CDs doubled in price and DVDs
halved in price.
c As it happens, the total price Claudia ends up paying is the same in both situations. Given
that CDs cost $12 and books cost $20 (so c = 12 and b = 20), how much do DVDs cost?
14 A shop charges $c for a box of tissues.
a Write an expression for the total cost, in dollars, of buying n boxes of tissues.
b If the original price is tripled, write an expression for the total cost of buying n boxes of
tissues.
c If the original price is tripled and twice as many boxes are bought, write an expression for
the total cost.
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Chapter 8 Algebraic 394 techniques 1
15 To hire a basketball court costs $10 for a booking fee, plus $30 per hour.
a Write an expression for the total cost, in dollars, to hire the court for x hours.
b For the cost of $40, you could hire the court for 1 hour. How long could you hire the court
for the cost of $80?
c Explain why it is not the case that hiring the court for twice as long costs twice as much.
d Find the average cost per hour if the court is hired for a 5-hour basketball tournament.
e Describe what would happen to the average cost per hour if the court is hired for many
hours (e.g. more than 50 hours).
Enrichment: Mobile phone mayhem
16 Rochelle and Emma are on different mobile phone plans, as shown below.
Connection Cost per minute
Rochelle 20 cents 60 cents
Emma 80 cents 40 cents
a Write an expression for the cost, in dollars, of making a t-minute call using Rochelle’s phone.
b Write an expression for the cost of making a t-minute call using Emma’s phone.
c Whose phone plan would be cheaper for a 7-minute call?
d What is the length of call for which it would cost exactly the same for both phones?
e Investigate current mobile phone plans and describe how they compare to those of
Rochelle’s and Emma’s plans.
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number and algebra 395
Substitution involving negative numbers and
mixed operations
The process known as substitution involves replacing a
pronumeral or letter with a number. As a car
accelerates, its speed can be modelled by the rule
10 + 4t. So, after 8 seconds we can calculate the car’s
speed by substituting t = 8 into 10 + 4t.
So 10 + 4t = 10 + 4 × 8 = 42 metres per second.
We can also look at the car’s speed before time
t = 0. So at 2 seconds before t = 0 (i.e. t = -2), the speed
would be 10 + 4t = 10 + 4 × (-2) = 2 metres
per second.
let’s start: Order matters
Two students substitute the values a = -2, b = 5 and c = -7 into the expression ac - bc. Some of the
different answers received are 21, -49, -21 and 49.
• Which answer is correct and what errors were made in the computation of the three incorrect answers?
8H
We can use pronumerals to work out this car’s speed at
a given time.
■■ Substitute into an expression by replacing pronumerals
(or letters) with numbers.
■■ Use brackets around negative numbers to avoid confusion with
other symbols.
If a = -3 then
3 - 7a = 3 - 7 × (-3)
= 3 - (-21)
= 3 + 21
= 24
Key ideas
Example 17 Substituting integers
Evaluate the following expressions using a = 3 and b = -5.
a 2 + 4a b 7 - 4b c b ÷ 5 - a
SoluTion ExplanaTion
a 2 + 4a = 2 + 4 × 3
= 2 + 12
= 14
Replace a with 3 and evaluate the multiplication fi rst.
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Chapter 8 Algebraic 396 techniques 1
b 7 - 4b = 7 - 4 × (-5)
= 7 - (-20)
= 7 + 20
= 27
Replace the b with -5 and evaluate the multiplication
before the subtraction.
c b ÷ 5 - a = -5 ÷ 5 - 3
= -1 - 3
= - 4
Replace b with -5 and a with 3, and then evaluate.
Exercise 8H
1 Which of the following shows the correct substitution of a = -2 into the expression a - 5?
a 2 - 5 B -2 + 5 C -2 - 5 D 2 + 5
2 Which of the following shows the correct substitution of x = -3 into the expression 2 - x ?
a -2 - (-3) B 2 - (-3) C -2 + 3 D -3 + 2
3 Rafe substitutes c = -10 into 10 - c and gets 0. Is he correct? If not, what is the correct answer?
4 Evaluate the following expressions using a = 6 and b = -2.
a 5 + 2a b -7 + 5a c b - 6 d b + 10
e 4 - b f 7 - 2b g 3b - 1 h -2b + 2
i 5 - 12 ÷ a j 1 - 60 ÷ a k 10 ÷ b - 4 l 3 - 6 ÷ b
5 Evaluate the following expressions using a = -5 and b = -3.
a a + b b a - b c b - a d 2a + b
e 5b + 2a f 6b - 7a g -7a + b + 4 h -3b - 2a - 1
6 Evaluate these expressions for the values given.
a 26 - 4x (x = -3) b -2 - 7k (k = -1)
c 10 ÷ n + 6 (n = -5) d -3x + 2y (x = 3, y = -2)
e 18 ÷ y - x (x = -2, y = -3) f -36 ÷ a - ab (a = -18, b = -1)
7 These expressions contain brackets. Evaluate them for the values given. (Remember that
ab means a × b.)
a 2 × (a + b) (a = -1, b = 6) b 10 ÷ (a - b) + 1 (a = -6, b = -1)
c ab × (b - 1) (a = -4, b = 3) d (a - b) × bc (a = 1, b = -1, c = 3)
8 The area of a triangle, in m2, for a fi xed base of 4 metres is given by the rule 2h, where
h metres is the height of the triangle. Find the area of such a triangle with these heights.
a 3 m b 8 m
Example 17a,b
Example 17c
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Number and Algebra 397
9 A motorcycle’s speed, in metres per second, after a particular point on a racing track is
given by the expression 20 + 3t, where t is in seconds.
a Find the motorcycle’s speed after 4 seconds.
b Find the motorcycle’s speed at t = -2 seconds (i.e. 2 seconds before passing the t = 0 point).
c Find the motorcycle’s speed at t = -6 seconds.
10 The formula for the perimeter, P, of a rectangle is P = 2l + 2b, where l and b are the length and
the breadth, respectively.
a Use the given formula to find the perimeter of a rectangle with:
i l = 3 and b = 5 ii l = 7 and b = -8
b What problems are there with part a ii above?
11 Write two different expressions involving x that give an answer of -10 if x = -5.
12 Write an expression involving the pronumeral a combined with other integers, so if a = -4
the expression would equal these answers.
a -3 b 0 c 10
13 If a and b are any non-zero integer, explain why these expressions will always give the result of zero.
a a − b + b − a b a
a
− 1 c (a a)
b
− d ab
b
− a
Enrichment: Celsius/Fahrenheit
14 The Fahrenheit temperature scale (°F) is
still used today in some countries, but most
countries use the Celsius scale (°C).
32°F is the freezing point for water (0°C).
212°F is the boiling point for water
(100°C).
The formula for converting F to C is
C = 5
9
× (F − 32).
a Convert these temperatures from
F to C.
i 41°F ii 5°F iii -13°F
b Can you work out the formula that
converts from C to F?
c Use your rule from part b to check your
answers to part a.
The water temperature is 100°C and 212°F.
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Chapter 8 Algebraic 398 techniques 1
number patterns
Mathematicians commonly look at lists of numbers in an attempt to discover a pattern. They also aim to
fi nd a rule that describes the number pattern to allow them to predict future numbers in the sequence.
Here is a list of professional careers that all involve a high degree of mathematics and, in particular,
involve looking at data so that comments can be made about past, current or future trends.
Statistician, economist, accountant, market researcher, fi nancial analyst, cost estimator, actuary, stock
broker, data analyst, research scientist, fi nancial advisor, medical scientist, budget analyst, insurance
underwriter and mathematics teacher!
let’s start: What’s next?
A number sequence consisting of fi ve terms is placed on the board. Four gaps are placed after the
last number.
20, 12, 16, 8, 12, ___, ___, ___, ___
• Can you work out and describe the number pattern?
This number pattern involves a repeated process of subtracting 8 and then adding 4.
• Make up your own number pattern and test it on a class member.
8I
There are many careers that
involve using mathematics
and data.
EXTENSION
■■ Number patterns are also known as sequences, and each number in a sequence is called a term.
– Each number pattern has a particular starting number and terms are generated by following a
particular rule.
■■ Strategies to determine the pattern involved in a number sequence include:
– Looking for a common difference
Are terms increasing or decreasing by a constant amount?
For example: 2, 6, 10, 14, 18, … Each term is increasing by 4.
Key ideas
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Number and Algebra 399
Example 18 Identifying patterns with a common difference
Find the next three terms for these number patterns, which have a common difference.
a 6, 18, 30, 42, ___, ___, ___ b 99, 92, 85, 78, ___, ___, ___
Solution Explanation
a 54, 66, 78 The common difference is 12. Continue adding 12 to
generate the next three terms.
b 71, 64, 57 The pattern indicates the common difference is 7.
Continue subtracting 7 to generate the next three terms.
– Looking for a common ratio
Is each term being multiplied or divided by a constant amount?
For example: 2, 4, 8, 16, 32, … Each term is being multiplied by 2.
– Looking for an increasing/decreasing difference
Is there a pattern in the difference between pairs of terms?
For example: 1, 3, 6, 10, 15, … The difference increases by 1 each term.
– Looking for two interlinked patterns
Is there a pattern in the odd-numbered terms, and another pattern in the even-numbered
terms?
For example: 2, 8, 4, 7, 6, 6, … The odd-numbered terms increase by 2, the evennumbered
terms decrease by 1.
– Looking for a special type of pattern
Could it be a list of square numbers, prime numbers, Fibonacci numbers etc.?
For example: 1, 8, 27, 64, 125, … This is the pattern of cube numbers: 13, 23, 33, …
Key ideas
Example 19 Identifying patterns with a common ratio
Find the next three terms for the following number patterns, which have a common ratio.
a 2, 6, 18, 54, ___, ___, ___ b 256, 128, 64, 32, ___, ___, ___
Solution Explanation
a 162, 486, 1458 The common ratio is 3. Continue multiplying by 3 to
generate the next three terms.
b 16, 8, 4 The common ratio is
1
2
. Continue dividing by 2 to
generate the next three terms.
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Chapter 8 Algebraic 400 techniques 1
1 Generate the fi rst fi ve terms of the following number patterns.
a starting number of 8, common difference of adding 3
b starting number of 32, common difference of subtracting 1
c starting number of 2, common difference of subtracting 4
d starting number of 123, common difference of adding 7
2 Generate the fi rst fi ve terms of the following number patterns.
a starting number of 3, common ratio of 2 (multiply by 2 each time)
b starting number of 5, common ratio of 4
c starting number of 240, common ratio of
1
2
(divide by 2 each time)
d starting number of 625, common ratio of
1
5
3 State whether the following number patterns have a common difference (+ or -), a common
ratio (× or ÷) or neither.
a 4, 12, 36, 108, 324, … b 19, 17, 15, 13, 11, …
c 212, 223, 234, 245, 256, … d 8, 10, 13, 17, 22, …
e 64, 32, 16, 8, 4, … f 5, 15, 5, 15, 5, …
g 2, 3, 5, 7, 11, … h 75, 72, 69, 66, 63, …
4 Find the next three terms for the following number patterns, which have a common difference.
a 3, 8, 13, 18, ___, ___, ___ b 4, 14, 24, 34, ___, ___, ___
c 26, 23, 20, 17, ___, ___, ___ d 106, 108, 110, 112, ___, ___, ___
e 63, 54, 45, 36, ___, ___, ___ f 4, 3, 2, 1, ___, ___, ___
g 101, 202, 303, 404, ___, ___, ___ h 17, 11, 5, -1, ___, ___, ___
5 Find the next three terms for the following number patterns, which have a common ratio.
a 2, 4, 8, 16, ___, ___, ___ b 5, 10, 20, 40, ___, ___, ___
c 96, 48, 24, ___, ___, ___ d 1215, 405, 135, ___, ___, ___
e 11, 22, 44, 88, ___, ___, ___ f 7, 70, 700, 7000, ___, ___, ___
g 256, 128, 64, 32, ___, ___, ___ h 1216, 608, 304, 152, ___, ___, ___
6 Find the missing numbers in each of the following number patterns.
a 62, 56, ___, 44, 38, ___, ___ b 15, ___, 35, ___, ___, 65, 75
c 4, 8, 16, ___, ___, 128, ___ d 3, 6, ___, 12, ___, 18, ___
e 88, 77, 66, ___, ___, ___, 22 f 2997, 999, ___, ___, 37
g 14, 42, ___, ___, 126, ___, 182 h 14, 42, ___, ___, 1134, ___, 10 206
7 Write the next three terms in each of the following sequences.
a 3, 5, 8, 12, ___, ___, ___ b 1, 2, 4, 7, 11, ___, ___, ___
c 1, 4, 9, 16, 25, ___, ___, ___ d 27, 27, 26, 24, 21, ___, ___, ___
e 2, 3, 5, 7, 11, 13, ___, ___, ___ f 2, 5, 11, 23, ___, ___, ___
g 2, 10, 3, 9, 4, 8, ___, ___, ___ h 14, 100, 20, 80, 26, 60, ___, ___, ___
Example 18
Example 19
Exercise 8I E X T E N S I O N
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Number and Algebra 401
8 Generate the next three terms for the following number sequences and give an appropriate
name to the sequence.
a 1, 4, 9, 16, 25, 36, ___, ___, ___ b 1, 1, 2, 3, 5, 8, 13, ___, ___, ___
c 1, 8, 27, 64, 125, ___, ___, ___ d 2, 3, 5, 7, 11, 13, 17, ___, ___, ___
e 4, 6, 8, 9, 10, 12, 14, 15, ___, ___, ___ f 121, 131, 141, 151, ___, ___, ___
9 Complete the next three terms for the following challenging number patterns.
a 101, 103, 106, 110, ___, ___, ___ b 162, 54, 108, 36, 72, ___, ___, ___
c 3, 2, 6, 5, 15, 14, ___, ___, ___ d 0, 3, 0, 4, 1, 6, 3, ___, ___, ___
10 When making human pyramids, there is one less person on each row above, and it is complete
when there is a row of only one person on the top.
Write down a number pattern for a human pyramid with 10 students on the bottom row. How
many people are needed to make this pyramid?
11 The table below represents a seating plan with specific seat numbering for a section of a
grandstand at a soccer ground. It continues upwards for another 20 rows.
Row 4 25 26 27 28 29 30 31 32
Row 3 17 18 19 20 21 22 23 24
Row 2 9 10 11 12 13 14 15 16
Row 1 1 2 3 4 5 6 7 8
a What is the number of the seat directly above seat number 31?
b What is the number of the seat on the left-hand edge of row 8?
c What is the third seat from the right in row 14?
d How many seats are in the grandstand?
12 Find the next five numbers in the following number pattern.
1, 4, 9, 1, 6, 2, 5, 3, 6, 4, 9, 6, 4, 8, 1, ___, ___, ___, ___, ___
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Chapter 8 Algebraic 402 techniques 1
13 Jemima writes down the following number sequence: 7, 7, 7, 7, 7, 7, 7, …
Her friend Peta declares that this is not really a number pattern. Jemima defends her number
pattern, stating that it is most defi nitely a number pattern as it has a common difference and also
has a common ratio. What are the common difference and the common ratio for the number
sequence above? Do you agree with Jemima or Peta?
14 Find the sum of the following number sequences.
a 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10
b 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19
c 1 + 2 + 3 + 4 + 5 + . . . + 67 + 68 + 69 + 70
d 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 + 32 + 35 + 38
15 The great handshake problem. There are a certain number of people in a room and they must all
shake one another’s hand. How many handshakes will there be if there are:
a 3 people in the room?
b 5 people in the room?
c 10 people in the room?
d 24 people in a classroom?
e n people in the room?
Enrichment: What number am i?
16 Read the following clues to work out the mystery number.
a I have three digits.
I am divisible by 5.
I am odd.
The product of my digits is 15.
The sum of my digits is less than 10.
I am less than 12 × 12.
b I have three digits.
The sum of my digits is 12.
My digits are all even.
My digits are all different.
I am divisible by 4.
The sum of my units and tens digits equals my hundreds digit.
c I have three digits.
I am odd and divisible by 5 and 9.
The product of my digits is 180.
The sum of my digits is less than 20.
I am greater than 302.
d Make up two of your own mystery number puzzles and submit your clues to your teacher.
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number and algebra 403
Spatial patterns
Patterns can also be found in geometric shapes. Mathematicians
examine patterns carefully to determine how the next term in
the sequence is created. Ideally, a rule is formed that shows
the relationship between the geometric shape and the number
of objects (e.g. tiles, sticks or counters) required to make such
a shape. Once a rule is established it can be used to make
predictions about future terms in the sequence.
let’s start: Stick patterns
Materials required: One box of toothpicks/matches per student.
• Generate a spatial pattern using your sticks.
• You must be able to make at least three terms in your pattern.
For example:
• Ask your partner how many sticks would be required to make the next term in the pattern.
• Repeat the process with a different spatial design.
8J
A pattern rule can be created to show how
these shapes can be constructed.
EXTENSION
■■ A spatial pattern is a sequence of geometrical shapes that can be described by a number
pattern.
For example:
4 8 12
spatial pattern:
number pattern:
■■ A spatial pattern starts with a simple geometric design. Future terms are created by adding
on repeated shapes of the same design. If designs connect with an edge, the repetitive shape
added on will be a subset of the original design, as the connecting edge does not need to be
repeated.
For example:
starting
design
repeating
design
■■ To help describe a spatial pattern, it is generally converted to a number pattern and a
common difference is observed.
■■ The common difference is the number of objects (e.g. sticks) that need to be added on to
create the next term.
Key ideas
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Chapter 8 Algebraic 404 techniques 1
Example 20 Drawing and describing spatial patterns
a Draw the next two shapes in the spatial pattern shown.
b Write the spatial pattern above as a number pattern in regard to the number of sticks required to
make each shape.
c Describe the pattern by stating how many sticks are required to make the first term, and how many
sticks are required to make the next term in the pattern.
Solution Explanation
a Follow the pattern.
b 5, 8, 11, 14, 17 Count the number of sticks in each term. Look for
a pattern.
c 5 matches are required to start the pattern, and
an additional 3 matches are required to make
the next term in the pattern.
■■ Rules can be found that connect the number of objects (e.g. sticks) required to produce the
number of designs.
For example: hexagon design
Rule is: Number of sticks used = 6 × number of hexagons formed
Key ideas
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number and algebra 405
Example 21 Finding a general rule for a spatial pattern
a Draw the next two shapes in this spatial pattern.
b Complete the table.
number of triangles 1 2 3 4 5
number of sticks required 3
c Describe a rule connecting the number of sticks required to the number of triangles produced.
d Use your rule to predict how many sticks would be required to make 20 triangles.
SoluTion ExplanaTion
a Follow the pattern by adding one triangle
each time.
b
no. of
triangles
1 2 3 4 5
no. of
sticks
3 6 9 12 15
An extra 3 sticks are required to make each
new triangle.
c Number of sticks = 3 × number of triangles 3 sticks are required per triangle.
d Number of sticks = 3 × 20 triangles = 60 sticks 20 triangles × 3 sticks each
Exercise 8J E X T E N S I O N
1 Draw the next two terms for each of these spatial patterns.
a
b
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Chapter 8 Algebraic 406 techniques 1
c
d
e
2 Draw the following geometrical designs in sequential ascending (i.e. increasing) order and draw
the next term in the sequence.
3 For each of the following spatial patterns, draw the starting geometrical design and also the
geometrical design that is added on repetitively to create new terms. (For some patterns the
repetitive design is the same as the starting design.)
a
b
c
d
e
f
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4 For each of the spatial patterns below:
i Draw the next two shapes.
ii Write the spatial pattern as a number pattern.
iii Describe the pattern by stating how many sticks are required to make the first term and how
many more sticks are required to make the next term in the pattern.
a
b
c
d
e
f
5 a Draw the next two shapes in this spatial pattern.
b Copy and complete the table.
Number of crosses 1 2 3 4 5
Number of sticks required
c Describe a rule connecting the number of sticks required to the number of crosses produced.
d Use your rule to predict how many sticks would be required to make 20 crosses.
Example 20
Example 21
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Chapter 8 Algebraic 408 techniques 1
6 a Draw the next two shapes in this spatial pattern.
b Copy and complete the table. Planks are vertical and horizontal.
number of fence sections 1 2 3 4 5
number of planks required
c Describe a rule connecting the number of planks required to the number of fence sections
produced.
d Use your rule to predict how many planks would be required to make 20 fence sections.
7 At North Park Primary School, the classrooms have trapezium-shaped tables. Mrs Greene
arranges her classroom’s tables in straight lines, as shown.
a Draw a table of results showing the relationship between the number of tables in a row and
the number of students that can sit at the tables. Include results for up to fi ve tables in a row.
b Describe a rule that connects the number of tables placed in a straight row to the number of
students that can sit around the tables.
c The room allows seven tables to be arranged in a straight line. How many students can sit
around the tables?
d There are 65 students in Grade 6 at North Park Primary School. Mrs Greene would like to
arrange the tables in one straight line for an outside picnic lunch. How many tables will she
need?
8 The number of tiles required to pave around a spa is related to the size of the spa.
The approach is to use large tiles that are the same size as that of a small spa.
A spa of length 1 unit requires 8 tiles to pave around its perimeter, whereas a spa of length
4 units requires 14 tiles to pave around its perimeter.
a Complete a table of values relating length of spa and number of tiles required, for values up
to and including a spa of length 6 units.
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Number and Algebra 409
b Describe a rule that connects the number of tiles required for the length of the spa.
c The largest size spa manufactured is 15 units long. How many tiles would be required to
pave around its perimeter?
d A paving company has only 30 tiles left. What is the largest spa they would be able to tile
around?
9 Present your answers to either Question 7 or 8 in an A4 or A3 poster form. Express your
findings and justifications clearly.
10 Which rule correctly describes this spatial pattern?
A Number of sticks = 7 × number of ‘hats’
B Number of sticks = 7 × number of ‘hats’ + 1
C Number of sticks = 6 × number of ‘hats’ + 2
D Number of sticks = 6 × number of ‘hats’
11 Which rule correctly describes this spatial pattern?
A Number of sticks = 5 × number of houses + 1
B Number of sticks = 6 × number of houses + 1
C Number of sticks = 6 × number of houses
D Number of sticks = 5 × number of houses
12 Design a spatial pattern to fit the following number patterns.
a 4, 7, 10, 13, … b 4, 8, 12, 16, …
c 3, 5, 7, 9, … d 3, 6, 9, 12, …
e 5, 8, 11, 14, … f 6, 11, 16, 21, …
13 A rule to describe a special window spatial pattern is written as y = 4 × x + 1, where y represents
the number of ‘sticks’ required and x is the number of windows created.
a How many sticks are required to make one window?
b How many sticks are required to make 10 windows?
c How many sticks are required to make g windows?
d How many windows can be made from 65 sticks?
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Chapter 8 Algebraic 410 techniques 1
14 A rule to describe a special fence
spatial pattern is written
as y = m × x + n, where y represents
the number of pieces of timber
required and x represents the number
of fencing panels created.
a How many pieces of timber are
required to make one panel?
b What does m represent?
c Draw the fi rst three terms of the
fence spatial pattern for m = 4 and
n = 1.
Enrichment: Cutting up a circle
15 What is the greatest number of sections into which you can divide a circle, using only a
particular number of straight line cuts?
a Explore the problem above.
Note: The greatest number of sections is required and, hence, only one of the two diagrams
below is correct for three straight line cuts.
5 4
Incorrect.
Not the maximum
number of sections.
1
6 3
2
3
5
4
1
2
6
7
Correct.
The maximum
number of
sections.
b Copy and complete this table of values.
number of straight cuts 1 2 3 4 5 6 7
number of sections
created
7
c Can you discover a pattern for the maximum number of sections created? What is the
maximum number of sections that could be created with 10 straight line cuts?
d The formula for determining the maximum number of cuts is quite complex.
Sections = cuts2 + cuts + 1
2
1
2
1
Verify that this formula works for the values you listed in the table above.
Using the formula, how many sections could be created with 20 straight cuts?
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number and algebra 411
Tables and rules
In the previous section on spatial patterns, it was observed that rules can be used to connect the number of
objects (e.g. sticks) required to make particular designs.
A table of values can be created for any spatial pattern. Consider this spatial pattern and the
corresponding table of values.
What values would go in the next row of the table?
A rule that produces this table of values is:
Number of sticks = 4 × number of diamonds
Alternatively, if we consider the number of diamonds as the input and the number of sticks as the
output then the rule could be written as:
Output = 4 × input
If a rule is provided, a table of values can be created.
If a table of values is provided, often a rule can be found.
8K
number of diamonds
(input )
number of sticks
(output )
1 4
2 8
3 12
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Chapter 8 Algebraic 412 techniques 1
Let’s start: Guess the output
• A table of values is drawn on the board with three completed rows of data.
• Additional values are placed in the input column. What output values should be in the output column?
• After adding output values, decide which rule fits (models) the values in the table and check that it
works for each input and output pair.
Four sample tables are listed below.
Input Output Input Output Input Output Input Output
2 6 12 36 2 3 6 1
5 9 5 15 3 5 20 8
6 10 8 24 9 17 12 4
1 ? 0 ? 7 ? 42 ?
8 ? 23 ? 12 ? 4 ?
■■ A rule shows the relation between two varying quantities.
For example: output = input + 3 is a rule connecting the two quantities input and output.
The values of the input and the output can vary, but we know from the rule that the value of the
output will always be 3 more than the value of the input.
■■ A table of values can be created from any given rule. To complete a table of values, the input
(one of the quantities) is replaced by a number. This is known as substitution. After substitution
the value of the other quantity, the output, is calculated.
For example: If input = 4, then
Output = input + 3
= 4 + 3
= 7
■■ Often, a rule can be determined from a table of values. On close inspection of the values, a
relationship may be observed. Each of the four operations should be considered when looking for
a connection.
Input 1 2 3 4 5 6
Output 6 7 8 9 10 11
By inspection, it can be observed that every output value is 5 more than the corresponding input
value. The rule can be written as: output = input + 5.
Key ideas
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Number and Algebra 413
Example 22 Completing a table of values
Complete each table for the given rule.
a Output = input – 2 b Output = (3 × input ) + 1
Input 3 5 7 12 20 Input 4 2 9 12 0
Output Output
Solution Explanation
a Output = input – 2
Input 3 5 7 12 20
Output 1 3 5 10 18
Replace each input value in turn into the rule.
e.g. When input is 3:
Output = 3 – 2 = 1
b Output = (3 × input ) + 1
Input 4 2 9 12 0
Output 13 7 28 37 1
Replace each input value in turn into the rule.
e.g. When input is 4:
Output = (3 × 4) + 1 = 13
Example 23 Finding a rule from a table of values
Find the rule for each of these tables of values.
a b
Input 3 4 5 6 7 Input 1 2 3 4 5
Output 12 13 14 15 16 Output 7 14 21 28 35
Solution Explanation
a Output = input + 9 Each output value is 9 more than the input
value.
b Output = input × 7 or Output = 7 × input By inspection, it can be observed that each
output value is 7 times bigger than the
input value.
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Chapter 8 Algebraic 414 techniques 1
1 State whether each of the following statements is true or false.
a If output = input × 2, then when input = 7, output = 14.
b If output = input – 2, then when input = 5, output = 7.
c If output = input + 2, then when input = 0, output = 2.
d If output = input ÷ 2, then when input = 20, output = 10.
2 Which table of values matches the rule output = input – 3?
a B
Input 10 11 12 Input 5 6 7
Output 13 14 15 Output 15 18 21
C D
Input 8 9 10 Input 4 3 2
Output 5 6 7 Output 1 1 1
3 Which table of values matches the rule output = input ÷ 2?
a B
Input 20 14 6 Input 8 10 12
Output 18 12 4 Output 4 5 6
C D
Input 4 5 6 Input 4 3 2
Output 8 10 12 Output 6 5 4
4 Match each rule (A to D) with the correct table of values (a to d).
Rule A: output = input - 5 Rule B: output = input + 1
Rule C: output = 4 × input Rule D: output = 5 + input
a b
Input 20 14 6 Input 8 10 12
Output 15 9 1 Output 13 15 17
c d
Input 4 5 6 Input 4 3 2
Output 5 6 7 Output 16 12 8
Exercise 8K E X T E N S I O N
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Number and Algebra 415
5 Copy and complete each table for the given rule.
a Output = input + 3 b Output = input × 2
Input 4 5 6 7 10 Input 5 1 3 21 0
Output Output
c Output = input – 8 d Output = input ÷ 5
Input 11 18 9 44 100 Input 5 15 55 0 100
Output Output
6 Copy and complete each table for the given rule.
a Output = (10 × input ) - 3 b Output = (input ÷ 2) + 4
Input 1 2 3 4 5 Input 6 8 10 12 14
Output Output
c Output = (3 × input ) + 1 d Output = (2 × input ) – 4
Input 5 12 2 9 0 Input 3 10 11 7 50
Output Output
7 State the rule for each of these tables of values.
a b
Input 4 5 6 7 8 Input 1 2 3 4 5
Output 5 6 7 8 9 Output 4 8 12 16 20
c d
Input 10 8 3 1 14 Input 6 18 30 24 66
Output 21 19 14 12 25 Output 1 3 5 4 11
8 Copy and complete the missing values in the table and state the rule.
Input 4 10 13 24 5 11 2
Output 39 42 9 15 6
9 Copy and complete the missing values in the table and state the rule.
Input 12 93 14 17 10
Output 3 8 12 1 34 0 200
Example 22a
Example 22b
Example 23
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Chapter 8 Algebraic 416 techniques 1
10 Copy and complete each table for the given rule.
a Output = input × input – 2 b Output = (24 ÷ input ) + 1
Input 3 6 8 12 2 Input 6 12 1 3 8
Output Output
c Output = input2 + input d Output = 2 × input × input – input
Input 5 12 2 9 0 Input 3 10 11 7 50
Output Output
11 Copy and complete each table for the given rule.
a Output = input + 6 b Output = 3 × input – 2
Input c d 2p b 2 www Input t k p 2 2f ab
Output Output
12 Copy and complete the missing values in the table and state the rule.
Input b e g 2 x c 1
Output cd cmn xc 0 c
13 It is known that for an input value of 3, the output value is 7.
a State two different rules that work for these values.
b How many different rules are possible? Explain.
Enrichment: Finding harder rules
14 a The following rules all involve two operations. Find the rule for each of these tables
of values.
i ii
Input 4 5 6 7 8 Input 1 2 3 4 5
Output 5 7 9 11 13 Output 5 9 13 17 21
iii iv
Input 10 8 3 1 14 Input 6 18 30 24 66
Output 49 39 14 4 69 Output 3 5 7 6 13
v vi
Input 4 5 6 7 8 Input 1 2 3 4 5
Output 43 53 63 73 83 Output 0 4 8 12 16
b Write three of your own two-operation rules and produce a table of values for each rule.
c Swap your tables of values with those of a classmate and attempt to fi nd one another’s rules.
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number and algebra 417
The Cartesian plane and graphs
We are already familiar with number lines.
A number line is used to locate a position in one
dimension (i.e. along the line).
A Cartesian plane is used to locate a position in two
dimensions (i.e. within the plane). A number plane uses
two number lines to form a grid system, so that points can
be located precisely. A rule can then be illustrated visually
using a Cartesian plane by forming a graph.
let’s start: Estimate your location
Consider the door as ‘the origin’ of your classroom.
• Describe the position you are sitting in within the classroom in reference to the door.
• Can you think of different ways of describing your position? Which is the best way?
Submit a copy of your location description to your teacher.
Can you locate a classmate correctly when location descriptions are read out by your teacher?
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2
3
1
1 2 3 4 5
x
y
O
4
5
What is the
position of this
point on the
Cartesian plane?
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Chapter 8 Algebraic 418 techniques 1
Example 24 Plotting points on a Cartesian plane
Plot these points on a Cartesian plane.
A(2, 5) B (4, 3) C (0, 2)
Solution Explanation
2
3
1
O 1 2 3 4 5
4
A
B
C
5
y
x
Draw a Cartesian plane, with both axes labelled
from 0 to 5.
The first coordinate is the x coordinate.
The second coordinate is the y coordinate.
To plot point A, go along the horizontal axis to
the number 2, then move vertically up 5 units.
Place a dot at this point, which is the intersection
of the line passing through the point 2 on the
horizontal axis and the line passing through the
point 5 on the vertical axis.
■■ A number plane is used to represent position in two dimensions, therefore it requires two
coordinates.
■■ In Mathematics, a number plane is generally referred to as a Cartesian plane, named after the
famous French mathematician, René Descartes (1596–1650).
■■ A number plane consists of two straight perpendicular number lines, called axes.
– The horizontal number line is known as the x-axis.
– The vertical number line is known as the y-axis.
■■ For a rule describing a pattern with input and output, the x value is the input and the y value is
the output.
■■ The point at which the two axes intersect is called the origin, and is often labelled O.
■■ The position of a point on a number plane
is given as a pair of numbers, known as the
coordinates of the point. Coordinates are
always written in brackets and the numbers are
separated by a comma. For example: (2, 4).
– The x coordinate (input ) is always written
first. The x coordinate indicates how far to go
from the origin in the horizontal direction.
– The y coordinate (output ) is always written
second. The y coordinate indicates how far to
go from the origin in the vertical direction.
Key ideas
2
3
1
2
the origin
the vertical,
y-axis
the horizontal,
x-axis
O 1 3 4 5
4
5
y
This dot is
represented by
the coordinates
(2, 4).
x
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number and algebra 419
Example 25 Drawing a graph
For the given rule output = input + 1:
a Complete the given table of values.
b Plot each pair of points in the table to form a graph.
SoluTion ExplanaTion
a Use the given rule to fi nd each output value for each
input value. The rule is:
Output = input + 1, so add 1 to each input value.
b
2
3
1
O 1 2 3
4
Output
Input
x
y Plot each (x, y) pair.
The pairs are (0, 1), (1, 2), (2, 3) and (3, 4).
Input ( x ) Output ( y )
0 1
1
2
3
Input ( x ) Output ( y )
0 1
1 2
2 3
3 4
Exercise 8L E X T E N S I O N
1 Draw a number plane, with the numbers 0 to 6 marked on each axis.
2 Draw a Cartesian plane, with the numbers 0 to 4 marked on both axes.
3 Which of the following is the correct way to describe point A?
a 2 1
B 2, 1
C (2, 1)
D (x 2, y1)
E (2x, 1y)
2
3
1 A
O 1 2 3
x
y
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Chapter 8 Algebraic 420 techniques 1
4 Which of the following is the correct set of coordinates for point B ?
a (2, 4)
B 4, 2
C (4, 2)
D (2 4)
E x = 4, y = 2
5 Copy and complete the following sentences.
a The horizontal axis is known as the .
b The is the vertical axis.
c The point at which the axes intersect is called the .
d The x coordinate is always written .
e The second coordinate is always the ______________.
f comes before in the dictionary, and the coordinate comes before
the coordinate on the Cartesian plane.
6 Plot the following points on a Cartesian plane.
a A(4, 2) b B (1, 1) c C (5, 3) d D (0, 2)
e E (3, 1) f F (5, 4) g G (5, 0) h H (0, 0)
7 Write down the coordinates of each of these labelled points.
a
2
3 G
A
D
B
E
1
O 1 2 3 4 5 6
4
5
6
y
x
C
H
F
b
2
3
M
Q
T
R
N
1
O 1 2 3 4 5 6
4
5
6
y
x
U
P
S
8 For the given rule output = input + 2:
a Copy and complete the given table of values.
b Plot each pair of points in the table to form a graph.
Input (x ) Output ( y )
0 2
1
2
3
2
3
1
O 1 2 3 4
B
y
x
Example 24
Example 25
Output
Input
2
3
1
O 1 2 3 4
4
5
y
x
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Number and Algebra 421
9 For the given rule output = input – 1:
a Copy and complete the given table of values.
b Plot each pair of points in the table to form a graph.
Input (x ) Output ( y )
1
2
3
4
2
3
1
2 3
Input
Output
O 1 4
y
x
10 For the given rule output = input × 2:
a Copy and complete the given table of values.
b Plot each pair of points in the table to form a graph.
Input (x ) Output ( y )
0
1
2
3
Input
Output
2
3
1
O 1 2 3
4
5
6
y
x
11 Draw a Cartesian plane from 0 to 5 on both axes. Place a cross on each pair of coordinates that
have the same x and y value.
12 Draw a Cartesian plane from 0 to 8 on both axes. Plot the following points on the grid and join
them in the order they are given.
(2, 7), (6, 7), (5, 5), (7, 5), (6, 2), (5, 2), (4, 1), (3, 2), (2, 2), (1, 5), (3, 5), (2, 7)
13 a Plot the following points on a Cartesian plane and join the points in the order given, to
draw the basic shape of a house.
(1, 5), (0, 5), (5, 10), (10, 5), (1, 5), (1, 0), (9, 0), (9, 5)
b Describe a set of four points to draw a door.
c Describe two sets of four points to draw two windows.
d Describe a set of four points to draw a chimney.
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Chapter 8 Algebraic 422 techniques 1
14 Point A(1, 1) is the bottom left-hand corner of a square of side length 3.
a State the other three coordinates of the square.
b Draw the square on a Cartesian plane and shade in half of the square where the
x coordinates are greater than the y coordinates.
15 A grid system can be used to make secret messages. Jake decides to arrange the letters of the
alphabet on a Cartesian plane in the following manner.
2
3
1
O 1 2 3 4 5
4
5
y
U V W X Y
P Q R S T
K L M N O
F G H I J
A B C D E
x
a Decode Jake’s message:
(3, 2), (5, 1), (2, 3), (1, 4)
b Code the word ‘secret’.
c To increase the diffi culty of the code, Jake does not include brackets or commas and he uses
the origin to indicate the end of a word.
What do the following numbers mean?
13515500154341513400145354001423114354.
d Code the phrase: ‘Be here at seven’.
16 ABCD is a rectangle. The coordinates of A, B and C are given below. Draw each rectangle on a
Cartesian plane and state the coordinates of the missing corner, D.
a A(0, 5) B (0, 3) C (4, 3) D (?, ?)
b A(4, 4) B (1, 4) C (1, 1) D (?, ?)
c A(0, 2) B (3, 2) C (3, 0) D (?, ?)
d A(4, 1) B (8, 4) C (5, 8) D (?, ?)
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Number and Algebra 423
17 Write a rule (e.g. output = input × 2) that would give these graphs.
a
2
3
1
O 1 2 3 4
4
5
6
Output
Input
x
y b
4
6
2
O 1 2 3
8
10
Output
Input
x
y
c
2
3
1
O 1 2 3 4 5 6
Output
Input
x
y
18 A(1, 0) and B (5, 0) are the base points of an isosceles triangle.
a Find the coordinates of a possible third vertex.
b Show on a Cartesian plane that the possible number of answers for this third vertex is
infinite.
c Write a sentence to explain why the possible number of answers for this third vertex is
infinite.
d The area of the isosceles triangle is 10 square units. State the coordinates of the third vertex.
Enrichment: Locating midpoints
19 a Plot the points A(1, 4) and B (5, 0) on a Cartesian plane. Draw the line segment AB. Find the
coordinates of M, the midpoint of AB, and mark it on the grid.
b Find the midpoint, M, of the line segment AB, which has coordinates A(2, 4) and B (0, 0).
c Determine a method for locating the midpoint of a line segment without having to draw the
points on a Cartesian plane.
d Find the midpoint, M, of the line segment AB, which has coordinates A(6, 3) and B (2, 1).
e Find the midpoint, M, of the line segment AB, which has coordinates A(1, 4) and B (4, 3).
f Find the midpoint, M, of the line segment AB, which has coordinates A(-3, 2) and B (2, -3).
g M (3, 4) is the midpoint of AB and the coordinates of A are (1, 5). What are the coordinates
of B ?
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Chapter 8 Algebraic 424 techniques 1
Fencing paddocks
A farmer is interested in fencing off a large number of 1 m × 1 m foraging regions for the chickens.
Consider the pattern below.
n = 1 n = 2 n = 3 n = 4
a For n = 2, the outside perimeter is 8 m, the area is 4 m2 and the total length of fencing required
is 12 m. Copy and complete the following table.
n 1 2 3 4 5 6
outside perimeter (m) 8
area (m2) 4
Fencing required 12
b Write an expression for:
i the total outside perimeter of the fenced section
ii the total area of the fenced section
c The farmer knows that the expression for the total amount of fencing is one of the following.
Which one is correct? Prove to the farmer that the others are incorrect.
i 6n ii (n + 1)2 iii n × 2 × (n + 1)
d Use the correct formula to work out the total amount of fencing required if the farmer wishes to
have a total area of 100 m2 fenced off.
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Number and Algebra 425
In a spreadsheet application these calculations can be made automatically. Set up a spreadsheet as
follows.
Drag down the cells until you have all the rows from n = 0 to n = 30.
e Find the amount of fencing needed if the farmer wants the total area to be at least:
i 25 m2 ii 121 m2 iii 400 m2 iv 500 m2
f If the farmer has 144 m of fencing, what is the maximum area his grid could have?
g For each of the following lengths of fencing, give the maximum area, in m2, that the farmer
could contain in the grid.
i 50 m ii 200 m iii 1 km iv 40 km
h In the end, the farmer decides that the overall grid does not need to be a square, but could be
any rectangular shape. Design rectangular paddocks with the following properties.
i perimeter = 20 m and area = 21 m2
ii perimeter = 16 m and fencing required = 38 m2
iii area = 1200 m2 and fencing required = 148 m
iv perimeter = 1 km and fencing required is less than 1.5 km
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Chapter 8 Algebraic 426 techniques 1
1 Find the values of the pronumerals below in the following sum/product tables.
a
Sum
a b c
d 24 32
Sum 12 e 48
b
product
a b 18
2 c d
product 12 e 180
2 Copy and complete the following table, in which x and y are always whole numbers.
x 2
y 7 6 12
3x 6 9
x + 2y 9 7
xy 0 5
3 What is the coeffi cient of x once the expression
x + 2(x + 1) + 3(x + 2) + 4(x + 3) + … + 100(x + 99) is simplifi ed completely?
4 In a mini-Sudoku, the digits 1 to 4 occupy each square such that no row, column or 2 × 2 block has
the same digit twice. Find the value of each of the pronumerals in the following mini-Sudoku.
a 3 2 c
c d e f
2 g d + 1 h
i 1 j k
5 In a magic square the sum of each row, column and diagonal is the same. Find the value of the
pronumerals to make the following into magic squares. Confi rm your answer by writing out the
magic square as a grid of numbers.
a
A B C
A - 1 A + 1 B - C
B - 1 C - 1 A + C
b
2D
E
2
F 3G - 2
G 2G D + 3 D
4F + 1 E F + G EF
G - 1
EG
2
2(F + G ) F - 1
6 Think of any number and then perform the following operations: Add 5, then double the result, then
subtract 12, then subtract the original number, then add 2. Use algebra to explain why you now have
the original number again. Then design a puzzle like this yourself and try it on friends.
puzzles and challenges
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number and algebra 427
Chapter summary
Always equal when pronumerals
are substituted
e.g. 2x + 3 and 3 + 2x are equivalent.
4(3x ) and 12x are equivalent.
To simplify an expression, find a
simpler expression that is equivalent.
Algebra
Combination of numbers, pronumerals
and operations
e.g. 2xy + 3yz, – 3
3(a + 4) = 3a + 12
5k(10 – 2j ) = 50k – 10kj
Using the distributive law gives an
equivalent expression.
Algebraic expressions
Equivalent expressions
Expanding brackets
Cost is 50 + 90x
b

call-out
hourly
Applications
e.g. 4a + 2 + 3a = 7a + 2
3b + 5c + 2b – c = 5b + 4c
12xy + 3x – 5yx = 7xy + 3x
Combining like terms gives a
way to simplify.
Replacing pronumerals with values
e.g. 5x + 2y when x =10 & y = 3
becomes 5(10) + 2(3) = 50 + 6
= 56
e.g. q2 when q = 7 becomes 72 = 49
Substitution
are letters used to represent numbers
e.g. g: number of grapes in a bunch
d: distance travelled by a hockey ball
Pronumerals
combined with × or ÷ .
e.g. 4x, 10y, , 12 a3
Terms are pronumerals and numbers
6 more than k : k + 6
Product of 4 and x : 4x
10 less than b: b – 10
Half of q:
The sum of a and b is
tripled: 3(a + b)
q
2
3a means 3 × a
b means b ÷ 10
10
Mathematical convention
Creating expressions
Like terms have exactly the same
Expressions are used widely
A =  × b
P = 2 + 2b
12
x
fee
rate
pronumerals.
5a and 3a
2ab and 12ba
7ab and 2a
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Chapter 8 Algebraic 428 techniques 1
Multiple-choice questions
1 In the expression 3x + 2y + 4xy + 7yz the coefficient of y is:
A 3 B 2 C 4 D 7 E 16
2 If t = 5 and u = 7, then 2t + u is equal to:
A 17 B 32 C 24 D 257 E 70
3 If x = 2, then 3x 2 is equal to:
A 32 B 34 C 12 D 25 E 36
4 Which of the following pairs does not consist of two like terms?
A 3x and 5x B 3y and 12y C 3ab and 2ab
D 3cd and 5c E 3xy and yx
5 A fully simplified expression equivalent to 2a + 4 + 3b + 5a is:
A 4 B 5a + 5b + 4 C 10ab + 4
D 7a + 3b + 4 E 11ab
6 The simplified form of 4x × 3yz is:
A 43xyz B 12xy C 12xyz D 12yz E 4x3yz
7 The simplified form of
21
3
ab
ac
is:
A 7b
c
B
7ab
ac
C
21
3
b
c
D 7 E
b
7c
8 When brackets are expanded, 4(2x + 3y) becomes:
A 8x + 3y B 2x + 12y C 8x + 8y D 24x E 8x + 12y
9 The fully simplified form of 2(a + 7b) – 4b is:
A 2a + 10b B 2a + 3b C a + 3b
D 2a + 14b – 4b E 2a + 18b
10 A number is doubled and then 5 is added. The result is then tripled. If the number is represented by k,
then an expression for this description is:
A 3(2k + 5) B 6(k + 5) C 2k + 5
D 2k + 15 E 30k
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Number and Algebra 429
Short-answer questions
1 a List the four individual terms in the expression 5a + 3b + 7c + 12.
b What is the constant term in the expression above?
2 Write an expression for each of the following.
a 7 is added to u b k is tripled c 7 is added to half of r
d 10 is subtracted from h e the product of x and y f x is subtracted from 12
3 If u = 12, find the value of:
a u + 3 b 2u c
24
u
d 3u - 4
4 If p = 3 and q = -5, find the value of:
a pq b p + q c 2(q – p) d 4p + 3q
5 If t = 4 and u = 10, find the value of:
a t 2 b 2u 2 c 3 + t d 10tu
6 For each of the following pairs of expressions, state whether they are equivalent (E) or not
equivalent (N).
a 5x and 2x + 3x b 7a + 2b and 9ab
c 3c – c and 2c d 3(x + 2y) and 3x + 2y
7 Classify the following pairs as like terms (L) or not like terms (N).
a 2x and 5x b 7ab and 2a c 3p and p
d 9xy and 2yx e 4ab and 4aba f 8t and 2t
g 3p and 3 h 12k and 120k
8 Simplify the following by collecting like terms.
a 2x + 3 + 5x b 12p – 3p + 2p c 12b + 4a + 2b + 3a + 4
d 12mn + 3m + 2n + 5nm e 1 + 2c + 4h – 3o + 5c f 7u + 3v + 2uv – 3u
9 Simplify the following expressions involving products.
a 3a × 4b b 2xy × 3z c 12f × g × 3h d 8k × 2 × 4lm
10 Simplify the following expressions involving quotients.
a
3
2
u
u
b
12
20
y
y
c
2
6
ab
b
d
12
9
xy
yz
11 Expand the following expressions using the distributive law.
a 3(x + 2) b 4(p – 3) c 7(2a + 3) d 12(2k + 3l )
12 Give two examples of expressions that expand to give 12b + 18c.
13 If tins of paints weigh 9 kg, write an expression for the weight of t tins of paint.
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Chapter 8 Algebraic 430 techniques 1
14 If there are g girls and b boys in a room, write an expression for the total number of children in
the room.
15 Write an expression for the total number of books that Analena owns if she has x fiction books and
twice as many non-fiction books.
Extended-response questions
1 A taxi driver charges $3.50 to pick up passengers and then $2.10 per kilometre travelled.
a State the total cost if the trip length is:
i 10 km
ii 20 km
iii 100 km
b Write an expression for the total cost, in dollars, of travelling a distance of k kilometres.
c Use your expression to find the total cost of travelling 40 km.
d Prove that your expression is not equivalent to 2.1 + 3.5 k by substituting a value for k.
e Another taxi driver charges $6 to pick up passengers and then $1.20 per kilometre. Write an
expression for the total cost of travelling k kilometres in this taxi.
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Number and Algebra 431
2 An architect has designed a room, shown opposite, for which x and y are unknown. (All
measurements are in metres.)
a Find the perimeter of this room if x = 3 and y = 2.
b It costs $3 per metre to install skirting boards around the
perimeter of the room. Find the total cost of installing skirting
boards if the room’s perimeter is x = 3 and y = 2.
c Write an expression for the perimeter of the room and simplify
it completely.
d Write an expanded expression for the total cost, in dollars, of
installing skirting boards along the room’s perimeter.
e Write an expression for the total area of the floor in this room.
3
x + 2
x + 5
x + y
y
x
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Chapter Equations 1 9
What you will learn
Introduction to equations
Solving equations by inspection
Equivalent equations
Solving equations systematically
Equations with fractions
Equations with brackets
Formulas and relationships EXTENSION
Using equations to solve problems EXTENSION
9A
9B
9C
9D
9E
9F
9G
9H
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theme park equations
Equations are used widely in mathematics and in
many other fi elds. Whenever two things are equal, or
should be equal, there is the potential to use the study
of equations to help deal with such a situation.
Knowledge of mathematics and physics is
vitally important when designing theme park rides.
Engineers use equations to ‘build’ model rides on a
computer so that safety limits can be determined in a
virtual reality in which no one gets injured.
Algebraic equations are solved to determine the
dimensions and strengths of structures required to
deal safely with the combined forces of weight, speed
and varying movement. Passengers might scream
with a mixture of terror and excitement but they must
return unharmed to earth!
At Dreamworld on the Gold Coast, Queensland,
‘The Claw’ swings 32 people upwards at 75 km/h to a
maximum height of 27.1 m (9 storeys), simultaneously
spinning 360° at 5 r.p.m. (revolutions per minute).
‘The Claw’ is the most powerful pendulum ride on the
planet. It is built to scare!
nSW Syllabus
for the australian
Curriculum
Strand: number and algebra
Substrand: EQuatIonS
Outcome
A student uses algebraic techniques
to solve simple linear and quadratic
equations.
(MA4–10NA)
433
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Chapter 434 9 Equations 1
1 Fill in the missing number.
a + 3 = 10 b 41 - = 21 c × 3 = 48 d 100 ÷ = 20
2 If = 5, state whether each of these equations is true or false.
a - 2 = 5 b × 3 = 15 c 20 ÷ 4 = d × = 36
3 If x = 3, fi nd the value of:
a x + 4 b 8 - x c 5x d 3 + 7x
4 If n = 6, state the value of:
a n ÷ 2 b 4n + 3 c 8 - n d 12 ÷ n + 4
5 The expression n + 3 can be described as ‘the sum of n and 3’. Write expressions for:
a the sum of k and 5 b double p
c the product of 7 and y d one-half of q
6 Simplify each of the following algebraic expressions.
a 3x + 2x b 7 × 4b c 2a + 7b + 3a d 4 + 12a - 2a + 3
7 State the missing values in the tables below.
a
n 1 3 c d 12
5 × n a b 20 35 e
b
n 2 4 c 12 e
n - 2 a b 10 d 39
c
n a b 3 0 e
2n + 1 5 11 c d 27
d
n a 4 c d 2
6 - n 3 b 5 6 e
8 For each of the following, state the opposite operation.
a × b + c ÷ d -
pre-test
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number and algebra 435
Introduction to equations
An equation is a mathematical statement used to
say that two expressions have the same value. It
will always consist of two expressions that are
separated by an equals sign (=).
Sample equations include:
3 + 3 = 6
30 = 2 × 15
100 - 30 = 60 + 10
which are all true equations.
An equation does not have to be true.
For instance:
2 + 2 = 17
5 = 3 - 1 and
10 + 15 = 12 + 3 are all false equations.
If an equation contains pronumerals, one cannot tell whether the equation is true or false until values
are substituted for the pronumerals.
For example, 5 + x = 7 could be true (if x is 2) or it could be false (if x is 15).
let’s start: Equations – True or false?
Rearrange the following fi ve symbols to make as many different equations as possible.
5, 2, 3, +, =
• Which of them are true? Which are false?
• Is it always possible to rearrange numbers and operations to make true equations?
9A
This equation was proposed by the famous scientist Albert
Einstein (1879–1955). It explains the special theory of relativity.
■■ An expression is a collection of pronumerals, numbers and operators without an equals sign
(e.g. 2x + 3).
■■ An equation is a mathematical statement stating that two things are equal (e.g. 2x + 3 = 4y - 2).
■■ Equations have a left-hand side (LHS), a right-hand side (RHS) and an equals sign in between.
2x + 3
LHS
= 4y - 2
RHS
equals sign
■■ Equations are mathematical statements that can be true (e.g. 2 + 3 = 5) or false
(e.g. 5 + 7 = 21).
■■ If a pronumeral is included in an equation, you need to know the value to substitute before
deciding whether the equation is true.
For example, 3x = 12 would be true if 4 is substituted for x, but it would be false if 10 is substituted.
■■ The value(s) that make an equation true are called solutions.
For example, the solution of 3x = 12 is x = 4.
Key ideas
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Chapter 436 9 Equations 1
Example 1 Identifying equations
Which of the following are equations?
a 3 + 5 = 8 b 7 + 7 = 18 c 2 + 12 d 4 = 12 - x e 3 + u
Solution Explanation
a 3 + 5 = 8 is an equation. There are two expressions (i.e. 3 + 5 and 8) separated
by an equals sign.
b 7 + 7 = 18 is an equation. There are two expressions separated by an equals sign.
Although this equation is false, it is still an equation.
c 2 + 12 is not an equation. This is just a single expression. There is no equals sign.
d 4 = 12 - x is an equation. There are two expressions separated by an equals sign.
e 3 + u is not an equation. There is no equals sign, so this is not an equation.
Example 2 Classifying equations
For each of the following equations, state whether it is true or false.
a 7 + 5 = 12 b 5 + 3 = 2 × 4
c 12 × (2 - 1) = 14 + 5 d 3 + 9x = 60 + 6, if x = 7
e 10 + b = 3b + 1, if b = 4 f 3 + 2x = 21 - y, if x = 5 and y = 8
Solution Explanation
a true The left-hand side (LHS) and right-hand side (RHS)
are both equal to 12, so the equation is true.
b true LHS = 5 + 3 = 8 and RHS = 2 × 4 = 8, so both sides
are equal.
c false LHS = 12 and RHS = 19, so the equation is false.
d true If x is 7, then:
LHS = 3 + 9 × 7 = 66
RHS = 60 + 6 = 66
e false If b is 4, then:
LHS = 10 + 4 = 14
RHS = 3(4) + 1 = 13
f true If x = 5 and y = 8, then:
LHS = 3 + 2(5) = 13
RHS = 21 - 8 = 13
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number and algebra 437
Example 3 Writing equations from a description
Write equations for each of the following scenarios.
a The sum of x and 5 is 22.
b The number of cards in a deck is x. In 7 decks there are 91 cards.
c Priya’s age is currently j. In 5 years’ time her age will equal 17.
d Corey earns $w per year. He spends
1
12
on sport and
2
13
on food. The total amount Corey spends
on sport and food is $15 000.
SolutIon ExplanatIon
a x + 5 = 22 The sum of x and 5 is written x + 5.
b 7x = 91 7x means 7 × x and this number must equal the 91 cards.
c j + 5 = 17 In 5 years’ time Priya’s age will be 5 more than her current
age, so j + 5 must be 17.
d 1
12
2
13
× w + × w = 15000
1
12
of Corey’s wage is
1
12
× w and
2
13
of his wage is
2
13
× w.
1 Classify each of the following as an equation (E) or not an equation (N).
a 7 + x = 9 b 2 + 2 c 2 × 5 = t
d 10 = 5 + x e 2 = 2 f 7 × u
g 10 ÷ 4 = 3p h 3 = e + 2 i x + 5
2 Classify each of these equations as true or false.
a 2 + 3 = 5 b 3 + 2 = 6 c 5 - 1 = 6
3 Consider the equation 4 + 3x = 2x + 9.
a If x = 5, state the value of the left-hand side (LHS).
b If x = 5, state the value of the right-hand side (RHS).
c Is the equation 4 + 3x = 2x + 9 true or false when x = 5?
4 If x = 2, is 10 + x = 12 true or false?
Example 1
Example 2a–c
Example 2d,e
Exercise 9A
WORKING
MATHEMATICALLY
U F
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C
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Chapter 438 9 Equations 1
5 For each of the following equations, state whether it is true or false.
a 10 × 2 = 20 b 12 × 11 = 144 c 3 × 2 = 5 + 1
d 100 - 90 = 2 × 5 e 30 × 2 = 32 f 12 - 4 = 4
g 2(3 - 1) = 4 h 5 - (2 + 1) = 7 - 4 i 3 = 3
j 2 = 17 - 14 - 1 k 10 + 2 = 12 - 4 l 1 × 2 × 3 = 1 + 2 + 3
m 2 × 3 × 4 = 2 + 3 + 4 n 100 - 5 × 5 = 20 × 5 o 3 - 1 = 2 + 5 - 5
6 If x = 3, state whether each of these equations is true or false.
a 5 + x = 7 b x + 1 = 4 c 13 - x = 10 + x d 6 = 2x
7 If b = 4, state whether each of the following equations is true or false.
a 5b + 2 = 22 b 10 × (b - 3) = b + b + 2
c 12 - 3b = 5 - b d b × (b + 1) = 20
8 If a = 10 and b = 7, state whether each of these equations is true or false.
a a + b = 17 b a × b = 3 c a × (a - b) = 30
d b × b = 59 - a e 3a = 5b - 5 f b × (a - b) = 20
g 21 - a = b h 10 - a = 7 - b i 1 + a - b = 2b - a
9 Write equations for each of the following.
a The sum of 3 and x is equal to 10.
b When k is multiplied by 5, the result is 1005.
c The sum of a and b is 22.
d When d is doubled, the result is 78.
e The product of 8 and x is 56.
f When p is tripled, the result is 21.
g One-quarter of t is 12.
h The sum of q and p is equal to the product of q and p.
Example 2f
Example 3a
WORKING
MATHEMATICALLY
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10 Write true equations for each of these problems. You do not
need to solve them.
a Chairs cost $c at a store. The cost of 6 chairs is $546.
b Patrick works for x hours each day. In a 5-day working week,
he works 37
1
2
hours in total.
c Pens cost $a each and pencils cost $b. Twelve pens and three
pencils cost $28 in total.
d Amy is f years old. In 10 years’ time her age will be 27.
e Andrew’s age is j and Hailey’s age is m. In 10 years’ time
their combined age will be 80.
Example 3b–d
WORKING
MATHEMATICALLY
U F
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9A
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Number and Algebra 439
11 Find a value of m that would make this equation true: 10 = m + 7.
12 Find two possible values of k that would make this equation true: k × (8 - k) = 12.
13 If the equation x + y = 6 is true, and x and y are both whole numbers between 1 and 5, what values
could they have?
14 Equations involving pronumerals can be split into three groups:
A: Always true, no matter what values are substituted.
N: Never true, no matter what values are substituted.
S: Sometimes true but sometimes false, depending on the values substituted.
Categorise each of these equations as either A, N or S.
a x + 5 = 11 b 12 - x = x c a = a d 5 + b = b + 5
e a = a + 7 f 5 + b = b - 5 g 0 × b = 0 h a × a = 100
i 2x + x = 3x j 2x + x = 4x k 2x + x = 3x + 1 l a × a + 100 = 0
WORKING
MATHEMATICALLY
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Enrichment: Equation permutations
15 For each of the following, rearrange the symbols to make a true equation.
a 6, 2, 3, ×, = b 1, 4, 5, -, =
c 2, 2, 7, 10, -, ÷, = d 2, 4, 5, 10, -, ÷, =
16 a How many different equations can be produced using the symbols 2, 3, 5, +, = ?
b How many of these equations are true?
c Is it possible to change just one of the numbers above and still produce true equations by
rearranging the symbols?
d Is it possible to change just the operation above (i.e. +) and still produce true equations?
Many mathematical equations need to be solved in order to build and
launch space stations into orbit.
WORK ING
MATHEMATICALLY
U F
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C
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Chapter 440 9 Equations 1
Solving equations by inspection
Solving an equation is the process of fi nding the values that pronumerals must take in order to make the
equation true. Pronumerals are also called unknowns when solving equations. For simple equations, it
is possible to fi nd a solution by trying a few values for the pronumeral until the equation is true. This
method does not guarantee that we have found all the solutions (if there is more than one) and it will not
help if there are no solutions, but it can be a useful and quick method for simple equations.
let’s start: Finding the missing value
• Find the missing values to make the
following equations true.
10 × - 17 = 13
27 = 15 + 3 ×
2 × + 4 = 17 +
• Can you always fi nd a value to put in
the place of in any equation?
9B
■■ Solving an equation means fi nding the values of any pronumerals to make the equation true.
These values are called solutions.
■■ An unknown in an equation is a pronumeral whose value needs to be found in order to make the
equation true.
■■ One method of solving equations is by inspection (also called trial and error or guess and
check), which involves inspecting (or trying) different values and seeing which ones make the
equation true.
Key ideas
Example 4 Finding the missing number
For each of these equations, fi nd the value of the missing number that would make it true.
a × 7 = 35 b 20 - = 14
SolutIon ExplanatIon
a 5 5 × 7 = 35 is a true equation.
b 6 20 - 6 = 14 is a true equation.
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number and algebra 441
Example 5 Solving equations
Solve each of the following equations by inspection.
a c + 12 = 30 b 5b = 20 c 2x + 13 = 21 d x2 = 9
SolutIon ExplanatIon
a c + 12 = 30
c = 18
The unknown variable here is c.
18 + 12 = 30 is a true equation.
b 5b = 20
b = 4
The unknown variable here is b.
Recall that 5b means 5 × b, so if b = 4,
5b = 5 × 4 = 20.
c 2x + 13 = 21
x = 4
The unknown variable here is x.
Trying a few values:
x = 10 makes LHS = 20 + 13 = 33, which is too large.
x = 3 makes LHS = 6 + 13 = 19, which is too small.
x = 4 makes LHS = 21.
d x2 = 9
x = -3, x = 3
x = ±3
(-3)2 = 9 is a true equation
and
(3)2 = 9 is also a true equation.
This equation has two solutions.
1 If the missing number is 5, classify each of the following equations as true or false.
a + 3 = 8 b 10 × + 2 = 46
c 10 - = 5 d 12 = 6 + × 2
2 For the equation + 7 = 13:
a Find the value of the LHS (left-hand side) if = 5.
b Find the value of the LHS if = 10.
c Find the value of the LHS if = 6.
d What value of would make the LHS equal to 13?
3 Find the value of the missing numbers.
a 4 + = 7 b 2 × = 12
c 13 = + 3 d 10 = 6 +
e 42 = × 7 f 100 - = 30
g 15 + 6 = + 1 h + 11 = 49 -
4 Name the unknown pronumeral in each of the following equations?
a 4 + x = 12 b 50 - c = 3
c 4b + 2 = 35 d 5 - 10d = 2
Example 4
Exercise 9B
WORKING
MATHEMATICALLY
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C
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Chapter 442 9 Equations 1
5 Solve the following equations by inspection.
a 8 × y = 64 b 6 ÷ l = 3 c l × 3 = 18
d 4 - d = 2 e l + 2 = 14 f a - 2 = 4
g s + 7 = 19 h x ÷ 8 = 1 i 12 = e + 4
j r ÷ 10 = 1 k 13 = 5 + s l 0 = 3 - z
6 Solve the following equations by inspection.
a 2p - 1 = 5 b 3p + 2 = 14 c 4q - 4 = 8
d 4v + 4 = 24 e 2b - 1 = 1 f 5u + 1 = 21
g 5g + 5 = 20 h 4(e - 2) = 4 i 45 = 5(d + 5)
j 3d - 5 = 13 k 8 = 3m - 4 l 8 = 3o - 1
7 Solve the following equations by inspection. (All solutions are whole numbers between 1 and 10.)
a 4 × (x + 1) - 5 = 11 b 7 + x = 2 × x c (3x + 1) ÷ 2 = 8
d 10 - x = x + 2 e 2 × (x + 3) + 4 = 12 f 15 - 2x = x
g x2 = 4 h x2 = 100 i 36 = x2
Example 5a,b
Example 5c
Example 5d
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8 Find the value of the number in each of these examples.
a A number is doubled and the result is 22.
b 3 less than a number is 9.
c Half of a number is 8.
d 7 more than a number is 40.
e A number is divided by 10, giving a result of 3
f 10 is divided by a number, giving a result of 5.
9 Justine is paid $10 an hour for x hours. During a particular week, she earns $180.
a Write an equation involving x to describe this situation.
b Solve the equation by inspection to fi nd the value of x.
10 Karim’s weight is w kg and his brother is twice as heavy, weighing 70 kg.
a Write an equation involving w to describe this situation.
b Solve the equation by inspection to fi nd the value of w.
11 Taylah buys x kg of apples at $4.50 per kg. She
spends a total of $13.50.
a Write an equation involving x to describe this
situation.
b Solve the equation by inspection to fi nd x.
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9B
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Number and Algebra 443
12 Yanni’s current age is y years old. In 12 years’ time he will be three times as old.
a Write an equation involving y to describe this situation.
b Solve the equation by inspection to find y.
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13 a Solve the equation x + (x + 1) = 19 by inspection.
b The expression x + (x + 1) can be simplified to 2x + 1. Use this observation to solve
x + (x + 1) = 181 by inspection.
14 There are three consecutive whole numbers that add to 45.
a Solve the equation x + (x + 1) + (x + 2) = 45 by inspection to find the three numbers.
b An equation of the form x + (x + 1) + (x + 2) = ? has a whole number solution only if the
right-hand side is a multiple of 3. Explain why this is the case. (Hint: Simplify the LHS.)
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Enrichment: Multiple variables
15 When multiple variables are involved, inspection can still be used to find a solution. For each of the
following equations find, by inspection, one pair of values for x and y that make them true.
a x + y = 8 b x - y = 2 c 3 = 2x + y
d x × y = 6 e 12 = 2 + x + y f x + y = x × y
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Chapter 444 9 Equations 1
Equivalent equations
Sometimes, two equations essentially express the same thing. For example, the equations x + 5 = 14,
x + 6 = 15 and x + 7 = 16 are all made true by the same value of x. Each time, we have added one to both
sides of the equation.
We can pretend that true equations are about different objects that have the same weight. For instance,
to say that 3 + 5 = 8 means that a 3 kg block added to a 5 kg block weighs the same as an 8 kg block.
double
both sides
subtract 3 from
both sides
add 1 to
both sides
x + 6 = 15
2x + 10 = 28
x + 2 = 11
initial equation
x + 5 = 14
14
14
14 14
11
x
1
1
1
1
1
x
1
1
1
1
1
x
1
1
1
1
1
1
1
x
1
1
x
1
1
1
1
1
A true equation stays true if we ‘do the same thing to both sides’, such as adding a number or
multiplying by a number. The exception to this rule is that multiplying both sides of any equation by
zero will always make the equation true, and dividing both sides of any equation by zero is not permitted
because nothing can be divided by zero. If we do the same thing to both sides we will have an equivalent
equation.
9C
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Number and Algebra 445
Let’s start: Equations as scales
The scales in the diagram show 2 + 3x = 8.
• What would the scales look like if two ‘1 kg’ blocks were
removed from both sides?
• What would the scales look like if the two ‘1 kg’ blocks were
removed only from the left-hand side? (Try to show whether
they would be level.)
• Use scales to illustrate why 4x + 3 = 4 and 4x = 1 are
equivalent equations.
1
1
1
1
1
1
1
1
1
1
x
x
x
■■ Two equations are equivalent if you can get from one to the other by repeatedly:
–– Adding a number to both sides
–– Subtracting a number from both sides
–– Multiplying both sides by a number other than zero
–– Dividing both sides by a number other than zero
–– Swapping the left-hand side with the right-hand side of the equation
Key ideas
Example 6 Applying an operation
For each equation, find the result of applying the given operation to both sides and then simplify.
a 2 + x = 5 [add 4 to both sides] b 7x = 10 [multiply both sides by 2]
c 30 = 20b [divide both sides by 10] d 7q - 4 = 10 [add 4 to both sides]
Solution Explanation
a 2 + x = 5
2 + x + 4 = 5 + 4
x + 6 = 9
The equation is written out, and 4 is added to both sides.
Simplify the expressions on each side.
b 7x = 10
7x × 2 = 10 × 2
14x = 20
The equation is written out, and both sides are multiplied by 2.
Simplify the expressions on each side.
c 30 = 20b
30
10
20
10
= b
3 = 2b
The equation is written out, and both sides are divided by 10.
Simplify the expressions on each side.
d 7q - 4 = 10
7q - 4 + 4 = 10 + 4
7q = 14
The equation is written out, and 4 is added to both sides.
Simplify the expressions on each side.
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Chapter 446 9 Equations 1
Example 7 Showing that equations are equivalent
Show that these pairs of equations are equivalent by stating the operation used.
a 2x + 10 = 15 and 2x = 5
b 5 = 7 - x and 10 = 2(7 - x)
c 10(b + 3) = 20 and b + 3 = 2
SolutIon ExplanatIon
a Both sides have had 10 subtracted.
- 10 - 10
2x + 10 = 15
2x = 5
2x + 10 - 10 simplifi es to 2x, so we get the second
equation by subtracting 10.
b Both sides have been multiplied by 2.
5 = 7 - x
10 = 2(7 - x)
× 2 × 2
2(7 - x) represents the RHS; i.e. 7 - x, being multiplied
by 2.
c Both sides have been divided by 10.
10(b + 3) = 20
b + 3 = 2
÷ 10 ÷ 10
Remember 10(b + 3) means 10 × (b + 3).
If we have 10(b + 3), we get b + 3 when dividing by 10.
1 Write an equation that results from adding 10 to both sides of each of these equations.
a 10d + 5 = 20 b 7e = 31 c 2a = 12 d x = 12
2 Match up each of these equations (a to e) with its equivalent equation (i.e. a to E), where 3 has
been added.
a 10 + x = 14 a 12x + 3 = 123
b x + 1 = 13 B x + 13 = 11x + 3
c 12 = x + 5 C 13 + x = 17
d x + 10 = 11x D x + 4 = 16
e 12x = 120 E 15 = x + 8
Example 6a
Exercise 9C
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3 For each equation, show the result of applying the listed operations to both sides.
(Note: [+ 1] means ‘add 1 to both sides’.)
a 5 + x = 10 [+ 1] b 3x = 7 [× 2] c 12 = 8q [÷ 4]
d 9 + a = 13 [- 3] e 7 + b = 10 [+ 5] f 5 = 3b + 7 [- 5]
g 2 = 5 + a [+ 2] h 12x - 3 = 3 [+ 5] i 7p - 2 = 10 [+ 2]
Example 6b–d
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Number and Algebra 447
4 Show that these pairs of equations are equivalent by stating the operation used.
a 4x + 2 = 10 and 4x = 8 b 7 + 3b = 12 and 9 + 3b = 14
c 20a = 10 and 2a = 1 d 4 = 12 - x and 8 = 2(12 - x)
e 18 = 3x and 6 = x f 12 + x = 3 and 15 + x = 6
g 4(10 + b) = 80 and 10 + b = 20 h 12x = 5 and 12x + 4 = 9
5 For each of the following equations, show the equivalent equation that is the result of adding 4 to both
sides and then multiplying both sides by 3.
a x = 5 b 2 = a + 1 c d - 4 = 2
d 7 + a = 8 e 3y - 2 = 7 f 2x = 6
Example 7
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6 Match up each of these equations (a to e) with its equivalent equation (i.e. A to E), stating
the operation used.
a m + 10 = 12 A 7 - m = 6
b 3 - m = 2 B 5m = 18
c 12m = 30 C 6m = 10
d 5m + 2 = 20 D 6m = 15
e 3m = 5 E m + 12 = 14
7 For each of the following pairs of equations, show that they are equivalent by listing the two steps
required to transform the first equation to the second.
a x = 5 and 3x + 2 = 17 b m = 2 and 10m - 3 = 17
c 5(2 + x) = 15 and x = 1 d 10 = 3x + 10 and 0 = x
8 For each of the following equations, write an equivalent equation that you can get in one operation.
Your equation should be simpler (i.e. smaller) than the original.
a 2q + 7 = 9 b 10x + 3 = 10 c 2(3 + x) = 40 d x ÷ 12 = 5
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9 Sometimes two equations that look quite different can be equivalent.
a Show that 3x + 2 = 14 and 10x + 1 = 41 are equivalent by copying and completing the
following.
3x + 2 = 14
− 2 − 2
3x = 12
÷ 3 ÷ 3
___ = ___
× 10 × 10
___ = ___
+ 1 + 1
10x + 1 = 41
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Chapter 448 9 Equations 1
b Show that 5x - 3 = 32 and x + 2 = 9 are equivalent. (Hint: Try to go via the equation x = 7.)
c Show that (x ÷ 2) + 4 = 9 and (x + 8) ÷ 2 = 9 are equivalent.
10 As stated in the rules for equivalence, which are listed in Key ideas, multiplying both sides by
zero is not permitted.
a Write the result of multiplying both sides of the following equations by zero.
i 3 + x = 5 ii 2 + 2 = 4 iii 2 + 2 = 5
b Explain in a sentence why multiplying by zero does not give a useful equivalent equation.
11 Substituting pronumerals can be thought of as fi nding equivalent equations. Show how you can
start with the equation x = 3 and fi nd an equivalent equation with:
a 7x + 2 on the LHS b 8 + 2x on the LHS
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Enrichment: Equivalence relations
12 Classify each of the following statements as true or false, justifying your answer.
a Every equation is equivalent to itself.
b If equation 1 and equation 2 are equivalent, then equation 2 and equation 1 are equivalent.
c If equation 1 and equation 2 are equivalent, and equation 2 and equation 3 are equivalent, then
equation 1 and equation 3 are equivalent.
d If equation 1 and equation 2 are not equivalent, and equation 2 and equation 3 are not equivalent,
then equation 1 is not equivalent to equation 3.
9C
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number and algebra 449
Solving equations systematically
A soccer player preparing for a game will put on shin pads, then
socks and, fi nally, soccer boots. When the game is over, these
items are removed in reverse order: fi rst the boots, then the
socks and, fi nally, the shin pads. Nobody takes their socks off
before their shoes. A similar reversal of procedures occurs with
equivalent equations.
Here are three equivalent equations.
x = 3
2x = 6
2x + 4 = 10
× 2
+ 4
× 2
+ 4
We can undo the operations around x by doing the opposite
operation in the reverse order.
2x + 4 = 10
2x = 6
x = 3
- 4
÷ 2 ÷ 2
- 4
Because these equations are equivalent, this means that the solution to 2x + 4 = 10 is x = 3. An
advantage with this method is that solving equations by inspection can be very diffi cult if the solution is
not just a small whole number.
let’s start: Attempting solutions
Georgia, Khartik and Lucas try to solve the equation 4x + 8 = 40. They present their attempted solutions
below.
Georgia Khartik Lucas
4x + 8 = 40
x + 8 = 10
x = 2
÷ 4
- 8
÷ 4
- 8
4x + 8 = 40
4x = 48
x = 12
- 8
÷ 4
+ 8
÷ 4
4x + 8 = 40
4x = 32
x = 8
- 8
÷ 4
- 8
÷ 4
• Which of the students has the correct solution to the equation? Justify your answer by substituting
each student’s fi nal answer.
• For each of the two students with the incorrect answer, explain the mistake they have made in their
attempt to have equivalent equations.
• What operations would you do to both sides if the original equation was 7x - 10 = 11?
9D
The order in which things are done matters
in both sports and maths.
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Chapter 450 9 Equations 1
■■ Sometimes it is very difficult to solve an equation by inspection, so a systematic approach is
required.
■■ To solve an equation, find a simpler equation that is equivalent. Repeat this until the solution is
found.
■■ A simpler equation can be found by applying the opposite operations in reverse order.
e.g. For 5x + 2 = 17, we have:
x 5x 5x + 2
× 5 + 2
So we solve the equation by ‘undoing’ them in reverse order.
5x + 2 5x x
- 2 ÷ 5
This gives the solution:
■■ A solution can be checked by substituting the value to see if the equation is true.
e.g. LHS = 5(3) + 2 = 17 and RHS = 17.
5x + 2 = 17
5x = 15
x = 3
- 2
÷ 5 ÷ 5
- 2
Key ideas
Example 8 Solving one-step equations
Solve each of the following equations systematically.
a 5x = 30 b 17 = y - 21 c 10
3
= q
Solution Explanation
a
So the solution is x = 6.
The opposite of × 5 is ÷ 5.
By dividing both sides by 5, we get an equivalent
equation. Recall that 5x ÷ 5 simplifies to x.
b
So the solution is y = 38.
The opposite of - 21 is + 21.
Write the pronumeral on the LHS.
c
So the solution is q = 30.
Multiplying both sides by 3 gives an equivalent equation
that is simpler. Note that
q
q
3
× 3 = .
Write the pronumeral on the LHS.
5x = 30
x = 6
÷ 5 ÷ 5
17 = y - 21
38 = y
+ 21 + 21
10 =
q
3
30 = q
× 3 × 3
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Number and Algebra 451
Example 9 Solving two-step equations
Solve each of the following equations systematically and check the solution.
a 7 + 4a = 23 b d
3
− 2 = 4 c 12 = 2(e + 1)
Solution Explanation
a
At each step, try to make the equation simpler by applying
an operation to both sides.
Choose the opposite operations based on 7 + 4a:
a 4a 7 + 4a
× 4 + 7
Opposite operations: - 7, then ÷ 4.
Check:
LHS = 7 + 4a RHS = 23 ✓
= 7 + 4 × 4
= 7 + 16
= 23 ✓
Check that our equation is true by substituting a = 4 back
into the equation to show LHS = RHS.
b
At each step, try to make the equation simpler by
applying an operation to both sides.
The opposite of ‑ 2 is + 2 and the opposite of ÷ 3 is × 3.
Check:
LHS = d ÷ 3 - 2 RHS = 4 ✓
= 18 ÷ 3 - 2
= 6 - 2
= 4 ✓
Check that our equation is true by substituting d = 18
back into the equation to show LHS = RHS.
c
So the solution is e = 5.
At each step, try to make the equation simpler by
applying an operation to both sides.
The opposite of × 2 is ÷ 2 and the opposite of + 1 is - 1.
Write the solution on the LHS.
Check:
LHS = 12 ✓ RHS = 2(e + 1)
= 2(5 + 1)
= 2 × 6
= 12 ✓
Check that our equation is true by substituting e = 5 back
into the equation to show LHS = RHS.
7 + 4a = 23
4a = 16
a = 4
- 7
÷ 4
- 7
÷ 4
d
3
- 2 = 4
d
3
= 6
d = 18
+ 2
× 3
+ 2
× 3
12 = 2(e + 1)
6 = e + 1
5 = e
÷ 2
- 1
÷ 2
- 1
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Chapter 452 9 Equations 1
1 State whether each of the following equations is true or false.
a x + 4 = 7, if x = 3 b b - 2 = 7, if b = 5
c 7(d - 6) = d, if d = 7 d g + 5 = 3g, if g = 2
e f × 4 = 20, if f = 3
2 Consider the equation 7x = 42.
a Copy and complete the following.
b What is the solution to the equation 7x = 42?
3 The equations g = 2 and 12(g + 3) = 60 are equivalent. What is the solution to the equation
12(g + 3) = 60?
4 Copy and complete the following, showing which operation was used. Remember that the same
operation must be used for both sides.
a 5 + a = 30
a = 25
? ?
b 10b = 72
b = 7.2
? ?
c 12 =
c
4
48 = c
? ?
d 8 = c - 12
20 = c
? ?
7x = 42
x = __
÷ 7 ÷ 7
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5 Solve the following equations systematically.
a 6m = 54 b g - 9 = 2 c s × 9 = 81 d i - 9 = 1
e 7 + t = 9 f 8 + q = 11 g 4y = 48 h 7 + s = 19
i 24 = j × 6 j 12 = l + 8 k 1 = v ÷ 2 l 19 = 7 + y
m k ÷ 5 = 1 n 2 = y - 7 o 8z = 56 p 13 = 3 + t
q b × 10 = 120 r p - 2 = 9 s 5 + a = 13 t n - 2 = 1
6 Copy and complete the following to solve the given equations systematically.
a 7a + 3 = 38
7a = 35
__ = __
?
?
?
?
b 4b - 10 = 14
__ = __
__ = __
+ 10
?
+ 10
?
c 2(q + 6) = 20
q + 6 = __
__ = __
÷ 2
?
÷ 2
?
d 5 = x
10
+ 3
__ =
x
10
__ = __
- 3
?
- 3
?
7 For each of these equations, state the fi rst operation you would apply to both sides to solve it.
a 2x + 3 = 9 b 4x - 7 = 33
c 5(a + 3) = 50 d 22 = 2(b - 17)
Example 8
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Exercise 9D
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Number and Algebra 453
8 For each of the following equations:
i Solve the equation systematically, showing your steps.
ii Check your solution by substituting the value into the LHS and RHS.
a 6f - 2 = 64 b
k
4
+ 9 = 10 c 5x - 4 = 41
d 3(a - 8) = 3 e 5k - 9 = 31 f
a
3
+ 6 = 8
g 2n - 8 = 14 h
n
4
+ 6 = 8 i 1 = 2g - 7
j 30 = 3q - 3 k 3z - 4 = 26 l 17 = 9 + 8p
m 10d + 7 = 47 n 38 = 6t - 10 o 9u + 2 = 47
p 7 = 10c - 3 q 10 + 8q = 98 r 80 = 4( y + 8)
s 4(q + 8) = 40 t 7 + 6u = 67
9 Solve the following equations, giving your solutions as fractions.
a 4x + 5 = 8 b 3 + 5k = 27 c 22 = (3w + 7) × 2
d 10 = 3 × (2 + x) e 3 = (8x + 1) ÷ 2 f 3(x + 2) = 7
10 Solve the following equations systematically. (Note: The solutions for these equations are negative
numbers.)
a 4r + 30 = 2 b 2x + 12 = 6 c 10 +
t
2
= 2
d
y
4
+ 10 = 4 e -3x = 15 f 4 = 2k + 22
g 2x = -12 h 5x + 20 = 0 i 0 = 2x + 3
Example 9
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11 For each of the following, write an equation and solve it systematically.
a The sum of x and 5 is 12.
b The product of 2 and y is 10.
c When b is doubled and then 6 is added, the result is 44.
d 7 is subtracted from k. This result is tripled, giving 18.
e 3 is added to one-quarter of b, giving a result of 6.
f 10 is subtracted from half of k, giving a result of 1.
12 Danny gets paid $12 per hour, plus a bonus of $50 for each
week. In one week he earned $410.
a Write an equation to describe this, using n for the number
of hours worked.
b Solve the equation systematically and state the number of
hours worked.
13 Jenny buys 12 pencils and 5 pens for the new school year.
The pencils cost $1.00 each.
a If pens cost $x each, write an expression for the total cost, in dollars.
b The total cost was $14.50. Write an equation to describe this.
c Solve the equation systematically, to find the total cost of each pen.
d Check your solution by substituting your value of x into 12 + 5x.
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Chapter 454 9 Equations 1
14 Write equations and solve them systematically to fi nd the unknown value in each of the
following diagrams.
a b
3 area = 15
b
x
4
area = 12
c 10
x perimeter = 28
d
b perimeter = 28
15 Solve the following equations systematically.
a 7(3 + 5x) - 21 = 210
b (100x + 13) ÷ 3 = 271
c 3(12 + 2x) - 4 = 62
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16 Write fi ve different equations that give a solution of x = 2.
17 a Show that 2x + 5 = 13 and 5x = 20 are equivalent by fi lling in the
missing steps.
b Show that 10 + 2x = 20 and 2(x - 3) = 4 are equivalent.
c If two equations have exactly the same solutions, does this
guarantee they are equivalent? Justify your answer.
d If two equations have different solutions, does this guarantee they
are not equivalent? Justify your answer.
18 Nicola has attempted to solve four equations. Describe the error she has made in
each case.
a 4x + 2 = 36
x + 2 = 9
x = 7
÷ 4
- 2
÷ 4
- 2
b 3x + 10 = 43
3x = 33
x = 30
- 10
- 3
- 10
- 3
c 2a + 5 = 11
2a = 16
a = 8
- 5
÷ 2
- 5
÷ 2
d 7 + 12a = 43
7 + a = 31
a = 24
- 12
- 7
- 12
- 7
2x + 5 = 13
__ = __
x = 4
5x = 20
- 5
?
?
- 5
?
?
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9D
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Number and Algebra 455
Enrichment: Equations with pronumerals on both sides
19 If an equation has a pronumeral on both sides, you can subtract it from one side and then use the
same method as before. For example:
Solve the following equations using this method.
a 5x + 2 = 3x + 10 b 8x - 1 = 4x + 3
c 5 + 12l = 20 + 7l d 2 + 5t = 4t + 3
e 12s + 4 = 9 + 11s f 9b - 10 = 8b + 9
g 5j + 4 = 10 + 2j h 3 + 5d = 6 + 2d
Just like dance steps, a strict order must be followed when solving equations
systematically.
5x + 4 = 3x + 10
2x + 4 = 10
2x = 6
x = 3
- 3x
- 4
÷ 2
- 3x
- 4
÷ 2
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Chapter 456 9 Equations 1
Equations with fractions
Solving equations that involve fractions is straightforward once we recall that, in algebra, a
b
means a ÷ b.
This means that if we have a fraction with b on the denominator, we can multiply both sides by b to get a
simpler, equivalent equation.
let’s start: Fractional differences
Consider these three equations.
a
2 3
5
7
x + = b
2
5
3 7
x + = c 2
5
3 7
x  
 
+ =
• Solve each of them (by inspection or systematically).
• Compare your solutions with those of your classmates.
• Why do two of the equations have the same solution?
9E
■■ Recall that
a
b
means a ÷ b.
■■ The expressions
x
3
+ 2 and
x + 2
3
are different, as demonstrated in these fl ow charts.
x vs
x
3
x
3
+ 2
÷ 3 + 2
x x + 2
x + 2
3
+ 2 ÷ 3
■■ To solve an equation that has a fraction on one side, multiply both sides by the denominator.
■■ If neither side of an equation is a fraction, do not multiply by the denominator.
■■ Sometimes it is wise to swap the LHS and RHS.
For example: 12
3
= +1 x becomes x
3
+1 = 12, which is easier to solve.
x
5
= 4
x = 20
× 5 × 5
x
3
+ 5 = 8
. . .
× 3 × 3 ✗ Do not do this
x
3
+ 5 = 8
x
3
= 3
x = 9
- 5
× 3
- 5
× 3 ✓ Do this
Key ideas
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number and algebra 457
Example 10 Solving equations with fractions
Solve each of the following equations.
a
a
7
= 3 b
5
3
10
y = c
3
4
7 13
x + = d
2 3
5
3
x − =
SolutIon ExplanatIon
a
Multiplying both sides by 7 removes the denominator
of 7.
b
Multiplying both sides by 3 removes the denominator
of 3.
Then the equation 5y = 30 can be solved normally.
c
First, we subtract 7 because we do not have a fraction by
itself on the LHS.
Once there is a fraction by itself, multiply by its
denominator (in this case, 4) and solve the equation
3x = 24 as you would normally.
d
First, multiply both sides by 5 to remove the denominator.
Then solve the equation 2x - 3 = 15 as you would
normally.
a
7
= 3
a = 21
× 7 × 7
5
3
y
= 10
5y = 30
y = 6
× 3
÷ 5
× 3
÷ 5
3
4
x
+ 7 = 13
3
4
x
= 6
3x = 24
x = 8
- 7
× 4
÷ 3
- 7
× 4
÷ 3
2 3
5
x −
= 3
2x - 3 = 15
2x = 18
x = 9
× 5
+ 3
÷ 2
× 5
+ 3
÷ 2
1 Classify each of the following as true or false.
a
a
5
means a ÷ 5. b
q
12
means 12 ÷ q.
c
4
3
+ a
means (4 + a) ÷ 3. d
4
3
+ a
means 4 + (a ÷ 3).
e
12 3
4
+ q
means (12 + 3q) ÷ 4. f 2
5
+ x
means (2 + x) ÷ 5.
Exercise 9E
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Chapter 458 9 Equations 1
2 a If x = 10, fi nd the value of
x + 4
2
.
b If x = 10, fi nd the value of
x
2
+ 4.
c State whether the following is true or false.
x + 4
2
and
x
2
+ 4 are equivalent expressions.
3 Fill in the missing steps to solve each of these equations.
a b
4
= 11
b = __
× 4 × 4
b d
5
= 3
__ = __
× 5 × 5
c h
4
= 7
__ = __
? ?
d p
13
= 2
__ = __
? ?
4 For each of the following equations (a to d), choose the appropriate fi rst step (i.e. a to D) needed
to solve it.
a
x
3
= 10 a Multiply both sides by 2.
b
x
3
+ 2 = 5 B Multiply both sides by 3.
c
x − = 3
2
1 C Subtract 2 from both sides.
d
x
2
− 3 = 5 D Add 3 to both sides.
Example 10a
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5 Solve the following equations systematically.
a
m
6
= 2 b
c
9
= 2 c
s
8
= 2 d
r
5
= 2
e
3
5
u
= 12 f
2
9
y
= 4 g
5
2
x = 10 h
3
8
a
= 6
i
4
5
h
= 8 j
3
5
j
= 9 k
5
9
v = 5 l
3
4
q
= 6
6 Solve the following equations systematically. Check your solutions using substitution.
a
h + = 15
12
2 b
y + = 5
11
1 c
j + = 8
11
1 d
b − = 2
2
1
e
7 12
9
1
u − = f 14
4
9
+ = 18 t
g 1
5
11
= w +
h 1
4 13
3
= r −
i
2
9
2 4
q + = j
s + = 2
5
1 k
3
2
9 21
l + = l 12
2
7
= +10 z
Example 10b
Example 10c,d
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Number and Algebra 459
m 1
4
7
= v −
n
f − = 2
7
1 o 9 4
5
2
= + x
p 3
7 4
9
= + d
q
7
5
14 21
n + = r
7 7
4
21
m+ = s 3
7
4
= −11 p
t
4 6
5
6
a − =
7 Solve the following equations systematically. (Note: The solutions to these equations are negative
numbers.)
a
y + = 4
3
1 b
a
10
+ 2 = 1 c
2
5
10 6
x + = d
x
4
+ 12 = 0
e 0 12
2
5
= + u
f
3
5
8 2
y + = g 1
2 3
5
= − u −
h −2 = +
4
5
2
d
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8 In each of the following cases, write an equation and solve it to find the number.
a A number, t, is halved and the result is 9.
b One-third of q is 14.
c A number, r, is doubled and then divided by 5. The result is 6.
d 4 is subtracted from q and this is halved, giving a result of 3.
e 3 is added to x and the result is divided by 4, giving a result of 2.
f A number, y, is divided by 4 and then 3 is added, giving a result of 5.
9 A group of five people go out for dinner and then split the bill evenly. They each pay $31.50.
a If b represents the total cost of the bill, in dollars, write an equation to describe this situation.
b Solve this equation systematically.
c What is the total cost of the bill?
10 Lee and Theo hired a tennis court for a cost of $x, which
they split evenly. Out of his own pocket, Lee also bought
some tennis balls for $5.
a Write an expression for the total amount of money that
Lee paid.
b Given that Lee paid $11 in total, write an equation and
solve it to find the total cost of hiring the court.
c State how much money Theo paid for his share of
hiring the tennis court.
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Chapter 460 9 Equations 1
11 Sometimes the solution for an equation will be a fraction. For example, 2x = 1 has the
solution x = 1
2
.
a Give another equation that has x = 1
2
as its solution.
b Find an equation that has the solution x = 5
7
.
c Could an equation have the solution x = − 1
2
? Justify your answer.
12 a Explain, in one sentence, the difference between the expressions
2 3
5
x +
and
2
5
3
x + .
b What is the fi rst operation you would apply to both sides to solve the equation
2 3
5
7
x + = ?
c What is the fi rst operation you would apply to both sides to solve the equation
2
5
3 7
x + = ?
d Are there any values of x for which the expressions
2 3
5
x +
and
2
5
3
x + are equal to each other?
13 Dividing by 2 and multiplying by
1
2
have the same effect.
For example, 6 ÷ 2 = 3 and 6 ×
1
2
= 3.
a Show how each of these equations can be solved systematically.
i
x
2
= 5 ii
1
2
× x = 5
b Solve the two equations
x + = 4
3
10 and
1
3
(x + 4) = 10 systematically, showing clearly the
steps you would use at each stage.
c How does rewriting divisions as multiplications change the fi rst step when solving equations?
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Enrichment: Fractional solutions
14 Solve each of the following equations, giving your solutions as a fraction.
a
2 5
4
3
x + = b
3 4
6
3
4
x − =
c
7 2
4
3 10
+  
 
× = x
d
1
2
3 1
5
= x −
15 Consider the equation 5 3
7
6
x − = . The solution is x = 9. Change one number or one operator
(i.e. ×, - or ÷) in the equation so that the solution will be x = 12.
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9E
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number and algebra 461
Equations with brackets
Recall from Chapter 8 that expressions with brackets can be expanded
using the observation shown at right about rectangles’ areas.
So 3(x + 5) is equivalent to 3x + 15.
When solving 3(x + 5) = 21, we could fi rst divide both sides by 3
or we could fi rst expand the brackets, giving 3x + 15 = 21, and then
subtract 15. For some equations, the brackets must be expanded fi rst.
let’s start: Removing brackets
a Draw two rectangles, with areas 4(x + 3) and 5(x + 2).
b Use these to show that 4(x + 3) + 5(x + 2) is equivalent to 9x + 22.
c Can you solve the equation 4(x + 3) + 5(x + 2) = 130?
9F
3 × x = 3x
x 5
3 3 × 5 = 15
■■ To expand brackets, use the distributive law, which states that:
a(b + c) = ab + ac e.g. 3(x + 4) = 3x + 12
a(b - c) = ab - ac e.g. 4(b - 2) = 4b - 8
■■ Like terms are terms that contain exactly the same pronumerals and can be collected to
simplify expressions. For example, 3x + 4 + 2x can be simplifi ed to 5x + 4.
■■ Equations involving brackets can be solved by fi rst expanding brackets and collecting like
terms.
Key ideas
Example 11 Expanding brackets
Expand each of the following.
a 4(x + 3) b 6(q - 4) c 5(3a + 4)
SolutIon ExplanatIon
a 4(x + 3) = 4x + 12 Using the distributive law:
4(x + 3) = 4x + 12
b 6(q - 4) = 6q - 24 Using the distributive law:
6(q - 4) = 6q - 24
c 5(3a + 4) = 15a + 20 Using the distributive law:
5(3a + 4) = 5 × 3a + 20
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Chapter 462 9 Equations 1
Example 12 Simplifying expressions with like terms
Simplify each of these expressions.
a 2x + 5 + x b 3a + 8a + 2 - 2a + 5
Solution Explanation
a 2x + 5 + x = 3x + 5 Like terms are 2x and x.
These are combined to get 3x.
b 3a + 8a + 2 - 2a + 5 = 9a + 7 Like terms are combined:
3a + 8a - 2a = 9a
2 + 5 = 7
Example 13 Solving equations by expanding brackets
Solve each of these equations by expanding brackets first.
a 3(x + 2) = 18 b 7 = 7(4q - 3) c 3(b + 5) + 4b = 29
Solution Explanation
a 3(x + 2) = 18
3x + 6 = 18
3x = 12
x = 4
- 6
÷ 3
- 6
÷ 3
Use the distributive law to expand the brackets.
Solve the equation by performing the same operations to
both sides.
b 7 = 7(4q - 3)
7 = 28q - 21
28 = 28q
1 = q
+ 21
÷ 28
+ 21
÷ 28
So q = 1 is the solution.
Use the distributive law to expand brackets.
Solve the equation by performing the same operations to
both sides.
c 3(b + 5) + 4b = 29
3b + 15 + 4b = 29
7b + 15 = 29
7b = 14
b = 2
- 15
÷ 7
- 15
÷ 7
Use the distributive law to expand brackets.
Collect like terms to simplify the expression.
Solve the equation by performing the same operations to
both sides.
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number and algebra 463
1 Which of the following is the correct expansion of 5(x + 2)?
a 5 × x + 2 B 5x + 2 C 5x + 10 D 10x
2 Fill in the missing numbers.
a 3(x + 2) = 3x + b 4(3a + 1) = 12a +
c 2(b + 1) = b + 2 d 6(2c + 3) = c + 18
3 Answer true or false to each of the following.
a 4x + 3x can be simplifi ed to 7x.
b 2a + 4b can be simplifi ed to 6ab.
c 6p - 4p can be simplifi ed to 2p.
d 7a + 3 + 2a can be simplifi ed to 9a + 3.
e 2b + 3 can be simplifi ed to 5b.
f 20x - 12x + 3y can be simplifi ed to 32x + 3y.
Exercise 9F
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4 Expand each of the following.
a 2(x + 1) b 5(2b + 3) c 2(3a - 4) d 5(7a + 1)
e 4(3x + 4) f 3(8 - 3y) g 12(4a + 3) h 2(u - 4)
5 Simplify these expressions by collecting like terms.
a 3a + a + 2 b 5 + 2x + x c 2b - 4 + b d 5a + 12 - 2a
e 5x + 3 + x f 3k + 6 - 2k g 7 + 2b - 1 h 6k - k + 1
6 Solve the following equations by expanding the brackets fi rst. Check your solutions by
substituting them in.
a 2(10 + s) = 32 b 2(5 + l ) = 12 c 3( p - 7) = 6 d 8( y + 9) = 72
e 8(4 + q) = 40 f 7( p + 7) = 133 g 8(m + 7) = 96 h 22 = 2(b + 5)
i 25 = 5(2 + p) j 63 = 7( p + 2) k 9( y - 6) = 27 l 2(r + 8) = 32
7 Solve these equations by expanding the brackets fi rst.
a 6(3 + 2d ) = 54 b 8(7x - 7) = 56 c 3(2x - 4) = 18 d 27 = 3(3 + 6e)
e 44 = 4(3a + 8) f 30 = 6(5r - 10) g 10 = 5(9u - 7) h 3(2q - 9) = 39
8 Solve the following equations by fi rst expanding the brackets. You will need to simplify the
expanded expressions by collecting like terms.
a 5(4s + 4) + 4s = 44 b 5i + 5(2 + 2i ) = 25 c 3(4c - 5) + c = 50
d 3(4 + 3v) - 4v = 52 e 5(4k + 2) + k = 31 f 4q + 6(4q - 4) = 60
g 40 = 4y + 6(2y - 4) h 44 = 4 f + 4(2f + 2) i 40 = 5t + 6(4t - 3)
9 Solve the following equations. (Note: The solutions to these equations are negative numbers.)
a 3(u + 7) = 6 b 2(k + 3) = 0 c 6( p - 2) = -18
d 16 = 8(q + 4) e 5(2u + 3) = 5 f 3 = 2(x + 4) + 1
g 4( p - 3) + p = -32 h 3(r + 4) + 2r + 40 = 2
Example 11
Example 12
Example 13a
Example 13b
Example 13c
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Chapter 464 9 Equations 1
10 For each of the following problems:
i Write an equation.
ii Solve your equation by fi rst expanding any brackets.
a 5 is added to x and then this is doubled, giving a result of 14.
b 3 is subtracted from q and the result is tripled, giving a fi nal result of 30.
c A number, x, is doubled and then 3 is added. This number is doubled again to get a result of 46.
d 4 is added to y and this is doubled. Then the original number, y, is subtracted, giving a
result of 17.
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11 For each of the following equations, prove that there are no solutions.
a 2(x + 5) - 2x = 7
b 3(2x + 1) + 6(2 - x) = 4
c 4(2x + 1) - 10x + 2(x + 1) = 12
12 Consider the equation 2(3x + 4) - 6x + 1 = 9.
a Show that this equation is true if x = 0.
b Show that this equation is true if x = 3.
c Explain why this equation is always true.
d Give an example of another equation involving brackets that is always true, where one side
contains a pronumeral but the other side is just a number.
13 For equations like 4(3x + 2) = 44, you have been expanding the brackets fi rst.
Since 4(3x + 2) = 44 is the same as 4 × (3x + 2) = 44, you can just start by dividing both sides
by 4.
Without expanding brackets, solve the equations in Question 6 by dividing fi rst.
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Enrichment: Equations with multiple brackets
14 Solve each of the following equations.
a 6(2j - 4) + 4(4j - 3) = 20 b 3(4a + 5) + 5(1 + 3a) = 47
c 2(5a + 3) + 3(2a + 3) = 63 d 222 = 3(4a - 3) + 5(3a + 3)
e 77 = 2(3c - 5) + 3(4c + 5) f 240 = 4(3d + 3) + 6(3d - 2)
g 2(x + 3) + 4(x + 5) = 32 h 4(x + 5) + 4(x - 5) = 24
i 2(3x + 4) + 5(6x + 7) + 8(9x + 10) = 123
9F
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number and algebra 465
Formulas and relationships
Often, two or more variables are
closely related. For example, the speed
at which a car travels and the time it
takes to arrive at its destination are
related variable quantities. A formula
is an equation that contains two or
more pronumerals and shows how they
are related.
let’s start: Fahrenheit and Celsius
In Australia, we measure temperature in degrees Celsius, whereas in the USA it is measured in degrees
Fahrenheit. A formula to convert between them is F
C = + 9
5
32.
• At what temperature in degrees Fahrenheit does water freeze?
• At what temperature in degrees Fahrenheit does water boil?
• What temperature is 100° Fahrenheit in Celsius? Do you know what is signifi cant about this
temperature?
9G
A formula can show the relationship between a car’s speed and the distance
it travels.
■■ The subject of an equation is a pronumeral that occurs by itself on the left-hand side.
For example: T is the subject of T = 4x + 1.
■■ A formula or rule is an equation that contains two or more pronumerals, one of which is the
subject of the equation.
■■ To use a formula, fi rst substitute all the known values into the equation and then solve it to
fi nd the fi nal value.
Key ideas
Example 14 applying a formula involving two pronumerals
Consider the rule k = 3b + 2. Find the value of:
a k if b = 5
b k if b = 10
c b if k = 23
EXTENSION
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Chapter 466 9 Equations 1
SolutIon ExplanatIon
a k = 3 × 5 + 2
= 17
Substitute b = 5 into the equation.
b k = 3 × 10 + 2
= 32
Substitute b = 10 into the equation.
c 23 = 3b + 2
21 = 3b
7 = b
- 2
÷ 3
- 2
÷ 3
Therefore, b = 7.
Substitute k = 23 into the equation. Now solve the
equation to fi nd the value of b.
Example 15 applying a formula involving three pronumerals
Consider the rule Q = w(4 + t). Find the value of:
a Q if w = 10 and t = 3 b t if Q = 42 and w = 6
SolutIon ExplanatIon
a Q = 10(4 + 3)
= 10 × 7
= 70
Substitute w = 10 and t = 3 to evaluate.
b 42 = 6(4 + t )
42 = 24 + 6t
18 = 6t
3 = t
- 24
÷ 6
- 24
÷ 6
So, t = 3.
Substitute Q = 42 and w = 6.
Expand the brackets and then solve the equation.
1 State whether each of the following equations is a rule (R) or not a rule (N).
a 2x + 5 = 10 b y = 3x + 5 c F = ma
d 5 - q = 3 e w = 12 - v f P = I + k - 3
Exercise 9G EXTENSION
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Number and Algebra 467
2 Substitute:
a x = 3 into the expression 5x
b x = 7 into the expression 4(x + 2)
c y = 3 into the expression 20 - 4y
d y = 10 into the expression
y + 4
7
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3 Consider the rule h = 2m + 1. Find:
a h if m = 3
b h if m = 4
c m if h = 17. Set up an equation and solve it systematically.
d Find m if h = 21. Set up an equation and solve it systematically.
4 Consider the formula y = 5 + 3x. Find:
a y if x = 6 b x if y = 17 c x if y = 26
5 Consider the rule A = q + t. Find:
a A if q = 3 and t = 4 b q if A = 5 and t = 1 c t if A = 3 and q = 3
6 Consider the formula G = 7x + 2y. Find:
a G if x = 3 and y = 3 b x if y = 2 and G = 11 c y if G = 31 and x = 3
Example 14
Example 15
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7 A formula for the area of a rectangle is A = b × h, where b is the rectangle’s base and h is
the rectangle’s height.
a Set up and solve an equation to find the base of a rectangle with A = 20 and h = 4.
b A rectangle is drawn for which A = 25 and b = 5.
i Set up and solve an equation to find h.
ii What type of rectangle is this?
8 The perimeter for a rectangle is given by P = 2(b + h). Find the:
a perimeter when b = 3 and h = 5
b value of h when P = 10 and b = 2
c area of a rectangle if its perimeter is 20 and base is 4
9 To convert between temperatures in Celsius and Fahrenheit the rule is F
C = + 9
5
32.
a Find F if C = 20.
b Find the value of C if F = 50.
c Find the temperature in Celsius if it is 53.6° Fahrenheit.
d Marieko claims the temperature in her city varies between 68° Fahrenheit and
95° Fahrenheit. What is the difference, in Celsius, between these two temperatures?
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Chapter 468 9 Equations 1
10 Rearranging a formula involves fi nding an equivalent equation that
has a different variable on one side by itself. For example, as shown
at right, the formula S = 6g + b can be rearranged to make g by
itself.
Now we have a formula that can be used to fi nd g once S and b are
known.
a Rearrange S = 5d + 3b to make a rule where d is by itself.
b Rearrange the formula F
C = + 9
5
32 to make C by itself.
c Rearrange the formula Q = 3(x + 12) + x to make x by itself. (Hint: You will need to expand the
brackets fi rst.)
11 A taxi company charges different amounts of money based on how far the taxi travels and how long
the passenger is in the car. Although the company has not revealed the formula it uses, some sample
costs are shown below.
Distance ( D ) in km time ( t ) in minutes Cost (C ) in dollars
10 20 30
20 30 50
a Show that the rule C = D + t is consistent with the values above.
b Show that the rule C = 3D is not consistent with the values above.
c Show that the rule C = 2D + 10 is consistent with the values above.
d Try to fi nd at least two other formulas that the taxi company could be using, based on the values
shown.
S = 6g + b
S - b = 6g
S b
g
− =
6
- b
÷ 6
- b
÷ 6
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Enrichment: aFl equations
12 In Australian Rules Football (AFL), the score, S, is given by S = 6g + b, where g is the number of
goals scored and b is the number of ‘behinds’ (i.e. near misses).
a Which team is winning if the Abbotsford Apes have scored 11 goals (g = 11) and 9 behinds (b = 9),
and the Box Hill Baboons have scored 12 goals and 2 behinds?
b The Camperdown Chimpanzees have scored 7 behinds and their current score is S = 55. Solve an
equation systematically to fi nd how many goals the
team has scored.
c In some AFL competitions, a team can score a
‘supergoal’, which is worth 9 points. If q is the
number of supergoals that a team kicks, write a new
formula for the team’s score.
d For some rare combinations of goals and behinds,
the score equals the product of g and b. For example,
4 goals and 8 behinds gives a score of 6 × 4 + 8 =
32, and 4 × 8 = 32. Find all the other values of g and
b that make the equation 6g + b = gb true.
9G
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number and algebra 469
using equations to solve problems
Our methods for solving equations can be applied to many situations in which equations occur.
let’s start: Stationery shopping
Sylvia bought 10 pencils and 2 erasers for $20.40.
Edward bought 5 pencils and 3 erasers for $12.60.
• Use the information above to work out how
much Karl will pay for 6 pencils and 5 erasers.
• Describe how you got your answer.
• Is there more than one possible solution?
9H
■■ To solve a problem, follow these steps.
– Defi ne pronumerals to stand for unknown numbers.
(e.g. Let e = cost of an eraser.)
– Write an equation to describe the problem.
– Solve the equation by inspection, if possible, or systematically.
– Ensure you answer the original question, and include the correct units (e.g. dollars,
years, cm).
Key ideas
Example 16 Solving a problem using equations
When Kate adds her current age and her age next year, the result is 19. How old is Kate now?
SolutIon ExplanatIon
Let k = Kate’s current age. Defi ne a pronumeral to stand for the unknown number.
k + (k + 1) = 19 Write an equation to describe the situation. Note that
k + 1 is Kate’s age next year.
2k + 1 = 19
2k = 18
k = 9
- 1
÷ 2
- 1
÷ 2
Simplify the LHS and then solve the equation
systematically.
Kate is currently 9 years old. Answer the original question.
EXTENSION
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Chapter 470 9 Equations 1
1 For each of the following problems, choose the best pronumeral defi nition.
a Problem: Monique’s age next year is 12. How old is she now?
a Let m = Monique’s current age.
B Let m = Monique.
C Let m = 12.
D Let m = Monique’s age next year.
E Let m = this year.
b Problem: Callan has 15 boxes, which weigh a total of 300 kg. How much does each box weigh?
a Let w = 15.
B Let w = 300.
C Let w = the weight of one box.
D Let w = the number of boxes.
E Let w = the total weight.
c Problem: Jared’s family has a farm with cows and sheep. The total number of animals is 200 and
there are 71 cows. How many sheep are there?
a Let x = the size of a sheep.
B Let x = the total number of animals.
C Let x = the number of sheep.
D Let x = the number of cows.
E Let x = Jared’s age.
2 Solve the following equations by inspection or systematically.
a 5x = 30 b 7a + 2 = 16 c 2k - 3 = 15
Exercise 9H EXTENSION
WORKING
MATHEMATICALLY
U F
R PS
C
3 Launz buys a car and a trailer for a combined cost of $40 000. The trailer costs $2000.
a Defi ne a pronumeral for the car’s cost.
b Write an equation to describe the problem.
c Solve the equation systematically.
d Hence, state the cost of the car.
4 Meghan buys 12 pens for a total cost of $15.60.
a Defi ne a pronumeral for the cost of one pen.
b Write an equation to describe the problem.
c Solve the equation systematically.
d Hence, state the cost of one pen.
Example 16
WORKING
MATHEMATICALLY
U F
R PS
C
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Number and Algebra 471
5 Jonas is paid $17 per hour and gets paid a bonus of $65 each week. One particular week
he earned $643.
a Define a pronumeral for the number of hours Jonas worked.
b Write an equation to describe the problem.
c Solve the equation systematically.
d How many hours did Jonas work in that week?
6 This rectangular paddock has an area of 720 m2.
a Write an equation to describe the problem, using l for the
paddock’s length.
b Solve the equation systematically.
c How long is the paddock?
d What is the paddock’s perimeter?
24 m
 metres
WORKING
MATHEMATICALLY
U F
R PS
C
7 A number is doubled, then 3 is added and the result is doubled again. This gives a final result
of 34. Set up and solve an equation to find the original number, showing all the steps clearly.
8 The perimeter of the shape shown is 30.
Find the value of x.
9 Alexa watches some television on Monday, then twice as many hours on Tuesday, then twice
as many hours again on Wednesday. If she watches a total of 10
1
2
hours from Monday to
Wednesday, how much television did Alexa watch on Monday?
2x
x
6
12
WORKING
MATHEMATICALLY
U F
R PS
C
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Chapter 472 9 Equations 1
10 Marcus and Sara’s combined age is 30. Given that Sara is 2 years older than Marcus, write
an equation and fi nd Marcus’ age.
11 An isosceles triangle is shown below. Write an equation and solve it to fi nd x°, the unknown angle.
(Remember: The sum of angles in a triangle is 180°.)
x° x°
154°
12 Find the value of y in the triangle shown here, by fi rst writing an equation.
y°
(2y)°
13 A rectangle has base b and height h. The perimeter and area of the rectangle are equal. Write an
equation and solve it by inspection to fi nd some possible values for b and h. (Note: There are many
solutions to this equation. Try to fi nd a few.)
14 Find the values of x and y in the rectangle shown.
2x + 3
3y 6
10
WORKING
MATHEMATICALLY
U F
R PS
C
15 If photocopying costs 35 cents a page and p is the number of pages photocopied, which of the
following equations have possible solutions? Justify your answers. (Note: Fraction answers
are not possible because you must still pay 35 cents even if you photocopy only part of a page.)
a 0.35p = 4.20 b 0.35p = 2.90 c 0.35p = 2.80
16 Assume that an isosceles triangle is drawn so that each of its three angles is a whole number of
degrees. Prove that the angle a must be an even number of degrees.
a°
b° b°
WORKING
MATHEMATICALLY
U F
R PS
C
9H
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Number and Algebra 473
Enrichment: Strange triangles
17 Recall that the sum of angles in a triangle is 180°.
a David proposes the following triangle, which is not drawn to scale.
(x + 100)°
(x − 70)° 30°
i Find the value of x.
ii Explain what makes this triangle impossible.
b Helena proposes the following triangle, which is also not drawn to scale.
(60 − x)°
(70 + x)° 50°
i Explain why the information in the diagram is not enough to find x.
ii What are the possible values that x could take?
c Design a geometric puzzle, like the one in part a, for which the solution is impossible.
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Chapter 474 9 Equations 1
theme parks
There are thousands of theme parks all over the world
which offer a vast array of rides that are built to thrill.
By surfi ng the internet, you can discover the longest,
tallest, fastest and scariest rides. Although prices are
kept competitive, theme parks need to make a profi t
so that they can maintain safety standards and
continue to build new and more exciting rides.
Thrill World and Extreme Park are two theme
parks. Both charge different prices for entry and for
each ride. Their prices are:
• Thrill World: $20 entry and $5 per ride
• Extreme Park: $60 entry and $3 per ride
a Copy and complete the table below for each theme
park. The total cost for the day includes the entry
cost and cost of the rides.
number of rides ( n ) 1 2 3 4 5 6 7 8 … 20 21 22 23 24 25
thrill World total cost $T $25 …
Extreme park total cost $E $63 …
b Write an equation for:
i T, the total cost, in dollars, for n rides at Thrill World
ii E, the total cost, in dollars, for n rides at Extreme Park
c For each of these thrill seekers, use an equation to calculate how many rides they went on.
i Amanda, who spent $105 at Thrill World
ii George, who spent $117 at Extreme Park
d Refer to your completed table to determine the number of rides that will make the total cost for
the day the same at each theme park.
e A third theme park, Fun World, decides to charge no entry fee but charges $10 per ride. Find the
minimum number of rides that you could go on at Fun World before it becomes cheaper at:
i Thrill World
ii Extreme Park
f Fun World changes its pricing policy after it decides that it will be unable to compete by pricing
its rides so high, so it decides to charge an entry fee and then make all rides free.
Investigate how much Fun World should charge to attract customers while still making
profi ts that are similar to those of Thrill World and Extreme Park. Provide some mathematical
calculations to support your conclusions.
Investigation
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number and algebra 475
1 Find the unknown number in the following puzzles.
a A number is added to half of itself and the result is 39.
b A number is doubled, then tripled, then quadrupled. The result is 696.
c One-quarter of a number is subtracted from 100 and the result is 8.
d Half of a number is added to 47, and the result is the same as the original number doubled.
e A number is increased by 4, the result is doubled and then 4 is added again to give an answer
of 84.
2 Find the same values of x and y that will make both of these equations true.
x + y = 20 and x × y = 91
3 Find the same values of a, b and c, given the clues:
5(a + 2) + 3 = 38 and 2(b + 6) - 2 = 14 and 3a + 2b + c = 31
4 Find the values of a and b for each of these geometric fi gures.
a
(2b)°
(3a)°
(5a)°
b°
b
(7a + 4)°
(3(b + 2) − 1)°
32°
c
(3a)°
(2a)°
b° a°
(4a)°
d
35
(3(a − b))° (a + b))°
(2a)°
(3b)°
(
5 By solving equations, fi nd the answer to the question: What did the student expect when she
solved the puzzle?
If the solution is x = 1, then the letter is A. If the solution is x = 2, then the letter is B and so on.
3x + 2 = 5 16 = 2(x + 5) 5(x - 2) = 65 x
3
+ 1 = 7 2 =
x + 2
8
2x + 10 = 60 x
2
+ 3 = 8
2(x + 1) + x = 47 20 - x = 9 4x - 3 = 17
6 In a farmer’s paddock there are sheep and ducks. Farmer John says to his grandson, “There are
41 animals in this paddock.” Grandson James says to his grandpa, “There are 134 animal legs in
this paddock.”
How many sheep and how many ducks are in the paddock?
puzzles and challenges
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Chapter 476 9 Equations 1
Chapter summary
3x + 5 = 38
3x = 33
x = 11
Equations
Formulas or rules are types of
equations.
e.g. F = ma, A = b × h
P = 4x
To solve systematically, use equivalent
equations and opposite operations.
Main expression First step
Equations with fractions are solved
by multiplying by the denominator
when the fraction is by itself.
are obtained by performing
the same operation to
both sides
e.g.
Equivalent equations
+8 +8
3x + 5
12x
5x – 2
–5
÷12
+2
x4
x ×4
4
–5 –5
÷3 ÷3
e.g. = 7
x = 28
×4 ×4
x
7
e.g. = 4
x
7
= 2
2 +
x = 14
×7 ×7
–2 –2
3x + 9
4
e.g. = 6
×4 ×4
–9 –9
÷3 ÷3
12x = 48
x = 4
÷12 ÷12
3x + 9 = 24
3x = 15
x = 5
e.g.
5x – 2 = 33
5x = 35
x = 7
+2 +2
÷5 ÷5
e.g.
e.g. 4(x + 2) = 4x + 8
Brackets should be expanded
and like items collected.
Use the distributive law to
expand.
3x (2 + y ) = 6x + 3xy
5(x + 3) + 2x = 29
e.g.
5x + 15 + 2x = 29
7x + 15 = 29
7x = 14
x = 2
–15 –15
÷7 ÷7
e.g.
12 + 3x = 5
20 + 3x = 13
2 + 3 = 5
4 + x = 12
19 = 12 – 3q
7 + a
2 + 3 =
equations
not
equations
finding pronumeral values
to make equation true
e.g. 15 + x = 20
solution: x = 5
Solving an equation
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Number and Algebra 477
Multiple-choice questions
1 If x = 3, which one of the following equations is true?
A 4x = 21 B 2x + 4 = 12 C 9 - x = 6
D 2 = x + 1 E x - 3 = 4
2 When 11 is added to the product of 3 and x, the result is 53. This can be written as:
A 3x + 11 = 53 B 3(x + 11) = 53 C
x
3
+ 11 = 53
D
x + 11
3
= 53 E 3x - 11 = 53
3 Which of the following values of x make the equation 2(x + 4) = 3x true?
A 2 B 4 C 6
D 8 E 10
4 The equivalent equation that results from subtracting 3 from both sides of 12x - 3 = 27 is:
A 12x = 24 B 12x - 6 = 24 C 12x - 6 = 30
D 9x - 3 = 24 E 12x = 30
5 To solve 3a + 5 = 17, the first step to apply to both sides is to:
A add 5 B divide by 3 C subtract 17
D divide by 5 E subtract 5
6 The solution to 2t - 4 = 6 is:
A t = 1 B t = 3 C t = 5
D t = 7 E t = 9
7 The solution of
2
7
x
= 10 is:
A x = 35 B x = 70 C x = 20
D x = 30 E x = 5
8 The solution to the equation 10 =
3 5
2
p +
is:
A p = 5 B p = 20 C p = 15
D p = 7 E p = 1
9 The solution of 3(u + 1) = 15 is:
A u = 5 B u = 4 C u = 11
D u = 6 E u = 3
10 A formula relating A, p and t is A = 3p - t. If A = 24 and t = 6, then p equals:
A 18 B 4 C 30
D 2 E 10
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Chapter 478 9 Equations 1
Short-answer questions
1 Classify each of the following equations as true or false.
a 4 + 2 = 10 - 2 b 2(3 + 5) = 4(1 + 3) c 5w + 1 = 11, if w = 2
d 2x + 5 = 12, if x = 4 e y = 3y - 2, if y = 1 f 4 = z + 2, if z = 3
2 Write an equation for each of the following situations. You do not need to solve the equations.
a The sum of 2 and u is 22. b The product of k and 5 is 41.
c When z is tripled the result is 36. d The sum of a and b is 15.
3 Solve the following equations by inspection.
a x + 1 = 4 b x + 8 = 14 c 9 + y = 10
d y - 7 = 2 e 5a = 10 f
a
5
= 2
4 For each equation, find the result of applying the given operation to both sides and then simplify.
a 2x + 5 = 13 [- 5] b 7a + 4 = 32 [- 4]
c 12 = 3r - 3 [+ 3] d 15 = 8p - 1 [+ 1]
5 Solve each of the following equations systematically and check your solutions by substituting.
a 5x = 15 b r + 25 = 70 c 12p + 17 = 125 d 12 = 4b - 12
e 5
3
= + 2 x
f 13 = 2r + 5 g 10 = 4q + 2 h 8u + 2 = 66
6 Solve the following equations systematically.
a
3
4
6
u = b
8
3
8
p = c 3
2 1
3
= x +
d
5
2
10 30
y + = e 4
2 20
7
= y +
f
4
3
4 24
x + =
7 Expand the brackets in each of the following expressions.
a 2(3 + 2p) b 4(3x + 12) c 7(a + 5) d 9(2x + 1)
8 Solve each of these equations by expanding the brackets first. Check your solutions by substituting.
a 2(x - 3) = 10 b 27 = 3(x + 1) c 48 = 8(x - 1)
d 60 = 3y + 2(y + 5) e 7(2z + 1) + 3 = 80 f 2(5 + 3q) + 4q = 40
9 Consider the equation 4(x + 3) + 7x - 9 = 10.
a Is x = 2 a solution?
b Show that the solution to this equation is not a whole number.
10 a Does 3(2x + 2) - 6x + 4 = 15 have a solution? Justify your answer.
b State whether the following are solutions to 5(x + 3) - 3(x + 2) = 2x + 9.
i x = 2 ii x = 3
11 The formula for the area of a trapezium is A =
1
2
h(a + b), where h is the height of the trapezium, and a
and b represent the parallel sides.
a Set up and solve an equation to find the area of a trapezium with height 20 cm and parallel sides of
15 cm and 30 cm.
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Number and Algebra 479
b Find the height of a trapezium whose area is 55 cm2 and has parallel sides 6 cm and 5 cm,
respectively.
12 Consider the rule F = 3a + 2b. Find:
a F if a = 10 and b = 3
b b if F = 27 and a = 5
c a if F = 25 and b = 8
13 For each of the following problems, write an equation and solve it to find the unknown value.
a A number is added to three times itself and the result is 20. What is the number?
b The product of 5 and a number is 30. What is the number?
c Juanita’s mother is twice as old as Juanita. The sum of their ages is 60. How old is Juanita?
d A rectangle has a length of 21 cm and a perimeter of 54 cm. What is its breadth?
14 Find the value of y for each of these figures.
a
(y + 3)° (y + 2)°
y° (2y)°
b
(2y)°
(2y + 10)°
(4y + 20)°
Extended-response questions
1 Udhav’s mobile phone plan charges a 15-cents connection fee and then 2 cents per second for
every call.
a How much does a 30-second call cost?
b Write a rule for the total cost, C, in cents, for a call that lasts t seconds.
c Use your rule to find the cost of a call that lasts 80 seconds.
d If a call cost 39 cents, how long did it last? Solve an equation to find t.
e If a call cost $1.77, how long did it last?
f On a particular day, Udhav makes two calls - the second one lasting twice as long as the first,
with a total cost of $3.30. What was the total amount of time he spent on the phone?
2 Gemma is paid $x per hour from Monday to Friday, but earns an extra $2 per hour during weekends.
During a particular week, she worked 30 hours during the week and then 10 hours on the weekend.
a If x = 12, calculate the total wages Gemma was paid that week.
b Explain why her weekly wage is given by the rule W = 30x + 10(x + 2).
c Use the rule to find Gemma’s weekly wage if x = 16.
d If Gemma earns $620 in one week, find the value of x.
e If Gemma earns $860 in one week, how much did she earn from Monday to Friday?
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Chapter 10 Measurement and computation of length, 480 perimeter and area
Chapter
Measurement
and
computation
of length,
perimeter and
area
10
What you will learn
Measurement systems of the past and present FRINGE
Using and converting units of length REVISION
Perimeter of rectilinear fi gures
Pi and circumference of circles
Arc length and perimeter of sectors and composite fi gures
Units of area and area of rectangles
Area of triangles
Area of parallelograms
Area of composite fi gures
Mass and temperature REVISION
10A
10B
10C
10D
10E
10F
10G
10H
10I
10J
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Measurement everywhere
Imagine trying to describe facts about the world around
us without using any form of measurement. We use
units of length to describe distance and degrees
Celsius (°C) to describe temperature. Other units are
used for Area, Volume, Time, Capacity and Mass.
Here are some examples of facts that use different
units of measurement.
• The Eiffel Tower in France is painted with 50 tonnes
of paint every 7 years.
• The Great Wall of China is more than 6000 km long.
• The Great Pyramid of Giza was built around 2500 BC
and includes about 2 300 000 blocks of stone, each
weighing about 2500 kg.
• The world’s smallest country is Vatican City in
Rome, with an area of 0.44 km2.
• The maximum temperature during the day on Mars
is about 20°C.
• The distance between the orbits of Mars and the
Earth around the Sun is about 78 000 000 km.
• The volume of water in Sydney Harbour is about
500 gigalitres or fi ve hundred thousand million litres
or 0.5 km3.
nSW Syllabus
for the australian
Curriculum
Strand: Measurement and Geometry
Substrand: lEnGtH and aREa
Outcomes
A student calculates the perimeters of
plane shapes and the circumference of
circles.
(MA4–12MG)
A student uses formulas to calculate the
areas of quadrilaterals, and converts units
of area.
(MA4–13MG)
481
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482 Chapter 10 Measurement and computation of length, perimeter and area
1 Measure the length of these lines in millimetres.
a
b
c
2 Arrange these units from smallest to largest.
a centimetre (cm), kilometre (km), metre (m), millimetre (mm)
b gram (g), kilogram (kg), milligram (mg), tonne (t)
c hour (h), year, second (s), day, minute (min)
3 For each of the following, fi nd how many:
a millimetres are in a centimetre b centimetres are in a metre
c grams are in a kilogram d seconds are in a minute
e minutes are in an hour f millilitres are in a litre
4 Calculate the answer to each of the following.
a 2 × 1000 b 200 ÷ 100 c 56 000 ÷ 1000 d 2.5 × 1000
e 3 × 60 f 2 × 60 × 60 g 1400 ÷ 1000 h 27 ÷ 1000
5 Find the total distance around these shapes.
a
22 m
8 m
13 m
12 m
b
13 cm
7 cm c
7 cm
6 How many unit squares make up the area of these shapes?
a b c
7 Give the most appropriate unit (e.g. metres) for measuring each of the following.
a distance between two towns b your weight
c length of a school lesson d width of a large plate
pre-test
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Measurement and Geometry 483
Measurement systems of the past and present
From about 3000 bc a
unit of measure called the
cubit was used in ancient
Egypt to measure lengths.
It is known that the cubit,
which is the length of an
arm from the elbow to the
end of the middle fi nger,
was used to measure the
depth of the Nile River in
fl ood. Other unit measures
based on the human body
were also used. Some
include the digit (width of
a fi nger), palm (width of four fi ngers) and span (distance from the tip of the thumb to the tip of the little
fi nger in an outstretched hand).
Because each individual’s arm length and fi nger width is different, there was a need to develop a
standard unit of length. The Egyptians defi ned a standard royal cubit (about 524 mm), and this was
represented as a stone rod. From this cubit the following divisions were made: 28 digits in a cubit, 4 digits
in a palm and 14 digits in a span.
Many of these units of measurement were adapted and developed by the Babylonians, Greeks,
Romans, English and French over many centuries. The English imperial system, which was adapted from
the Roman and Greek systems, is commonly used in the United Kingdom and the United States today,
and was used in Australia until the 1970s. Many people today still prefer to describe lengths and other
measures using Imperial units, such as the inch and mile.
The metric system was developed in France in the 1790s and is the universally accepted system today.
The word metric comes from the Greek word metron, meaning ‘measure’. It is a decimal system where length
measures are based on the unit called the metre. The defi nition of the metre has changed over time. Originally
it was proposed to be the length of a pendulum that beats at a rate of one per second. It was later defi ned as
1/10 000 000 of the distance from the North Pole to the equator on a line on the Earth’s surface passing through
Paris. In 1960, a metre became 1 650 763.73 wave lengths of the spectrum of the krypton-86 atom in a vacuum.
In 1983, the metre was defi ned as the distance that light travels in 1/299 792 458 seconds inside a vacuum.
let’s start: Egyptian trader
Imagine you are in ancient Egypt and you are trading goods at a market. You use the Egyptian units: digit
(width of a fi nger), palm (width of four fi ngers) and span (distance from the top of the thumb to the tip of
the little fi nger in an outstretched hand).
• Use a ruler to fi nd the metric equivalent of your digit, palm and span.
10A
Egyptians used their arms and hands to measure different units.
Royal Cubits
1
1/3 1/4 1/2 1/2
1 1
FRINGE
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Chapter 10 Measurement and computation of length, 484 perimeter and area
• You purchase a wad of papyrus paper that is 1 digit thick. Which students in the class would get the
least paper if they used their own index finger width?
• You purchase a bowl of grain 1 span deep. Which student in the class gets the most grain?
• You purchase 5 cubits of cloth. Which student gets the most cloth?
■■ Ancient measurement systems that developed from about
3000 bc include the Egyptian, Babylonian, Greek and Roman
systems. The metric system is the commonly used system today.
■■ Roman system
– 1 foot = 12 inches = 16 digits = 4 palms
– 1 cubit = 6 palms
– 1 pace (double step) = 5 feet
– 1 mile = 1000 paces
■■ imperial system
– 1 foot = 12 inches (1 inch is about 2.5 cm)
– 1 yard = 3 feet (1 yard is about 91.5 cm)
– 1 rod = 16.5 feet
– 1 chain = 22 yards
– 1 furlong = 40 rods
– 1 mile = 8 furlongs = 1760 yards (1 mile is about 1.6 km)
■■ metric system
– 1 centimetre (cm) = 10 millimetres (mm)
– 1 metre (m) = 100 centimetres (cm)
– 1 kilometre (km) = 1000 metres (m)
Key ideas
Example 1 Using measurement systems
a How many feet are there in 1 mile, using the Roman measuring system?
b How many inches are there in 3 yards, using the imperial system?
Solution Explanation
a 1 mile = 1000 paces
= 5000 feet
There are 1000 paces in a Roman mile and 5 feet in a pace.
b 3 yards = 9 feet
= 108 inches
There are 3 feet in an imperial yard and 12 inches
in a foot.
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Measurement and Geometry 485
Example 2 Choosing metric lengths
Which metric unit would be the most appropriate for measuring these lengths?
a width of a large room b thickness of glass in a window
SolUtion Explanation
a metres (m) Using mm or cm would give a very large number, and
using km would give a number that is very small.
b millimetres (mm) The thickness of glass is likely to be around 5 mm.
1 Complete these number sentences.
a Roman system
i 1 = 12 inches = 16 =
palms
ii 1 = 1000 paces
b imperial system
i 1 foot = 12
ii 3 = 1 yard
iii = 1760 yards
c metric system
i 1 m = cm
ii 1 cm = mm
iii km = 1000 m
2 List the units of length (e.g. cubit), from smallest to largest, commonly used in the
Roman system.
3 List the units of length (e.g. inch), from smallest to largest, commonly used in the
imperial system.
4 List the units of length (e.g. centimetre), from smallest to largest, commonly used in the
metric system.
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5 Use the Roman system to state how many:
a feet are in 1 pace b feet are in 1 mile c palms are in 1 foot
d palms are in 1 pace e digits are in 1 foot f digits are in 1 pace
Example 1
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Exercise 10A F R I N G E
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Chapter 10 Measurement and computation of length, 486 perimeter and area
6 Use the imperial system to state how many:
a inches are in 1 foot b feet are in 1 yard c inches are in 1 yard
d yards are in 1 mile e yards are in 1 chain f rods are in 1 furlong
7 Use the metric system to state how many:
a millimetres are in 1 centimetre b centimetres are in 1 metre
c metres are in 1 kilometre d millimetres are in 1 metre
e centimetres are in 1 kilometre f millimetres are in 1 kilometre
8 Which metric unit would be the most appropriate for measuring the following?
a the distance between two towns b diameter of a small drill bit
c height of a fl ag pole d length of a garden hose
e width of a small desk f distance across a city
9 Choose which metric unit would be the most suitable for measuring the real-life length indicated
in these photos.
a b
c d
e f
Example 2
A drill bit
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10A
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Measurement and Geometry 487
10 A Roman offers you either 12 palms or 1 pace of cloth. Which option gives you the most
cloth?
11 The Roman army marches 5 Roman miles to the next post. How many paces is the journey?
12 An English cricketer runs 1 chain for each run made. How many yards will he run if he makes
20 runs?
13 Here is the length of 1 mm and 1 cm. Use these diagrams as a guide to
estimate the length of these lines.
a
b
c
d
e
14 Estimate the length of each line or curve, in centimetres.
a b
c
1 mm 1 cm
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15 Complete these tables.
a metric b imperial
mm cm m km
mm 1
1
10
cm 10 1
m 1000 1
km 1
inch feet yard mile
inch 1
1
36
feet 12 1
yard 1
mile 1760 1
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Chapter 10 Measurement and computation of length, 488 perimeter and area
c Roman
16 Why would it be more diffi cult to include the imperial units of chains and rods in the table
in Question 15b?
17 Generally speaking, why is the metric system easier to use than either the imperial or
Roman systems?
digit palm feet pace mile
digit 1
1
16
palm 4 1
feet 1
pace 1
mile 1
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Enrichment: Walking paces
18 The Roman pace involves 2 steps, and 1000 of these paces make up a Roman mile. These units
would have been used to estimate distances for the Roman armies that spread throughout much
of the world during that time.
a Estimate how many paces (i.e. double steps) you would take in 1 kilometre (1000).
b Calculate how many paces you would take to cover 1 kilometre.
c If each pace takes 1 second, fi nd how long it would take to walk from Sydney to Melbourne
(about 900 km) non-stop. Convert your answer to number of hours.
10A
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Measurement and Geometry 489
Using and converting units of length
To avoid the use of very large and very small numbers,
an appropriate unit is often chosen to measure a length or
distance. It may also be necessary to convert units of length.
For example, 150 pieces of timber, each measured
in centimetres, may need to be communicated as a total
length using metres. Another example might be that
5 millimetres is to be cut from a length of timber 1.4 metres
long because it is is too wide to fi t a door opening that is
139.5 centimetres wide.
let’s start: How good is your estimate?
In less than 3 seconds, guess the length of your desk,
in centimetres.
• Now use a ruler to fi nd the actual length in centimetres.
• Convert your answer to millimetres and metres.
• If you lined up all the class desks end to end, how many
desks would be needed to reach 1 kilometre? Explain
how you got your answer.
10B
A carpenter may need to measure lengths of wood
in metres, centimetres and millimetres.
R E V I S I O N
■■ The metre (m) is the basic metric unit of length.
– 1 km = 1000 m
– 1 m = 100 cm
– 1 cm = 10 mm
■■ Conversion
– When converting to a smaller unit, multiply by a power of 10 (i.e. 10, 100, 1000).
A smaller unit means you need more of them, so it makes sense that you multiply.
– When converting to a larger unit, divide by a power of 10 (i.e. 10, 100, 1000).
A larger unit means you need less of them, so it makes sense that you divide.
■■ When reading scales, be sure about what
units are showing on the scale. This scale
shows 36 mm.
×1000
÷1000
km m
×100
÷100
cm
×10
÷10
mm
mm
cm
1 2 3 4
Key ideas
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Chapter 10 Measurement and computation of length, 490 perimeter and area
Example 3 Converting metric units of length
Convert to the units given in brackets.
a 3 m (cm) b 25 600 cm (km)
SolUtion Explanation
a 3 m = 3 × 100 cm
= 300 cm
1 m = 100 cm
Multiply since you are converting to a smaller unit.
b 25 600 cm = 25 600 ÷ 100 000
= 0.256 km
There are 100 cm in 1 m and 1000 m in 1 km and
100 × 1000 = 100 000.
Example 4 Reading length scales
Read the scales on these rulers to measure the marked length.
a
1 cm 2 cm 3 cm
b
5 m 6 m 7 m
SolUtion Explanation
a 25 mm 2.5 cm is also accurate.
b 70 cm Each division is 1
10
of a metre, which is 10 cm.
1 Write down the missing number or word in these sentences.
a When converting from metres to centimetres, you multiply by .
b When converting from metres to kilometres, you divide by .
c When converting from centimetres to metres, you by 100.
d When converting from kilometres to metres, you by 1000.
e When converting to a smaller unit, you .
f When converting to a larger unit, you .
2 Calculate each of the following.
a 100 × 10 b 10 × 100 c 100 × 1000 d 10 × 100 × 1000
3 a When multiplying by a positive power of 10, in which direction does the decimal point
move – left or right?
b When dividing by a positive power of 10, in which direction does the decimal point move – left
or right?
Exercise 10B R E V I S I O N
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Measurement and Geometry 491
4 Convert these measurements to the units shown in brackets.
a 5 cm (mm) b 2 m (cm) c 3.5 km (m) d 26.1 m (cm)
e 40 mm (cm) f 500 cm (m) g 4200 m (km) h 472 mm (cm)
i 6.84 m (cm) j 0.02 km (m) k 9261 mm (cm) l 4230 m (km)
5 Add these lengths together and give the result in the units shown in brackets.
a 2 cm and 5 mm (cm) b 8 cm and 2 mm (mm) c 2 m and 50 cm (m)
d 7 m and 30 cm (cm) e 6 km and 200 m (m) f 25 km and 732 m (km)
6 Convert to the units shown in the brackets.
a 3 m (mm) b 6 km (cm) c 2.4 m (mm)
d 0.04 km (cm) e 47 000 cm (km) f 913 000 mm (m)
g 216 000 mm (km) h 0.5 mm (m)
7 These rulers show centimetres with millimetre divisions. Read the scale to measure the
marked length.
a
0 1 2
b
0 1 2 3 4 5
c
0 1 2
d
0 1 2 3
e
0 1 2 3 4
f
0 1 2 3 4 5
g
0 1 2 3
h
0 1 2 3 4
8 Read the scale on these diagrams. Be careful with the units shown!
a
m
0 1 2 3
b
11 12 13
km
9 Use subtraction to find the difference between the measurements, and give your answer with the units
shown in brackets.
a 9 km, 500 m (km) b 3.5 m, 40 cm (cm) c 0.2 m, 10 mm (cm)
Example 3a
Example 3b
Example 4
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Chapter 10 Measurement and computation of length, 492 perimeter and area
10 Arrange these measurements from smallest to largest.
a 38 cm, 540 mm, 0.5 m b 0.02 km, 25 m, 160 cm, 2100 mm
c 0.003 km, 20 cm, 3.1 m, 142 mm d 0.001 km, 0.1 m, 1000 cm, 10 mm
11 Joe widens a 1.2 m doorway by 50 mm. What is the new width of the doorway, in centimetres?
12 Three construction engineers individually have plans to build the world’s next tallest tower. The
Titan tower is to be 1.12 km tall, the Gigan tower is to be 109 500 cm tall and the Bigan tower
is to be 1210 m tall. Which tower will be the tallest?
13 Steel chain costs $8.20 per metre. How much does is cost to buy chain of the following lengths?
a 1 km b 80 cm c 50 mm
14 A house is 25 metres from a cliff above the sea. The cliff is eroding at a rate of 40 mm per year.
How many years will pass before the house starts to fall into the sea?
15 Mount Everest is moving with the Indo-Australian plate at a rate of about 10 cm per year. How
many years will it take to move 5 km?
16 A ream of 500 sheets of paper is 4 cm thick. How thick is 1 sheet of paper, in millimetres?
17 A snail slithers 2 mm every 5 seconds. How long will it take to slither 1 m?
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18 Copy this chart and fi ll in the missing
information.
19 Many tradespeople measure and communicate with millimetres, even for long measurements
like timber beams or pipes. Can you explain why this might be the case?
km
cm
mm
m
÷100
×10
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Enrichment: Very long and short lengths
20 When 1 metre is divided into 1 million parts, each part is
called a micrometre (μm). At the other end of the spectrum,
a light year is used to describe large distances in space.
a State how many micrometres there are in:
i 1 m ii 1 cm
iii 1 mm iv 1 km
b A virus is 0.000312 mm wide. How many micrometres
is this?
c Research the length called the light year. Explain what it
is and give examples of distances using light years, such
as to the nearest star other than the Sun.
10B
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Measurement and Geometry 493
perimeter of rectilinear fi gures
The distance around the outside of a twodimensional
shape is called the perimeter. The
word perimeter comes from the Greek words
peri, meaning ‘around’, and metron, meaning
‘measure’. We associate perimeter with the
outside of all sorts of regions and objects, like the
length of fencing surrounding a block of land or
the length of timber required to frame a picture.
Below is a list of words that are used to
describe measurements of shapes.
• dimensions
• length (l )
• width (w)
• breadth (b)
• base (b)
• height (h)
• side (s)
• perpendicular height (h)
The ‘length’ could be the longer side or the
shorter side of a rectangle.
let’s start: Is there enough information?
This diagram, which is not drawn to scale, includes only 90°
angles and only one side length is given. Discuss if there is enough
information given in the diagram to fi nd the perimeter of the shape.
What additional information, if any, is required?
10C
This fence marks the perimeter (i.e. the distance around the
outside) of a paddock.
■■ Perimeter, sometimes denoted as P, is the distance around the
outside of a two-dimensional shape.
■■ Sides with the same markings are of equal length.
■■ The unknown lengths of some sides can sometimes
be determined by considering the given lengths of
other sides.
1.6 cm
2.8 cm
4.1 cm
P = 1.6 + 1.6 + 2.8 + 4.1
= 10.1 cm
Key ideas
10 cm
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Chapter 10 Measurement and computation of length, 494 perimeter and area
Example 5 Finding the perimeter
Find the perimeter of each of these shapes.
a
3 cm
5 cm
b
6 m 5 m
3 m
2 m
SolUtion Explanation
a Perimeter = 2 × 5 + 3
= 13 cm
There are two equal lengths of 5 cm and one length
of 3 cm.
b Perimeter = 2 × 6 + 2 × 8
= 28 m
5 m
6 − 2 = 4 m
3 + 5 = 8m
6 m
3 m
2 m
1 These shapes are drawn on 1 cm grids. Give the perimeter of each.
a b
2 Use a ruler to measure the lengths of the sides of these shapes, and then fi nd the perimeter.
a b
c d
Exercise 10C
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Measurement and Geometry 495
3 Find the perimeter of these shapes. (Diagrams are not drawn to scale.)
a
7 cm
3 cm 5 cm
b 8 m
10 m
8 m
6 m
5 m
c
5 km
10 km
d 1 m
0.2 m
e
10 cm
6 cm
f
2.5 cm
4 Find the perimeter of these shapes. All corner angles are 90°.
a
10 cm
5 cm
7 cm
4 cm b
8 m
4 m
5 m
2 m
c
6 km
3 km
9 km
5 a A square has a side length of 2.1 cm. Find its perimeter.
b A rectangle has a length of 4.8 m and a width of 2.2 m. Find its perimeter.
c An equilateral triangle has all sides the same length. If each side is 15.5 mm, find
its perimeter.
Example 5a
Example 5b
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Chapter 10 Measurement and computation of length, 496 perimeter and area
6 A grazing paddock is to be fenced on all sides. It is rectangular in shape, with a length of
242 m and a breadth of 186 m. If fencing costs $25 per metre, fi nd the cost of fencing required.
7 A grass tennis court is lined with chalk.
All the measurements are shown in the
diagram and given in feet.
a Find the total number of feet of chalk
required to do all the lines of the given
tennis court.
b There are 0.305 metres in 1 foot.
Convert your answer to part a to
metres.
8 Only some side lengths are shown for these shapes. Find the perimeter.
a
15 mm
20 mm b
10 cm
4 cm
18 cm
9 Find the perimeter of each of these shapes. Give your answers in centimetres.
a
168 mm
7.1 cm
b
0.38 m
271 mm
c
430 mm
1.04 m
10 A square paddock has 100 equally-spaced posts that are 4 metres apart, including one in each corner.
What is the perimeter of the paddock?
11 The perimeter of each shape is given. Find the missing length of each.
a
4 cm
P = 11 cm
2 cm
?
b
?
P = 20 m
c
?
12 km
P = 38 km
12 A rectangle has a perimeter of 16 cm. Using only whole numbers for the length and width, how many
different rectangles can be drawn? Do not count rotations of the same rectangle.
27 feet
36 feet
39 feet
21 feet
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10C
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Measurement and Geometry 497
13 Write an equation (e.g. P = 2a + b) to describe the perimeter of each shape.
a
b
a b
a
c
b
a
d
a
e
b
a
f
a
c
b
14 Write an algebraic rule for the perimeter of each given shape.
a
c b
a b
a c
b
15 a A square has perimeter P. Write an expression for its side length.
b A rectangle has perimeter P and width a. Write an expression for its length.
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Enrichment: Picture frames
16 The amount of timber used to frame a picture depends on the outside
lengths of the overall frame. These are then cut at 45° to make the frame.
a A square painting of side length 30 cm is to be framed with
timber of width 5 cm. Find the total length of timber required for
the job.
b A rectangular photo with dimensions 50 cm by 30 cm is framed
with timber of width 7 cm. Find the total length of timber
required to complete the job.
c Kimberley uses 2 m of timber of width 5 cm to complete a square
picture frame. What is the side length of the picture?
d A square piece of embroidery has side length a cm and is framed by timber of width 4 cm.
Write an expression for the total amount of timber used.
30 cm
30 cm 5 cm
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Chapter 10 Measurement and computation of length, 498 perimeter and area
pi and circumference of circles
Since the ancient times, people have known about a special number that links a circle’s diameter to its
circumference. We know this number as pi (π). Pi is a mathematical constant that appears in formulas
relating to circles, but it is also important in many other areas of mathematics. The actual value of pi
has been studied and approximated by ancient and more modern civilisations over thousands of years.
The Egyptians knew pi was slightly more than 3 and
approximated it to be
356
81
≈ 3.16. The Babylonians used
25
8
≈ 3.125 and the ancient Indians used
339
108
≈ 3.139.
It is believed that Archimedes of Syracus (287–212 bc) was
the fi rst person to use a mathematical technique to evaluate pi.
He was able to prove that pi was greater than
223
71
and less
than
22
7
. In 480 ad, the Chinese mathematician Zu Chongzhi
showed that pi was close to
335
113
≈ 3.1415929, which is
accurate to 6 decimal places.
Before the use of calculators, the fraction
22
7
was
commonly used as a good and simple approximation to pi.
Interestingly, mathematicians have been able to prove that pi
is an irrational number, which means that there is no fraction
that can be found that is exactly equal to pi. If the exact value
of pi was written down as a decimal, the decimal places
would continue forever with no repeated pattern.
let’s start: Discovering pi
Given in the table are the diameters and
circumferences for three circles, correct to 2
decimal places. Use a calculator to work out the
value of circumference (C ) ÷ diameter (d ) and put
your results in the third column. Add your own
circle measurements by measuring the diameter
and circumference of circular objects, such as a
can or a wheel.
• What do you notice about the numbers C ÷ d in the third column?
• Why might the numbers in the third column vary slightly from one set of measurements to another?
• What rule can you write down which links C with d ?
10D
Sculpture of Archimedes at the Louvre, Paris
Diameter
d (mm)
Circumference
C (mm) C ÷ d
4.46
11.88
40.99
Add your own
14.01
37.32
128.76
Add your own
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Measurement and Geometry 499
■■ Features of a circle
• Diameter (d) is the distance across the centre of a circle.
• Radius (r) is the distance from the centre to the circle. Note: d = 2r.
• Chord: A line interval connecting two points on a circle.
• Tangent: A line that touches the circle at a point.
– A tangent to a circle is at right angles to the radius.
• Sector: A portion of a circle enclosed by two radii and a portion
of a circle (arc).
• Segment: An area of a circle ‘cut off’ by a chord.
■■ Circumference (C) is the distance around a circle.
• C = 2πr or C = πd
• Pi (π) is a constant numerical value and is an irrational number,
meaning that it cannot be expressed as a fraction.
– As a decimal, the digits have no pattern and continue forever.
• The ratio of the circumference to the diameter of any circle is
equal to pi (π); i.e. π = C
d
.
• π = 3.14159 (correct to 5 decimal places)
– Common approximations include 3, 3.14 and
22
7
.
– A more precise estimate for pi can be found on most calculators or on the internet.
circumumu fefef rence
diameter
radius
tangent
sector
segment
chord
Key ideas
Example 6 Finding the circumference with a calculator
Find the circumference of these circles, correct to 2 decimal places. Use a calculator for the
value of pi.
a
3.5 m
b
4 cm
SolUtion Explanation
a C = 2πr
= 2 × π × 3.5
= 7π
= 21.99 m
Since r is given, you can use C = 2πr.
Alternatively, use C = π d with d = 7.
b C = πd
= π × 4
= 4π
= 12.57 cm
Substitute d = 4 into the rule C = πd or use C = 2πr
with r = 2.
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Chapter 10 Measurement and computation of length, 500 perimeter and area
Example 7 Finding the circumference without a calculator
Calculate the circumference of these circles using the given approximation of π.
a
π = 3
10 m
b
 = 22
7
14 cm
SolUtion Explanation
a C = πd
= 3 × 10
= 30 m
Use π = 3 and multiply mentally. Move the decimal
point one place to the right.
Alternatively, use C = 2πr with r = 5.
b C = 2πr
= 2 ×
22
7
× 14
= 2 × 22 × 2
= 88 cm
Use π =
22
7
and cancel the 14 with the 7 before
calculating the fi nal answer.
1 Evaluate the following using a calculator and round to 2 decimal places.
a π × 5 b π × 13 c 2 × π × 3 d 2 × π × 37
2 Write down the value of π, correct to:
a 1 decimal place b 2 decimal places
c 3 decimal places
3 Name the features of the circles shown.
4 A circle has circumference (C ) 81.7 m and
diameter (d ) 26.0 m, correct to 1 decimal place.
Calculate C ÷ d. What do you notice?
a
b
c
Exercise 10D
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5 Answer true or false to the following questions. C
a The distance from the centre of a circle to its circumference is called the diameter.
b π = C
d
c C = πr
d Some reasonable approximations for pi include 3, 3.14 and
22
7
.
e A tangent to a circle will touch the circle only once.
f A tangent is at 100° to the radius.
g Pi can be expressed exactly as a fraction.
d
e
g
f
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Measurement and Geometry 501
6 Find the circumference of these circles, correct to 2 decimal places. Use a calculator for
the value of pi.
a
2 mm
b
18 m
c
39 cm
d
4 m
e
7 km
f
5 cm
7 Calculate the circumference of these circles using π = 3.
a
100 cm
b
20 m
c
3 km
8 Calculate the circumference of these circles using π =
22
7
.
a
7 mm
b
21 cm
c
70 m
Example 6
Example 7a
Example 7b
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9 A water tank has a diameter of 3.5 m. Find
its circumference, correct to 1 decimal
place.
10 An athlete trains on a circular track of
radius 40 m and jogs 10 laps each day,
5 days a week. How far does he jog each
week? Round the answer to the nearest
whole number of metres.
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Chapter 10 Measurement and computation of length, 502 perimeter and area
11 Here are some students’ approximate circle measurements.
Which students have incorrect measurements?
12 Which of these circles has the largest circumference?
a a circle with radius 5 cm
B a circle with diameter 11 cm
C a circle with diameter 9.69 cm
D a circle with radius 5.49 cm
13 What is the difference in the circumference of a circle with radius 2 cm and a circle with
diameter 3 cm? Use 3 as an approximation to pi.
14 Use a mental strategy to calculate the following.
a Using π = 3, fi nd the diameter of a circle if its circumference is:
i 6 m ii 15 cm iii 45 km
b Using π = 3, fi nd the radius of a circle if its circumference is:
i 12 mm ii 18 m iii 3 cm
r C
Mick 4 cm 25.1 cm
Svenya 3.5 m 44 m
Andre 1.1 m 13.8 m
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15 Explain why the rule C = 2πr is equivalent to (i.e. the same as) C = πd.
16 It is more precise in mathematics to give ‘exact’ values for circle calculations in terms of π.
For example, C = 2 × π × 3 gives C = 6π. This gives the fi nal exact answer and is not written as a
rounded decimal. Find the exact answers for Question 6 in terms of π.
17 We know that C = 2πr or C = πd.
a Rearrange these rules to write a rule for:
i r in terms of C ii d in terms of C
b Use the rules you found in part a to fi nd the following, correct to 2 decimal places.
i the radius of a circle with circumference 14 m
ii the diameter of a circle with circumference 20 cm
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Enrichment: Memorising pi
18 The box shows π, correct to 100 decimal places. The Guinness World record for the most number of
digits of π recited from memory is held by Lu Chao, a Chinese student. He recited 67 890 digits nonstop
over a 24-hour period.
3.1415926535 8979323846 2643383279 5028841971 6939937510
5820974944 5923078164 0628620899 8628034825 3421170679
Challenge your friends to see who can remember
the most number of digits in the decimal
representation of π.
Number of digits memorised Report
10+ A good show
20+ Great effort
35+ Superb
50+ Amazing memory
10D
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Measurement and Geometry 503
arc length and perimeter of sectors and
composite fi gures
Whenever a portion of a circle’s circumference is used in
a diagram or construction, an arc is formed. To determine
the arc’s length, the particular fraction of the circle is
calculated by considering the angle at the centre of the
circle that defi nes the arc.
let’s start
Complete this table to develop the rule for fi nding an arc
length (l ).
10E
The penalty arc on a soccer fi eld is drawn using the
penalty spot as the centre of a circle with radius
9.15 metres.
angle Fraction of circle arc length Diagram
180° 180
360
=
1
2
l =
1
2
× πd 180°
90° 90
360
= ___ l = ___ × ___ 90°
45°
30°
150°

■■ A circular arc is a portion of the circumference of a circle.
– The arc (l ) is subtended by an angle, , at the centre of the circle.
■■ Arc length formula
– fraction =

360
– l =

360
× 2πr or l =

360
× πd
■■ Perimeter of a sector
– P =

360
× 2πr + 2r


r
r
360
r r

 × 2r
Key ideas
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Chapter 10 Measurement and computation of length, 504 perimeter and area
Example 8 Finding an arc length
Find the length of these arcs for the given angles, correct to 2 decimal places.
a
50°
10 cm
b
230°
2 mm
Solution Explanation
a l = × ×
=
50
360
2 10
8 73

. cm
The fraction of the full circumference is
50
360
and the
full circumference is 2πr, where r = 10.
b l = × ×
=
230
360
2 2
8 03

. mm
The fraction of the full circumference is 230
360
and the
full circumference is 2πr.
Example 9 Finding the perimeter of a sector
Find the perimeter of these sectors, correct to 1 decimal place.
a
3 m
b
5 km
■■ Common circle portions
quadrant semicircle
r
r
P = × r + r 1
4
2 2
P = × r + r 1
2
2 2
■■ A composite figure is made up of more than one basic shape.
Key ideas
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Measurement and Geometry 505
SolUtion Explanation
a P = × × + ×
=
1
4
2 3 2 3
10 7

. m
The arc length is one-quarter of the circumference and
included are two radii, each of 3 m.
b P = × × +
=
1
2
5 5
12 9

. km
A semicircle’s perimeter consists of half the
circumference of a circle plus a full diameter.
Example 10 Finding the perimeter of a composite shape
Find the perimeter of the following composite shape, correct to 1 decimal place.
10 cm
5 cm
SolUtion Explanation
P = + + + + × ×
=
10 5 10 5
1
4
2 5
37 9

. cm
There are two straight sides of 10 cm and 5 cm shown
in the diagram. The radius of the circle is 5 cm, so the
straight edge at the base of the diagram is 15 cm long.
The arc is a quarter circle.
1 What fraction of a circle is shown in these diagrams? Name each shape.
a b
2 What fraction of a circle is shown in these sectors? Simplify your fraction.
a b c
60°
d
120°
e f
225°
Exercise 10E
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Chapter 10 Measurement and computation of length, 506 perimeter and area
3 Name the two basic shapes that make up these composite fi gures.
a b c
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4 Find the length of the following arcs for the given angles, correct to 2 decimal places.
a
60° 8 cm
b
80°
4 m
c
100°
7 mm
d
225°
0.5 m
e
300°
26 cm
f
330°
0.2 km
5 Find the perimeter of these sectors, correct to 1 decimal place.
a 4 cm b
10 m
c
20 mm
d
14 km
e
60° 4.3 m
f
115°
3.5 cm
Example 8
Example 9
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Measurement and Geometry 507
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6 Find the perimeter of these composite shapes, correct to 1 decimal place. C
a
3 m
4 m
b 2 cm c
30 cm
12 cm
d
10 km
6 km
e
10 mm
f
10 cm
8 cm
7 A window consists of a rectangular part of length 2 m and breadth 1 m, with a semicircular
top having a diameter of 1 m. Find its perimeter, correct to the nearest cm.
8 For these sectors, fi nd only the length of the arc, correct to 2 decimal places.
a
140°
7 m
b
330°
100 m
c
85°
1.3 m
9 Calculate the perimeter of these shapes, correct to 2 decimal places.
a
4 cm
b
9 m
c
10 m
5 m
Example 10
10 Give reasons why the circumference of this composite
shape can be found by simply using the rule P = 2πr + 4r. r
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Chapter 10 Measurement and computation of length, 508 perimeter and area
11 Explain why the perimeter of this shape is given by P = 2πr.
r
12 Find the radius of these sectors for the given arc lengths, correct to 1 decimal place.
a
30° r
4 m b
100° r
60 m c
45 km
r
Enrichment: Exact values and perimeters
13 The working to fi nd the exact perimeter of this composite shape is given by:
P = 2 × 8 + 4 + 1
2
π × 4
= 20 + 8π cm
8 cm
4 cm
Find the exact perimeter of the following composite shapes.
a
6 cm
b
5 cm
2 cm
14 Find the exact answers for Question 9 in terms of π.
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10E
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Measurement and Geometry 509
Units of area and area of rectangles
Area is measured in square units. It is often
referred to as the amount of space contained
inside a fl at (i.e. plane) shape; however,
curved three-dimensional (3D) solids also have
surface areas. The amount of paint needed to
paint a house and the amount of chemical
needed to spray a paddock are examples of
calculations of area.
let’s start: The 12 cm2 rectangle
A rectangle has an area of 12 square centimetres (12 cm2).
• Draw examples of rectangles that have this area, showing the length and width measurements.
• How many different rectangles with whole number dimensions are possible?
• How many different rectangles are possible if there is no restriction on the type of numbers allowed to
be used for length and width?
10F
The greater the area of the paddock, the more chemicals the
plane needs to carry.
■■ The metric units of area include:
– 1 square millimetre (1 mm2)
– 1 square centimetre (1 cm2)
There are 100 mm2 in 1 cm2.
– 1 square metre (1 m2)
There are 10 000 cm2 in 1 m2.
– 1 hectare (1 ha)
There are 10 000 m2 in 1 ha.
– 1 square kilometre (1 km2)
There are 1 000 000 m2 in 1 km2.
1 mm
1 mm
1 cm
1 cm
1 m
1 m
(Not drawn to scale.)
100 m
100 m
(Not drawn to scale.)
1 km
1 km
(Not drawn to scale.)
Key ideas
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Chapter 10 Measurement and computation of length, 510 perimeter and area
Example 11 Counting areas
Count the number of squares to find the area of the shape drawn
on this centimetre grid.
Solution Explanation
6 cm2 There are 5 full squares and half of
2 squares in the triangle, giving
1 more.
1
2
of 2 = 1
Example 12 Areas of rectangles and squares
Find the area of this rectangle and square.
a
4 mm
10 mm
b
2.5 cm
■■ The area of a rectangle is given by the number of
rows multiplied by the number of columns.
Written as a formula, this looks like A = l × b,
where l is the length and b is the breadth (or width).
This also works for numbers that are not whole.
■■ The area of a square is given by A = s × s = s 2.
b

A =  × b
s
A = s2
Key ideas
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Measurement and Geometry 511
SolUtion Explanation
a Area = l × b
= 10 × 4
= 40 mm2
The area of a rectangle is the product of the length and
breadth.
b Area = s 2
= 2.52
= 6.25 cm2
The breadth is the same as the length, so A = s × s = s 2.
(2.5)2 = 2.5 × 2.5
1 For this rectangle drawn on a 1 cm grid, fi nd each of the
following.
a the number of single 1 cm squares
b the length and the breadth
c length × breadth
2 For this square drawn on a centimetre grid, fi nd the following.
a the number of single 1 cm squares
b the length and the breadth
c length × breadth
3 Count the number of squares to fi nd the area of these shapes.
a b c
4 Which unit of area (mm2, cm2, m2, ha or km2) would you choose to measure these areas? Note that 1
km2 is much larger than 1 ha.
a area of an A4 piece of paper b area of a wall of a house
c area of a small farm d area of a large desert
e area of a large football oval f area of a nail head
Exercise 10F
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Chapter 10 Measurement and computation of length, 512 perimeter and area
5 Count the number of squares to fi nd the area of these shapes on centimetre grids.
a b
c d
e f
6 Find the area of these rectangles and squares. Diagrams are not drawn to scale.
a
10 cm
20 cm
b
11 mm
2 mm
c
2 cm
3.5 cm
Example 11
Example 12
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Measurement and Geometry 513
d
5 m
e
1.2 mm
f
2.5 mm
g
0.8 m
1.7 m
h
0.9 cm
i
17.6 km
10.2 km
7 Find the side length of a square with each of these areas. Use trial and error if you are unsure.
a 4 cm2 b 25 m2 c 144 km2
8 There are 10 000 m2 in one hectare (ha). Convert these measurements to hectares.
a 20 000 m2 b 100 000 m2 c 5000 m2
9 A rectangular soccer field is to be laid with new grass. The field is 100 m long and 50 m wide. Find
the area of grass to be laid.
10 Glass is to be cut for a square window of side length 50 cm. Find the area of glass required for the
window.
11 Two hundred square tiles, each measuring 10 cm by 10 cm, are used to tile an open floor area. Find the
area of flooring that is tiled.
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12 a A square has a perimeter of 20 cm. Find its area.
b A square has an area of 9 cm2. Find its perimeter.
c A square’s area and perimeter are the same number. How many units is the side length?
13 The carpet chosen for a room costs $70 per square metre. The room is rectangular and is 6 m long by
5 m wide. What is the cost of carpeting the room?
14 Troy wishes to paint a garden wall that is 11 m long and 3 m high. Two coats of paint are needed. The
paint suitable to do the job can be purchased only in whole numbers of litres and covers an area of
15 m2 per litre. How many litres of paint will Troy need to purchase?
15 A rectangular area of land measures 200 m by 400 m. Find its area in hectares.
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16 a Find the missing length for each of these rectangles.
i
5 cm
?
A = 50 cm2
ii
2.5 mm
?
A = 22.5 mm2
b Explain the method that you used for finding the missing lengths of the rectangles above.
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Chapter 10 Measurement and computation of length, 514 perimeter and area
17 Explain why the area shaded here is exactly 2 cm2.
18 A square has perimeter P cm.
a If P = 44 cm, fi nd the area of the square.
b If P is unknown, write an expression for the area of the
square, using P.
19 A square has all its side lengths doubled. How does this change the area? Investigate and justify your
answer.
1 cm
4 cm
P = 44 cm
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Enrichment: area conversions
20 a Use this diagram or similar to help answer the following.
i How many mm2 in 1 cm2?
ii How many cm2 in 1 m2?
iii How many m2 in 1 km2?
b Complete the diagram below.
×______
÷______
km2 m2
×_____
÷_____
cm2
×_____
÷_____
mm2
c Convert these units to the units shown in brackets.
i 2 cm2 (mm2) ii 10 m2 (cm2) iii 3.5 km2 (m2)
iv 300 mm2 (cm2) v 21 600 cm2 (m2) vi 4 200 000 m2 (km2)
vii 0.005 m2 (mm2) viii 1 km2 (ha) ix 40 000 000 cm2 (ha)
10 mm
10 mm
A = 1 cm2
Guess the area of the paddock in the foreground
in square millimetres.
10F
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Measurement and Geometry 515
area of triangles
Put simply, a triangle can be considered to be half a rectangle,
which is why the formula for the area of a triangle looks very
much like the formula for the area of a rectangle but with the
added factor of 1
2
. One of the sides of a triangle is called the
base (b), and the height (h) is the distance between the base
and the opposite vertex. This is illustrated using a line that is
perpendicular (i.e. at 90°) to the base.
Any shape with all straight sides (i.e. polygons) can be
divided up into a combination of rectangles and triangles.
This can help to fi nd areas of such shapes.
let’s start: Half a rectangle
Look at these triangles. For each one, discuss why the area could be considered as half a rectangle.
Give reasons for each case.
1 2 3
10G
height
base
Any area of fl oor with straight sides can be fi lled
with rectangular and triangular tiles.
■■ One side of a triangle is called the base, b.
■■ The perpendicular distance from the base to the opposite vertex is called the height, h. In
the third illustration below, the base needs to be extended to help show the height.
h
b
h
b
h
b
■■ The area of a triangle is given by the formula:
A =
1
2
bh =
1
2
× base × height
Note that
1
2
bh is equivalent to bh
2
.
Key ideas
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Chapter 10 Measurement and computation of length, 516 perimeter and area
Example 13 Finding areas of triangles
Find the area of each given triangle.
a
10 cm
9 cm
b
9 m
6 m
SolUtion Explanation
a Area = 1
2
bh
= 1
2
× 10 × 9
= 45 cm2
Use the formula and substitute the values for base length
and height.
b Area =
1
2
bh
= 1
2
× 6 × 9
= 27 m2
The length measure of 9 m is marked at 90° to the side
marked 6 m. So 6 m is the length of the base and 9 m is the
perpendicular height.
Any side can be used as the base.
1 For each of these triangles, what length would be used as the base?
a
8 cm
20 cm
b
2.1 m
7 m
c
3 m
2 m d
6.3 cm
5 cm
2 For each of these triangles, what length would be used as the height?
a
6 m
11 m b
15 mm
11 mm
Exercise 10G
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Measurement and Geometry 517
c
4.1 m 1.9 m
d
3.2 mm
4.7 mm
3 Find the value of A in A =
1
2
bh if:
a b = 5 and h = 4 b b = 7 and h = 16 c b = 2.5 and h = 10
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4 Find the area of each triangle given.
a
4 m
8 m
b
5 cm
12 cm
c
16 m
20 m
d
2.4 m
1 m
e
3 mm
2 mm
f
7 km
7 km g 1.3 cm
2 cm
h
10 m 4 m
i
5 m
1.7 m
Example 13
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Chapter 10 Measurement and computation of length, 518 perimeter and area
5 Find the area of these triangles, which have been drawn on 1 cm grids. Give your answer
in cm2.
a b
c d
6 A rectangular block of land measuring 40 m long by 24 m wide is cut in half along a diagonal. Find
the area of each triangular block of land.
7 A square pyramid has a base length of 120 m and
a triangular face of height 80 m. Find the area of
one triangular face of the pyramid.
80 m
120 m
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10G
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Measurement and Geometry 519
8 Each face of a 4-sided die is triangular, with a base of 2 cm and a height of 1.7 cm. Find the
total area of all 4 faces of the die.
9 A farmer uses fencing to divide a triangular piece of land
into two smaller triangles, as shown. What is the difference
in the two areas?
10 A yacht must have two of its sails
replaced as they have been damaged by a
recent storm. One sail has a base length of
2.5 m and a height of 8 m, and the bigger
sail has a base length of 4 m and a height of
16 m. If the cost of sail material is $150 per
square metre, find the total cost to replace
the yacht’s damaged sails.
11 a The area of a triangle is 10 cm2 and its
base length is 4 cm. Find its height.
b The area of a triangle is 44 mm2 and its
height is 20 mm. Find its base length.
18 m 26 m
40 m
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12 The midpoint, M, of the base of a triangle joins the opposite
vertex. Is the triangle area split in half exactly? Give reasons for
your answer.
M
h
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Chapter 10 Measurement and computation of length, 520 perimeter and area
13 If the vertex, C, for this triangle moves parallel to the base AB,
will the area of the triangle change? Justify your answer.
14 The area of a triangle can be found using the formula A =
1
2
bh. Write down the formula to fi nd the
base, b, if you are given the area, A, and height, h.
A
C
B
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Enrichment: Estimating areas with curves
15 This diagram shows a shaded region that is
1
2
of 3 cm2 = 1.5 cm2.
Using triangles like the one shown here, and by counting whole squares also, estimate the areas of
these shapes below.
a b
c d
10G
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Measurement and Geometry 521
area of parallelograms
Recall that a parallelogram is a quadrilateral with two pairs of
parallel sides. Opposite sides are of the same length and opposite
angles are equal.
Like a triangle, the area of a parallelogram is found by using
the length of one side (called the base) and the height (which is perpendicular to the base.)
let’s start: Developing the rule
Start this activity by drawing a large parallelogram on a loose piece of paper.
Ensure the opposite sides are parallel and then use scissors to cut it out. Label
one side as the base and label the height, as shown in the diagram.
• Cut along the dotted line.
• Now shift the triangle to the other end of the parallelogram to make a
rectangle.
• Now explain how to fi nd the area of a parallelogram.
10H
b
b
a
a
height
base
Example 14 Finding the area of a parallelogram
Find the area of these parallelograms.
a
12 m
5 m
b
2 cm
3 cm
■■ The area of a parallelogram is given by:
■ ■ ■ ■ ■ A = bh
■ where b is the base (one of the sides) and h is the height (perpendicular to the base).
■ ■ ■ ■ ■ ■ ■
h
b
■ ■ ■ ■ ■ ■ ■ h
b
Key ideas
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Chapter 10 Measurement and computation of length, 522 perimeter and area
SolUtion Explanation
a A = bh
= 12 × 5
= 60 m2
Choose the given side as the base (12 m) and note the
perpendicular height is 5 m.
b A = bh
= 2 × 3
= 6 cm2
Use the given side as the base (2 cm), noting that the height
is 3 cm.
1 Copy and complete the following, using the given values of b and h.
a b = 5, h = 7 b b = 20, h = 3 c b = 8, h = 2.5
A = bh A = ___ A = ___
= ___ × ___ = 20 × ___ = 8 × ___
= 35 = ___ = ___
2 For each of these parallelograms, state the side length of the base and the height that might be used to
fi nd the area.
a
6 cm
2 cm
b
4 m 10 m
c
7 m
5 m
d
5.8 cm
6.1 cm
e
2 cm
1.5 cm
5 cm f 1.3 m
0.9 m
1.8 m
Exercise 10H
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3 Find the area of these parallelograms.
a
10 m
4 m
b
7 m
4 m
c
12 km
3 km
Example 14
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Measurement and Geometry 523
d 7 m
2.5 m
e
15 m
2 m f
3 cm
2.1 cm
g
1.8 cm
1 cm
2.1 cm
h
2 cm
3 cm
7 cm
i
11 mm
16 mm 20 mm
4 These parallelograms are on 1 cm grids (not to scale). Find their area.
a b
c d
5 The floor of an office space is in the shape of a parallelogram. The longest sides are 9 m
and the distance between them is 6 m. Find the area of the office floor.
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6 Find the height of a parallelogram when its:
a area = 10 m2 and base = 5 m
b area = 28 cm2 and base = 4 cm
c area = 2.5 mm2 and base = 5 mm
7 Find the base of a parallelogram when its:
a area = 40 cm2 and height = 4 cm
b area = 150 m2 and height = 30 m
c area = 2.4 km2 and height = 1.2 km
8 A large wall in the shape of a parallelogram is to be painted with a special red paint, which costs
$20 per litre. Each litre of paint covers 5 m2. The wall has a base length of 30 m and a height of
10 m. Find the cost of painting the wall.
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Chapter 10 Measurement and computation of length, 524 perimeter and area
9 A proposed rectangular fl ag for a new country is yellow with a
red stripe in the shape of a parallelogram, as shown. Find:
a the area of the red stripe
b the yellow area 30 cm 70 cm
60 cm
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10 Explain why this parallelogram’s area will be less than the given rectangle’s area.
10 cm
5 cm
10 cm
5 cm
11 A parallelogram includes a green triangular area, as shown. What
fraction of the total area is the green area? Give reasons for your
answer.
12 The area of a parallelogram can be thought of as twice the area of a
triangle. Use this idea to complete this proof of the rule for the area of a parallelogram.
Area = twice triangle area
= 2 × –––––––––––
= –––––––––––
b
h
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Enrichment: Glass façade
13 The Puerta de Europa (Gate of Europe) towers are twin offi ce buildings in Madrid, Spain. They
look like normal rectangular glass-covered skyscrapers but they lean towards each other at an
angle of 15° to the vertical. Two sides are parallelograms and two sides are rectangles. Each
tower has a vertical height of 120 m, a slant height of 130 m and a square base of side 50 m.
All four sides are covered with glass. If the glass costs $180 per square metre, fi nd the cost of
covering one of the towers with glass. (Assume the glass covers the entire surface, ignoring the
beams.)
10H
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Measurement and Geometry 525
area of composite fi gures
The areas of more complicated shapes can be found by
dividing them up into more simple shapes, such as the
rectangle and triangle. We can see this in an aerial view
of any Australian city. Such a view will show that many
city streets are not parallel or at right angles to each other.
As a result, this causes city blocks to form interesting
shapes, many of which are composite fi gures made up of
rectangles and triangles.
let’s start: Dividing land to fi nd its area
Working out the area of this piece of land could be done by
dividing it into three rectangles, as shown.
• Can you work out the area using this method?
• What is another way of dividing the land to fi nd its area?
Can you use triangles?
• What is the easiest method to fi nd the area? Is there a way
that uses subtraction instead of addition?
10I
Streets, parks and buildings form complex shapes
that can be made up of triangles and rectangles.
20 m
30 m
20 m
10 m
30 m
Example 15 Finding the area of composite shapes
Find the area of each of these composite shapes.
a
9 m
12 m
5 m
b
6 cm
4 cm
■■ Composite fi gures are made up of more than one simple shape.
■■ The area of composite fi gures can be found by adding or subtracting the areas of simple shapes.
A square plus a triangle
A rectangle subtract a triangle
Key ideas
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Chapter 10 Measurement and computation of length, 526 perimeter and area
SolUtion Explanation
a A = l × b +
1
2
bh
= 9 × 5 +
1
2
× 3 × 5
= 45 + 7.5
= 52.5 m2
Divide the shape into a rectangle and triangle and fi nd the
missing lengths.
9 m
9 m 3 m
5 m 5 m
12 − 9 = 3m
b A = s 2 –
1
2
bh
= 62 –
1
2
× 6 × 4
= 36 – 12
= 24 cm2
Subtract the triangle (
1
2
× 6 × 4) at the top of the shape
from the larger square (6 × 6).
6 cm
6 cm
6 cm
4 cm
Exercise 10I
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1 Copy these diagrams and draw a dotted line where you might divide these shapes into two C
more simple shapes.
a b c
2 To fi nd the area of each of the following shapes, decide if the easiest method would involve the
addition of two shapes or the subtraction of one shape from another.
a b c
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Measurement and Geometry 527
3 Copy and complete the solutions for the areas of these shapes.
a 1 cm
1 cm
2 cm
3 cm
b
4 m
4 m
13 m
5 m
A = s 2 + ____ A = lb – ____
= 12 + 3 × ____ = ____ × 8 –
1
2
× ____ × ____
= ____ + ____ = ____ – ____
= ____ cm2 = ____ m2
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4 Find the area of these composite figures by adding together the area of simpler shapes.
a
4 m
6 m
3 m
b
10 mm
10 mm
40 mm
c
3 m
8 m
2 m d
3 m
5 m
9 m
e
9 cm
21 cm
17 cm
f
6 km
10 km
3 km
5 Use subtraction to find the area of these composite shapes.
a
6 m
7 m
2 m
3 m b
10 cm
3 cm
3 cm
Example 15a
Example 15b
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Chapter 10 Measurement and computation of length, 528 perimeter and area
c
2 m
5 m
6 m
d
9 m
9 m
8 m
4 m
e
20 cm
14 cm
f
1 m
4 m
2.5 m
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6 Find the areas of these composite fi gures.
a
4 cm
8 cm
5 cm
b
2 m
c
20 cm
4 cm 15 cm
7 By fi nding the missing lengths fi rst, calculate the area of these composite shapes.
a
12 cm
b
4 m
3.5 m
2 m
c
2 m
5 m
d
1 cm
2 cm
3.5 cm
6 cm
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10I
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Measurement and Geometry 529
8 A wall has three square holes cut into it
to allow for windows, as shown. Find the
remaining area of the wall.
9 A factory floor, with dimensions shown opposite, is to be
covered with linoleum. Including underlay and installation, the
linoleum will cost $25 per square metre. The budget for the job
is $3000. Is there enough money in the budget to cover the cost?
1.2 m
2.4 m
6 m
10 m
12 m
5 m
4 m
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10 Explain why using subtraction is sometimes quicker than using addition to
find the area of a composite shape. Refer to the diagram as an example.
11 The 4-sided shape called the trapezium has one pair of parallel sides.
a For the trapezium shown opposite, is it possible to find the
base length of each triangle on the sides? Justify your answer.
b Can you come up with a method for finding the area of a
trapezium using the rectangle and triangles shown in the
diagram? Use diagrams to explain your method.
6 m
10 m
5 m
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Enrichment: Adding to infinity
12 The square given opposite, which has an area of 1 unit, is divided to
show the areas of 1
2
, 1
4
, 1
8
, ...
a Similar to the one shown opposite, draw your own square,
showing as many fractions as you can. Try to follow the spiral
pattern shown. Note: The bigger the square you start with, the
more squares you will be able to show.
b i Write the next 10 numbers in this number pattern.
1
2
, 1
4
, 1
8
, ...
ii Will the pattern ever stop?
c What is the total area of the starting square?
d What do your answers to parts b ii and c tell you about the answer to the sum below?
1
2
+
1
4
+
1
8
+
1
16
+ ... (continues forever)
12
14
18
1
16
1
32
1
1
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Chapter 10 Measurement and computation of length, 530 perimeter and area
Mass and temperature
The scales for both mass and temperature are related to the
properties of water. In France in 1795, the gram was defi ned
as being the weight of 1 cm3 of water at 0°C. Later it was
redefi ned to be the weight at 4°C, as this is considered to be
the temperature at which water is the most dense. So, 1 litre
of water is very close to 1 kilogram, which is the basic unit
for mass. Other units for mass include the tonne, gram and
milligram. A small car has a mass of about 1 tonne and a
20-cent coin has a mass of about 11 grams.
Temperature tells us how hot or cold something is. Anders
Celsius (1701–1744), a Swedish scientist, worked to defi ne
a scale for temperature. After his death, temperature was
offi cially defi ned by:
• 0°C (0 degrees Celsius) – the freezing point of water.
• 100°C (100 degrees Celsius) – the boiling point of water
(at one standard degree of pressure).
This is still the common understanding of degrees
Celsius. As mentioned in Chapter 9, Fahrenheit is another
scale used for temperature.
let’s start: Choose a unit of mass
Name fi ve objects of which their mass would commonly be
measured in:
• tonnes
• kilograms
• grams
• milligrams
Is it possible for the temperature to drop below 0°C? How is this
measured and can you give examples of places or situations where
this might be the case?
10J
A small car has a mass of about 1 tonne.
REVISION
Ice melts at 0°C.
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Measurement and Geometry 531
Example 16 Converting units of mass
Convert to the units shown in brackets.
a 2.47 kg (g) b 170 000 kg (t)
SolUtion Explanation
a 2.47 kg = 2.47 × 1000 g
= 2470 g
1 kg = 1000 g
Multiply because you are changing to a smaller unit.
b 170 000 kg = 170 000 ÷ 1000 t
= 170 t
1 t = 1000 kg
Divide because you are changing to a larger unit.
1 Circle or write down which mass measurements are the same.
a 1 kg, 100 g, 1000 g, 10 t b 1000 mg, 10 kg, 1 g, 1000 t
2 From options a to F, choose the mass that best matches the given object.
a human hair a 300 g
b 10-cent coin B 40 kg
c bottle C 100 mg
d large book D 1.5 kg
e large bag of sand E 13 t
f truck F 5 g
3 From options a to D, choose the temperature that best matches the description.
a temperature of coffee a 15°C
b temperature of tap water B 50°C
c temperature of oven C −20°C
d temperature in Antarctica D 250°C
Exercise 10J R E V I S I O N
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■■ The basic unit for mass is the kilogram (kg). 1 litre of water has a mass that is very close to
1 kilogram.
■■ Metric units for mass include:
– 1 gram (g) = 1000 milligrams (mg)
– 1 kilogram (kg) = 1000 grams (g)
– 1 tonne (t) = 1000 kilograms (kg)
■■ The common unit for temperature is degrees Celsius (°C).
– 0°C is the freezing point of water.
– 100°C is the boiling point of water.
×1000
÷1000
t kg
×1000
÷1000
g
×1000
÷1000
mg
Key ideas
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Chapter 10 Measurement and computation of length, 532 perimeter and area
4 Convert to the units shown in brackets.
a 2 t (kg) b 70 kg (g) c 2.4 g (mg) d 2300 mg (g)
e 4620 mg (g) f 21 600 kg (t) g 0.47 t (kg) h 312 g (kg)
i 27 mg (g) j 3
4
t (kg) k 1
8
kg (g) l 10.5 g (kg)
m 210 000 kg (t) n 0.47 t (kg) o 592 000 mg (g) p 0.08 kg (g)
5 Read these mass scales.
a
1 2 3 4 5 6 7
kg
b
2
6
10
141620
g
c
6 Read these temperature scales.
a
5
10
15
°C b
34
36
38
40
°C c
10
15
20
°C
d
°C
0 300
50 250
100 200
150
e 1 2
°C
f
°C
9 18
27 36 45
7 A small truck delivers 0.06 t of stone for a garden. Write the mass of stones using these units.
a kg b g c mg
8 A box contains 20 blocks of cheese, each weighing 150 g. What is the approximate mass of the box in
the following units?
a g b kg
Example 16
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10J
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Measurement and Geometry 533
9 The temperature of water in a cup is initially 95°C.
After half an hour the temperature is 62°C. What is
the drop in temperature?
10 An oven is initially at a room temperature of 25°C.
The oven dial is turned to 172°C. What is the expected
increase in temperature?
11 Add all the mass measurements and give the result
in kg.
a 3 kg, 4000 g, 0.001 t
b 2.7 kg, 430 g, 930 000 mg, 0.0041 t
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12 Arrange these mass measurements from smallest to largest.
a 2.5 kg, 370 g, 0.1 t, 400 mg
b 0.000 32 t, 0.41 kg, 710 g, 290 000 mg
13 The highest and lowest temperatures recorded over a 7-day period are as follows.
Day 1 2 3 4 5 6 7
Lowest temperature (°C) 8 6 10 9 7 8 10
Highest temperature (°C) 24 27 31 32 21 19 29
a Which day had the largest temperature increase?
b What is the largest temperature drop from the highest temperature on one day to the
lowest temperature on the next day?
c What would have been the final temperature on Day 7 if the temperature increase had
been 16°C?
14 A 10 kg bag of flour is used at a rate of 200 g per day. How many days will the bag of flour last?
15 A boat has a weight limit of 3.5 t carrying capacity. Loaded onto the boat are 1500 tins of coffee
at 500 g each, 36 bags of grain at 20 kg each, 190 boxes of tobacco at 5.5 kg each and 15 people,
averaging 80 kg each. Is the load too much for the weight limit of the boat?
WORKING
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Chapter 10 Measurement and computation of length, 534 perimeter and area
16 A truck tare mass (i.e. mass with
no load) is 13.2 t. The truck’s gross
mass is 58.5 t. This is the total
maximum mass allowed, including
the load.
a What is the maximum
load the truck can carry?
b The truck is loaded with
120 timber beams at 400 kg each.
Will it exceed its gross
weight limit?
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17 Water weighs 1 kg per litre. What is the mass of these volumes of water?
a 1 mL b 1 kL c 1 ML
18 The containers shown below are fi lled with water. Calculate the mass of water in each container,
in kg.
a
20 cm
40 cm
15 cm
b
1 m
c
2 m 15 m
12 m
19 The kelvin (K) is a temperature unit used by many
scientists, where 273 K is approximately 0°C.
(The kelvin used to be called the ‘degree kelvin’ or ° K.)
An increase in 1 K is the same as an increase in 1°C.
a Write the following temperatures in °C.
i 283 K
ii 300 K
iii 1000 K
b Write the following temperatures in kelvins.
i 0°C
ii 40°C
iii −273°C
Liquid nitrogen freezes at −210°C and boils at −196°C.
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10J
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Measurement and Geometry 535
Enrichment: Fahrenheit
20 Daniel Fahrenheit (1686–1736) proposed the Fahrenheit temperature scale in 1724. It was commonly
used in Australia up until the mid 20th century, and is still used today in the United States.
32°F is the freezing point of water.
212°F is the boiling point of water.
a What is the difference between the temperature, in Fahrenheit, for the boiling point of water and
the freezing point of water?
b 1°F is what fraction of 1°C?
c 1°C is what fraction of 1°F?
d To convert from °F to °C, we use the formula C = (F – 32) × 5
9
. Convert these Fahrenheit
temperatures to °C.
i 32°F ii 68°F iii 140°F iv 221°F
e Find the rule to convert from °C to °F. Test your rule to see if it works and write it down.
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Chapter 10 Measurement and computation of length, 536 perimeter and area
opal mining
Greg, Sally and Alston apply for a mining licence to
look for opals at Coober Pedy in South Australia. They
are required to choose an area and mark it out with
special orange tape so that others will know which
areas are already taken. Their length of tape is 200 m.
Square mining areas
They fi rst decide to mark out an area as a square.
a Make a drawing of their square area.
b Calculate the side length and area and show this on your diagram. Also show any working.
Rectangular mining areas
They then change the mining area and experiment with different side lengths.
a Show three possible lengths and areas for rectangular mining sites.
b Complete a table similar to the one below. Fill in the missing numbers for the side lengths given,
then add your own rectangle measurements from part a above.
Length Breadth Perimeter Area
10
20
35
200
200
200
200
200
200
c Are there any rectangles that give a larger area than the square mining area from above?
Circular mining areas
They now decide to try to arrange the tape to form a circle. For this section you will need
the rule to calculate the distance around a circle (circumference). The circumference C is
given by C = 2 × π × r, where r is the length of the radius and π ≈ 3.14.
a Calculate the radius of the circle, correct to 1 decimal place. Use a trial and
error (guess and check) technique and remember that the circumference will be
200. Explain and show your method using a table of values.
b Calculate the area of the circular mining area, correct to the nearest square metre. Use the special
rule for the area of a circle, A, which is given by A = π × r 2.
The largest area
Compare the areas marked out with the 200 m tape by Greg, Sally and Alston. Comment on any
differences. Which shape gives the largest area for the given perimeter? Would your answer be the
same if any shape were allowed to be used? Explain.
r C
investigation
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Measurement and Geometry 537
1 Without measuring, state which line looks longer: A or B? Then measure to check your answer.
A B
OR
2 You have two sticks of length 3 m and 5 m, both with no scales. How might you mark a length
of 1 m?
3 Count squares to estimate the area of these circles, where one grid square = 1 cm.
a b
4 A house roof has 500 m2 of area. If there is 1 mm of rainfall, how much water, in litres, can be
collected from the roof?
5 Find the area of the shaded region.
6 These two rectangles overlap, as shown. Find the total area of the shaded region.
8 cm
5 cm
4 cm
6 cm
3 cm
2 cm
7 cm
3 cm
puzzles and challenges
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Chapter 10 Measurement and computation of length, 538 perimeter and area
Chapter summary
Measurement
Roman
Length
Area
16 digits = 12 inches = 1 foot
5 feet = 1 pace
1000 paces = 1 mile
Imperial
12 inches = 1 foot
3 feet = 1 yard
1760 yards = 1 mile
Units of length
10 mm = 1 cm
100 cm = 1 m
1000 m = 1 km
Perimeter
Temperature
°C (degrees Celsius)
P = 2 × 2.1 + 1.5 + 3
= 8.7 cm
2.1 cm
1.5 cm
3 cm
Mass
1 g = 1000 mg
1 kg = 1000 g
1 t = 1000 kg
Composite figures
A =  × b + bh
= 18 × 10 + × 10 × 9
= 225 m2
8 m
19 m
18 m
10 m
Triangle
3 m
2 m
Rectangle
A = b
= 4 × 2
= 8 cm2
4 cm
2 cm
1
2
1
2
A = bh
= × 3 × 2
= 3 m2
1
2
1
2
Parallelogram
2 cm
5 cm
A = bh
= 5 × 2
= 10 cm2
Square
9 m
Units of area
mm2
cm2
m2
ha
km2
1 ha = 10000 m2
A = s 2
= 92
= 81 m2
Circumference
C = 2r or d
= 2 ×  × 3
= 6 m2
3 m
Perimeter of a sector
P = 
360 × 2r + 2r
Arc length
Fraction =
 =

360

360 × 2r
 = 
360 × d
Circular arc
is a portion of the circumference
of a circle.
The arc () is subtended by an
angle, , at the centre of the circle.

l
r
r
360
r r

 × 2r
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Measurement and Geometry 539
Multiple-choice questions
1 Which of the following are imperial units?
A cm B pace C digit D yard E litre
2 Shonali buys 300 cm of wire that costs $2 per metre. How much does she pay for the wire?
A $150 B $600 C $1.50 D $3 E $6
3 The triangle given has a perimeter of 20 cm. What is the missing base length?
A 6 cm B 8 cm C 4 cm
D 16 cm E 12 cm
4 The area of a rectangle with length 2 m and width 5 m is:
A 10 m2 B 5 m2 C 5 m D 5 m3 E 10 m
5 A triangle has base length 3.2 cm and height 4 cm. What is its area?
A 25.6 cm2 B 12.8 cm C 12.8 cm2 D 6 cm E 6.4 cm2
6 The total area of this composite shape is:
A 56 km2 B 45.5 km2 C 35 km2
D 10.5 km2 E 24.5 km2
7 9 tonnes of iron ore is being loaded onto a ship at a rate of 20 kg per second. How
many minutes will it take to load all of the 9 tonnes of ore?
A 0.75 min B 45 min C 7.3 min D 450 min E 7.5 min
8 The base length of a parallelogram is 10 cm and its area is 30 cm2. The parallelogram’s height is:
A 10 cm B 3 cm C 30 cm D 3 cm2 E 10 m2
9 What is the exact circumference of a circle with radius 10 cm?
A 30 cm B 31.4 cm C 10p cm D 20p cm E p cm
10 What is the exact perimeter of a quadrant cut from a circle of radius 10 cm?
A 50 cm B 51.4 cm C (10p + 20) cm
D (5p + 20) cm E 100p cm
?
8 cm
3 km
7 km
5 km
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Chapter 10 Measurement and computation of length, 540 perimeter and area
Short-answer questions
1 a Using the Roman system, state how many:
i digits in 1 foot ii paces in 1 mile
b Using the imperial system, state how many:
i feet in 1 yard ii inches in 1 foot
c Using the metric system, state how many:
i millimetres in 1 cm ii centimetres in 1 km
2 Convert to the units shown in brackets.
a 5 cm (mm) b 200 cm (m)
c 3.7 km (m) d 421 000 cm (km)
e 7.1 kg (g) f 24 900 mg (g)
g 28 490 kg (t) h 0.009 t (g)
i 4000 mL (L) j 29 903 L (kL)
k 0.4 ML (kL) l 0.001 kL (mL)
m 1 day (min) n 3600 s (min)
o 84 h (days) p 2.5 h (s)
3 Read these scales.
a
cm 1 2 3
b
cm 1 2 3
c
1
2
3
kg
4
5
d
L
4
8
10
2
6
4 Find the perimeter of these shapes.
a
4 m
b
3.2 cm
7.1 cm
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Measurement and Geometry 541
c
9 m
5 m d
7 km
9 km
4 km
8 km
e
0.4 mm
f
6 m
5 Find the area of each of the following shapes.
a
4.9 cm
b 2 km
7 km
c
15 m
9 m
d 6 cm
4 cm
e 8 m
3.5 m
f
1 cm
1 cm
1 cm
3 cm
g
4 m
7 m
4 m
h
2 mm
5 mm
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Chapter 10 Measurement and computation of length, 542 perimeter and area
i
22 cm
15 cm
9 cm
9 cm
j
4 m
2 m
6 m
12 m
k
7 m
2 m l 1.5 km
0.6 km
6 Calculate the exact perimeter of these composite figures.
a
6 cm
b 5
11
7 Find the exact perimeter of this shape. The curved sections are semicircles.
10 cm 10 cm
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Measurement and Geometry 543
Extended-response questions
1 A regulation NBA basketball court is a rectangle with dimensions 94 feet by 50 feet. There are
12 inches in 1 foot. One inch is very close to 2.54 cm. Calculate the area of a basketball court in
square metres, correct to 2 decimal places.
2 Lachlan builds a race track around the
outside of his family house block.
The block combines a rectangular and
triangular area, as shown in the diagram.
a How far is one complete circuit of
the track?
b Lachlan can jog 10 laps at about
33 seconds each. What is the total time,
in minutes and seconds, that it takes him
to complete the 10 laps?
c What is the total area of the block?
d The house occupies 100 m2 and the rest of the block is to have instant turf, costing $12 per
square metre. What will be the cost for the instant turf?
18 m
36.3 m
10 m
15 m house
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Chapter
Introducing
indices 11
What you will learn
Divisibility tests
Prime numbers
Using indices
Prime decomposition
Squares, square roots, cubes and cube roots
The zero index and index laws
11A
11B
11C
11D
11E
11F
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number patterns around us:
Architecture
The Louvre Palace in Paris is the world’s largest
museum and is visited by over 8 million people a
year. Visitors enter the museum through a giant glass
pyramid that has a square base of length 35.4 metres
and is 21.6 metres in height. It is said that the
pyramid contains 666 glass panels.
To carefully count the number of glass panels,
we can observe how the sides of the pyramid are
constructed. Each triangular side of the pyramid has
17 rows of rhombus-shaped glass panels. The base
row is joined to the ground by triangular-shaped glass
panels. The trapezium-shaped entry has a height
of one rhombus and a width of six triangles. The
number of glass panels used in the Louvre Pyramid
can be determined using these facts and the related
properties and patterns.
Each rhombus panel is supported by four
segments of steel. Adjacent rhombuses share the
same steel segment for support. The number of steel
segments per row can be calculated, as well as the
total number of steel segments used.
nSW Syllabus
for the Australian
Curriculum
Strand: number and Algebra
Substrand: inDiCES
Outcome
A student operates with positive-integer
and zero indices of numerical bases.
(MA4–9NA)
545
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Chapter 11 546 Introducing indices
1 Find:
a 6 × 8 b 9 × 8 c 5 × 9
d 13 × 7 e 12 × 15 f 11 × 11
2 Which of the solutions to Question 1 is not divisible by 3?
3 Complete these statements.
a 8 = 4 × b 8 = 1 × c 16 = 4 × d 15 = × 3
e 12 × 2 = × 4 f 7 × 3 = 21 × g 15 × 2 = × 3 h 24 × 3 = 12 ×
4 State yes or no. Do the solutions to the following divisions contain remainders?
a 15 465 ÷ 2 b 15 465 ÷ 3
c 15 465 ÷ 5 d 15 465 ÷ 6
5 The number 5 is prime because it has only two factors (i.e. 1 and 5). The number 6 is not
prime. State whether the following numbers are prime (P) or not prime (N).
a 9 b 11 c 2 d 51
6 State whether each of the following is true or false.
a 6 × 4 × 5 = 30 × 4 b 20 + 5 = 4 × 5 + 5
c 3 × 2 × 5 = 6 × 5 d 40 ÷ 2 ÷ 2 = 40 ÷ 4
7 Find:
a 3 × 3 b 5 × 5 c 6 × 6
d 14 × 14 e 11 × 11 f 10 × 10
8 Complete:
a 9 × = 81 b 13 × = 169 c × 15 = 225
9 Find:
a 2 × 2 × 2 × 2 × 2 b 3 × 3 × 3 × 3
c 5 × 5 × 5 d 1 × 1 × 1 × 1 × 1 × 1 × 1 × 1
10 Complete these statements.
a 42 = 4 × = 16 b 72 = × = 49
c 62 = × 6 = 36 d 112 = × =
pre-test
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number and Algebra 547
Divisibility tests
It is useful to know whether a large number is exactly divisible by another number. Although we can
always carry out the division algorithm, this can be a diffi cult and tedious process for large numbers.
There are simple divisibility tests for each of the single-digit numbers, with the exception of 7. These
divisibility tests determine whether or not the number is divisible by the chosen divisor.
let’s start: Five questions in 5 minutes
In small groups, attempt to solve the following fi ve questions in 5 minutes.
1 Some numbers are only divisible by 1 and themselves. What are these numbers called?
2 Is 21 541 837 divisible by 3?
3 What two-digit number is the ‘most divisible’ (i.e. has the most factors)?
4 Find the smallest number that is divisible by 1, 2, 3, 4, 5 and 6.
5 Find a number that is divisible by 1, 2, 3, 4, 5, 6, 7 and 8.
11A
■■ A number is said to be divisible by another number if there is no remainder after the
division has occurred.
■■ If the divisor divides into the dividend exactly, then the divisor is said to be a factor of that
number.
■■ Division notation
■ Example: 27 ÷ 4 = 6 remainder 3
27
4
6 3 6 3
4 dividend = rem. =
remainder
quotient
divisor
Another way of representing this information is 27 = 4 × 6 + 3.
■■ Key terms:
Dividend The starting number; the total; the amount you have
Divisor The number doing the dividing; the number of groups
Quotient The number of times the divisor went into the dividend, also known as ‘the
answer’
Remainder The number left over; the number remaining (sometimes written as ‘rem.’)
Key ideas
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Chapter 11 548 Introducing indices
Example 1 Applying divisibility tests
Determine whether or not the following calculations are possible without leaving a remainder.
a 54 327 ÷ 3 b 765 146 ÷ 8
Solution Explanation
a Digit sum = 21
Yes, 54 327 is divisible by 3.
5 + 4 + 3 + 2 + 7 = 21
21 is divisible by 3.
b 8)14 6
18
6
rem. 2
No, 765 146 is not divisible by 8.
Check whether the last three digits are divisible by 8.
■■ Divisibility tests
1 All numbers are divisible by 1.
2 All even numbers are divisible by 2. Last digit must be a 0, 2, 4, 6 or 8.
3 The sum of the digits must be divisible by 3.
4 The number formed from the last two digits must be divisible by 4.
5 The last digit must be a 0 or 5.
6 Must pass the divisibility tests for 2 and 3.
7 There is no easy divisibility test for the numeral 7.
8 The number formed from the last three digits must be divisible by 8.
9 The sum of the digits must be divisible by 9.
10 The last digit must be 0.
Key ideas
Example 2 Testing divisibility
Carry out divisibility tests on the given number and fill in the table with ticks or crosses.
Number
Divisible
by 2
Divisible
by 3
Divisible
by 4
Divisible
by 5
Divisible
by 6
Divisible
by 8
Divisible
by 9
Divisible
by 10
48 569 412
Solution
Number
Divisible
by 2
Divisible
by 3
Divisible
by 4
Divisible
by 5
Divisible
by 6
Divisible
by 8
Divisible
by 9
Divisible
by 10
48 569 412 ✓ ✓ ✓ ✘ ✓ ✘ ✘ ✘
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number and Algebra 549
ExplAnAtion
48 569 412 is an even number and therefore is divisible by 2.
48 569 412 has a digit sum of 39 and therefore is divisible by 3, but not by 9.
48 569 412 is divisible by 2 and 3, therefore it is divisible by 6.
The last two digits are 12, which is divisible by 4.
The last three digits are 412, which is not divisible by 8.
The last digit is a 2 and therefore is not divisible by 5 or 10.
1 Give a reason why:
a 8631 is not divisible by 2 b 31 313 is not divisible by 3
c 426 is not divisible by 4 d 5044 is not divisible by 5
e 87 548 is not divisible by 6 f 214 125 is not divisible by 8
g 3 333 333 is not divisible by 9 h 56 405 is not divisible by 10
2 Give the remainder when:
a 326 is divided by 3 b 21 154 is divided into groups of four
c 72 is divided into six groups d 45 675 is shared into fi ve groups
3 Which three divisibility tests involve calculating the sum of the digits?
4 If you saw only the last digit of a 10-digit number, which three divisibility tests could you
still apply?
Exercise 11A
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5 i Without using a calculator, determine whether the following calculations are possible
without leaving a remainder.
a 23 562 ÷ 3 b 39 245 678 ÷ 4 c 1 295 676 ÷ 9
d 213 456 ÷ 8 e 3 193 457 ÷ 6 f 2 000 340 ÷ 10
g 51 345 678 ÷ 5 h 215 364 ÷ 6 i 9543 ÷ 6
j 25 756 ÷ 2 k 56 789 ÷ 9 l 324 534 565 ÷ 5
m 2 345 176 ÷ 8 n 329 541 ÷ 10 o 225 329 ÷ 3
p 356 781 276 ÷ 9 q 164 567 ÷ 8 r 2 002 002 002 ÷ 4
ii Repeat the process, using a calculator. Which way is quicker?
6 Write down fi ve two-digit numbers that are divisible by:
a 5 b 3 c 2 d 6
e 8 f 9 g 10 h 4
Example 1
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Chapter 11 550 Introducing indices
7 Carry out divisibility tests on the given numbers and fi ll in the table with ticks or crosses.
number
Divisible
by 2
Divisible
by 3
Divisible
by 4
Divisible
by 5
Divisible
by 6
Divisible
by 8
Divisible
by 9
Divisible
by 10
243 567
28 080
189 000
1 308 150
1 062 347
Example 2
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8 a Can Julie share $41.75
equally among her three
children?
b Julie fi nds one more dollar
on the fl oor and realises
that she can now share the
money equally among her
three children. How much
do they each receive?
9 The game of ‘clusters’ involves a group getting into smaller-sized groups as quickly as possible
once a particular cluster size has been called out. If a year level consists of 88 students, which
‘cluster’ sizes would ensure no students are left out of a group?
10 How many of the whole numbers between 1 and 250 inclusive are not divisible by 5? Explain,
in written form, how you arrived at your answer.
11 How many two-digit numbers are divisible by 2 and 3?
12 Find the largest three-digit number that is divisible by both 4 and 5.
13 Find the largest three-digit number that is divisible by both 6 and 7.
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14 a Is the number 968 362 396 392 139 963 359 divisible by 3?
b Many of the digits in the number above can actually be ignored when calculating the digit
sum. Which numbers can be ignored and why?
c To determine if the number above is divisible by 3, only fi ve of the 21 digits actually need to
be added together. Find this ‘reduced’ digit sum.
d Prepare a 1-minute talk to explain your answers to Question 14 a to c.
e Give your 1-minute talk to your partner and ask for feedback on how clearly you expressed
yourself.
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1111AA
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Number and Algebra 551
15 The divisibility test for the numeral 4 is to consider whether the number formed by the last
two digits is a multiple of 4. Complete the following sentences to make a more detailed
divisibility rule.
a If the second-last digit is even, the last digit must be either a , or .
b If the second-last digit is odd, the last digit must be either a or .
16 Blake’s age is a two-digit number. It is divisible by 2, 3, 6 and
9. How old is Blake if you know that he is older than 20 but
younger than 50?
17 Find the smallest number that satisfies each of the conditions
below.
The number must be larger than the divisor and leave:
a a remainder of 5 when divided by 6
b a remainder of 4 when divided by 5
c a remainder of 3 when divided by 4
d a remainder of 2 when divided by 3
e a remainder of 1 when divided by 2
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Enrichment: Divisible by 11?
18 a Write down the first nine multiples of the numeral 11.
b What is the difference between the two digits for each of these multiples?
c Write down some three-digit multiples of 11.
d What do you notice about the sum of the first digit and the last digit?
The following four-digit numbers are all divisible by 11:
1606, 2717, 6457, 9251, 9306
e Find the sum of the odd-placed digits and the sum of the even-placed digits. Then subtract the
smaller sum from the larger. What do you notice?
f Write down a divisibility rule for the number 11.
g Which of the following numbers are divisible by 11?
i 2 594 669 ii 45 384 559 iii 488 220
iv 14 641 v 1 358 024 679 vi 123 456 789 987 654 321
An alternative method is to alternate adding and subtracting each of the digits.
For example: 4 134 509 742 is divisible by 11.
Alternately adding and subtracting the digits will give the following result:
4 – 1 + 3 – 4 + 5 – 0 + 9 – 7 + 4 – 2 = 11
h Try this technique on some of your earlier numbers.
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Chapter 11 552 Introducing indices
prime numbers
It is believed that prime numbers
(i.e. positive whole numbers with two
factors) were fi rst studied by the ancient
Greeks. More recently, the introduction
of computers has allowed for huge
developments in this fi eld. Computers have
allowed mathematicians to determine which
large numbers are primes. Programs have
also been written to automatically generate
huge prime numbers that could not be
calculated previously by hand.
Remarkable fact: There are some
interesting prime numbers that have
patterns in their digits; for example,
12 345 678 901 234 567 891. This is known
as an ascending prime.
You can also get palindromic primes,
such as 111 191 111 and 123 494 321.
Below is a palindromic prime number
that reads the same upside down or when viewed in a mirror.
I88808I80888I
let’s start: How many primes?
How many numbers from 1 to 100 are prime?
You and a classmate have 4 minutes to come up with your answer.
11B
Computers allow mathematicians to examine and calculate with
extremely large numbers.
Example 3 Determining whether a number is a prime or composite
State whether each of these numbers is a prime or composite: 22, 35, 17, 11, 9, 5.
Solution ExplAnAtion
Prime: 5, 11, 17
Composite: 9, 22, 35
5, 11, 17 have only two factors (1 and itself).
9 = 3 × 3, 22 = 2 × 11, 35 = 5 × 7
■■ A prime number is a positive whole number that has only two factors: 1 and itself.
■■ A number that has more than two factors is called a composite number.
■■ The number 1 is neither a prime nor a composite number.
Key ideas
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number and Algebra 553
Example 4 Finding prime factors
Find the prime numbers that are factors of 30.
Solution ExplAnAtion
Factors of 30 are:
1, 2, 3, 5, 6, 10, 15, 30
Prime numbers from this list of factors
are 2, 3 and 5.
Find the entire set of factors fi rst.
Determine which factors are prime according to the
given defi nition.
1 The factors of 12 are 1, 2, 3, 4, 6 and 12. Is 12 a prime number?
2 The factors of 13 are 1 and 13. Is 13 a prime number?
3 List the fi rst 10 prime numbers.
4 List the fi rst 10 composite numbers.
5 What is the fi rst prime number greater than 100?
6 What is the fi rst prime number greater than 200?
Exercise 11B
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7 State whether each of the following is a prime (P) or composite (C) number.
a 14 b 23 c 70 d 37
e 51 f 27 g 29 h 3
i 8 j 49 k 99 l 59
m 2 n 31 o 39 p 89
8 Find the prime numbers that are factors of:
a 42 b 39 c 60
d 25 e 28 f 36
9 List the composite numbers between:
a 30 and 50 b 50 and 70 c 80 and 100
Example 3
Example 4
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10 The following are not prime numbers, yet they are the product (×) of two primes. Find the
two primes for each of the following numbers.
a 55 b 91 c 143
d 187 e 365 f 133
11 Which one of these numbers has factors that are only prime numbers, itself and 1?
12, 14, 16, 18, 20
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Chapter 11 554 Introducing indices
12 Twin primes are pairs of primes that are separated from each other by only one even number;
for example, 3 and 5 are twin primes. Find three more pairs of twin primes.
13 13 and 31 are known as a pair of ‘reverse numbers’. They are also both prime numbers. Find
any other two-digit pairs of prime reverse numbers.
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14 Find three different prime numbers that are less than 100 and which sum to a fourth different
prime number. Can you fi nd more than fi ve sets of such numbers?
15 Many mathematicians believe that every even number greater than 2 is the sum of two prime
numbers. Show this is true for even numbers between 30 and 50.
16 Give two examples of a pair of primes that add to a prime number. Explain why all possible
pairs of primes that add to a prime must contain the number 2.
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Enrichment: prime or not prime?
17 Design a spreadsheet that will check whether or not any number entered between 1 and 1000 is a
prime number.
If your spreadsheet is successful, someone should be able to enter the number 773 and very quickly
be informed whether or not this is a prime number.
You may choose to adapt your factor program (see Enrichment activity Exercise 4A, Question 16).
11B
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number and Algebra 555
using indices
When repeated multiplication of the same factor occurs, the expression can look quite cumbersome.
Mathematicians have a method for simplifying such expressions by writing them as powers. This
involves writing the repeated factor as the base number and then including an index number to indicate
how many times this factor must be multiplied by itself. This is also known as writing a number in index
form.
Powers are also used to represent very large and very small numbers. For example,
400 000 000 000 000 would be written as 4 × 1014. This way of writing a number is called standard form
or scientifi c notation.
let’s start: A better way...
• What is a better way of writing 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 (that is not the answer 20)?
• What is a better way of writing 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 (that is not the answer 1024)?
You may need to access the internet to fi nd out some of the following answers.
Computers have the capacity to store a lot of information. As you most likely know, computer memory
is given in bytes.
• How many bytes (B) are in a kilobyte (kB)?
• How many kilobytes are in a megabyte (MB)?
• How many megabytes are in a gigabyte (GB)?
• How many gigabytes are in a terabyte (TB)?
• How many bytes are in a gigabyte?
Hint: It is over 1 billion and it is far easier to write this number as a power!
• Why do computers frequently use base 2 (binary numbers)?
11C
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Chapter 11 556 Introducing indices
Example 5 Converting to index form
Simplify the following expressions by writing them in index form.
a 5 × 5 × 5 × 5 × 5 × 5 b 3 × 3 × 2 × 3 × 2 × 3
Solution Explanation
a 5 × 5 × 5 × 5 × 5 × 5 = 56 The number 5 is the repeated factor and it is repeated
six times.
b 3 × 3 × 2 × 3 × 2 × 3 = 22 × 34 2 is repeated two times.
3 is repeated four times.
■■ Powers are used to help write expressions involving repeated multiplication in index form.
expanded form 8 × 8 × 8 × 8 × 8 = 85 index form
■■ The expression 85 is written in index form. It is read as ‘8 to the power of 5’.
The 8 is called the base.
The 5 is called the index or exponent.
■■ In the statement 43 = 82, the 3 and 2 are called indices. This is the plural of ‘index’.
■■ 5 × 5 and 52 are read as ‘5 to the power of 2’ or ‘5 squared’.
5 × 5 × 5 and 53 are read as ‘5 to the power of 3’ or ‘5 cubed’.
5 × 5 × 5 × 5 and 54 are read as ‘5 to the power of 4’.
■■ 5b reads as ‘5 to the power of b’. In expanded form it would look like:
5 × 5 × 5 × 5 × 5 … × 5
The number 5 is written b times.
■■ Powers take priority in the order of operations.
For example: 3 + 2 × 42 = 3 + 2 × 16
= 3 × 32
= 96
■■ Note: 23 ≠ 2 × 3, therefore 23 ≠ 6. This is a common mistake that must be avoided.
Instead: 23 = 2 × 2 × 2 = 8.
Key ideas
Example 6 Expanding a power
Expand and evaluate the following terms.
a 24 b 23 × 52
Solution Explanation
a 24 = 2 × 2 × 2 × 2
= 16
Write 2 down four times and multiply.
b 23 × 52 = 2 × 2 × 2 × 5 × 5
= 8 × 25
= 200
Repeat the number 2 three times, and the number 5
two times.
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number and Algebra 557
Example 7 Evaluating expressions with powers
Evaluate:
a 72 – 62 b 2 × 33 + 102 + 17
Solution ExplAnAtion
a 72 – 62 = 7 × 7 – 6 × 6
= 49 – 36
= 13
Write in expanded form (optional).
Powers are evaluated before the subtraction occurs.
b 2 × 33 + 102 + 17
= 2 × 3 × 3 × 3 + 10 × 10 + 1 × 1 × 1
× 1 × 1 × 1 × 1
= 54 + 100 + 1
= 155
Write in expanded form (optional).
Follow order of operation rules.
Carry out the multiplication fi rst, then carry out the
addition.
1 Select the correct answer from the following alternatives.
37 means:
A 3 × 7 B 3 × 3 × 3 C 7 × 7 × 7
D 3 × 7 × 3 × 7 × 3 × 7 × 3 E 3 × 3 × 3 × 3 × 3 × 3 × 3 F 37
2 Select the correct answer from the following alternatives.
9 × 9 × 9 × 9 × 9 can be simplifi ed to:
A 9 × 5 B 5 × 9 C 59
D 95 E 99 999 F 95
3 Copy and complete the table.
index form Base number index number Basic numeral
23 2 3 8
52
104
27
112
121
05
Exercise 11C
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Chapter 11 558 Introducing indices
4 Simplify the following expressions by writing them as powers.
a 3 × 3 × 3 b 2 × 2 × 2 × 2 × 2 c 15 × 15 × 15 × 15
d 10 × 10 × 10 × 10 e 6 × 6 f 20 × 20 × 20
g 1 × 1 × 1 × 1 × 1 × 1 h 4 × 4 × 4 i 100 × 100
5 Simplify the following expressions by writing them as powers.
a 3 × 3 × 5 × 5 b 7 × 7 × 2 × 2 × 7 c 12 × 9 × 9 × 12
d 8 × 8 × 5 × 5 × 5 e 6 × 3 × 6 × 3 × 6 × 3 f 13 × 7 × 13 × 7 × 7 × 7
g 4 × 13 × 4 × 4 × 7 h 10 × 9 × 10 × 9 × 9 i 2 × 3 × 5 × 5 × 3 × 2 × 2
6 Simplify by writing using powers.
2 × 3 × 5 × 5 × 3 × 3 × 2 × 2 × 2 × 5 × 3 × 2 × 2 × 5 × 3
7 Expand these terms. (Do not evaluate.)
a 24 b 172 c 93 d 37
e 144 f 88 g 105 h 543
8 Expand these terms. (Do not evaluate.)
a 35 × 23 b 43 × 34 c 72 × 53 d 46 × 93
e 5 × 74 f 22 × 33 × 41 g 115 × 92 h 203 × 302
9 Evaluate:
a 25 b 82 c 103 d 32 × 23
e 104 f 23 × 53 g 16 × 26 h 112 × 18
10 Evaluate:
a 32 + 42 b 2 × 52 – 72 c 82 – 2 × 33
d (9 – 5)3 e 24 × 23 f 27 – 1 × 2 × 3 × 4 × 5
g 14 + 23 + 32 + 41 h 103 – 102 i (127 + 123) × 22
11 Use a calculator to evaluate the following.
a 152 − 132 b 93 + 34 c 54 × 52 + 53
d 222 + 193 e (123 − 172) × 43 f 1003 − 993
Example 5a
Example 5b
Example 6a
Example 6b
Example 7
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12 Determine the index number for the following basic numerals.
a 16 = 2? b 16 = 4? c 64 = 4? d 64 = 2?
e 27 = 3? f 100 = 10? g 49 = 7? h 625 = 5?
13 Write one of the symbols <, = or > in the box to make the following statements true.
a 26 29 b 83 82 c 24 42 d 32 42
e 64 53 f 122 34 g 112 27 h 18 23
14 A text message is sent to fi ve friends. Each of the fi ve friends then forwards it to fi ve other friends and
each of these people also sends it to fi ve other friends. How many people does the text message reach,
not including those who forwarded the message?
15 a Emily writes a chain email and sends it to fi ve friends. If each person who receives the email reads
it within 5 minutes of the email arriving and then sends it to fi ve other people, how many people
(including Emily) will have read the email 15 minutes after Emily fi rst sent it?
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11C
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Number and Algebra 559
b If the email always goes to a new person, and assuming every person in Australia has
an email address and access to email, how long would it take until everyone in Australia
has read the message? (Note: Australia’s population is approx.
23 million people.)
c How many people will read the email within 1 hour?
d Using the same assumptions as above, how long would it take
until everyone in the world has read the message? (Note: World
population is approx. 7 billion people.)
e How many people will have read the email in 2 hours?
f Write a 100-word newspaper report explaining the power of
powers! Make sure you have a catchy headline.
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16 Write the correct operation (+, −, ×, ÷) in the box to make the following equations true.
a 32 42 = 52 b 24 42 = 44 c 27 53 = 31
d 92 34 = 120 e 102 102 = 104 f 102 82 = 62
17 A chain email is initiated by an individual and sent to x number of recipients. This process is repeated
(i.e. is forwarded to x new recipients) y times. How many people receive the email, not including
those who forwarded the message?
18 Find a value for a and for b such that a ≠ b and ab = ba.
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Enrichment: Investigating factorials
19 In mathematics, the exclamation mark (!) is the symbol for factorials.
4! = 4 × 3 × 2 × 1 = 24
n! = n × (n – 1) × (n – 2) × (n – 3) × (n – 4) × ... × 6 × 5 × 4 × 3 × 2 × 1
a Evaluate 1!, 2!, 3!, 4!, 5! and 6!
Factorials can be written in prime factor form, which involves powers.
For example: 6! = 6 × 5 × 4 × 3 × 2 × 1
= (2 × 3) × 5 × (2 × 2) × 3 × 2 × 1
= 24 × 32 × 5
b Write these numbers in prime factor form.
i 7! ii 8! iii 9! iv 10!
c Write down the last digit of 12!
d Write down the last digit of 99!
e Find a method of working out how many consecutive zeros would occur on the right-hand end of
each of the following factorials if they were evaluated. Hint: Consider prime factor form.
i 5! ii 6! iii 15! iv 25!
f 10! = 3! × 5! × 7! is an example of one factorial equal to the product of three factorials. Express
24! as the product of two or more factorials.
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Chapter 11 560 Introducing indices
prime decomposition
All composite numbers can be broken down (i.e. decomposed) into a unique set of prime factors. A
common way of performing the decomposition into prime factors is using a factor tree. Starting with
the given number, ‘branches’ come down in pairs, representing a pair of factors that multiply to give the
number above it. This process continues until prime factors are reached.
let’s start: Composition of numbers from prime factors
‘Compose’ composite numbers from the following sets of prime factors. The fi rst one has been done
for you.
a 2 × 3 × 5 = 30 b 2 × 3 × 7 × 3 × 2
c 32 × 23 d 5 × 11 × 22
e 13 × 17 × 2 f 22 × 52 × 72
g 25 × 34 × 7 h 11 × 13 × 17
Note that this process is the reverse of decomposition.
11D
■■ Every composite number can be expressed as a product of its prime factors.
■■ A factor tree can be used to show the prime factors of a composite number.
■■ Each ‘branch’ of a factor tree eventually terminates in a prime factor.
■■ Powers are often used to effi ciently represent composite numbers in prime factor form.
For example:
48
starting composite
a pair of number
‘branches’
‘branches’
terminate on
prime factors  48 = 2 × 2 × 2 × 2 × 3
= 24 × 3
4 12
2
2 2
2 4 3
expressed with powers
■■ It does not matter with which pair of factors you start a factor tree. The fi nal set of prime factors
will always be the same.
■■ It is conventional to write the prime factors in ascending (i.e. increasing) order.
For example: 600 = 23 × 3 × 52
Key ideas
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number and Algebra 561
Example 8 Expressing composites in prime factor form
Express the number 60 in prime factor form.
Solution ExplAnAtion
 60 = 2 × 2 × 3 × 5
60 = 22 × 3 × 5
60
5 12
3 4
2 2
A pair of factors for 60 are 5 × 12.
The 5 branch terminates since 5 is a prime factor.
A pair of factors for 12 are 3 × 4.
The 3 branch terminates since 3 is a prime factor.
A pair of factors for 4 are 2 × 2.
Both these branches are now terminated.
Hence, the composite number 60 can be written as a
product of each terminating branch.
2 60
2 30
3 15
5 5
1
or
■■ Here is another way to decompose a number, using division by prime numbers.
Sometimes it may be necessary to divide by 2, 3, 5, 7, 11, 13 or any prime number.
Divide by 2 as
many times as
possible,
then 3,
then 5
etc.
2 600
2 300
2 150
3 75
5 25
5 5
5 1
Start with 600 and divide by
2 if there is no remainder.
Stop when this number is 1.
600 = 2 × 2 × 2 × 3 × 5 × 5
600 = 23 × 3 × 52
Key ideas
1 Sort the following list of numbers into two groups: composite numbers and prime numbers.
15, 13, 7, 5, 8, 9, 27, 23, 11, 4, 12, 2
Exercise 11D
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Chapter 11 562 Introducing indices
2 Fill in the gaps to complete the following factor trees.
a 40
8
2
2 2
b 30
5
3
c 100
10
2 5
3 Complete each of the following factor trees.
a 90
9 10
b 56
4 14
c 220
55 4
4 Write the following prime factors, using powers.
a 2 × 3 × 3 × 2 × 2 b 5 × 3 × 3 × 3 × 3 × 5
c 7 × 2 × 3 × 7 × 2 d 3 × 3 × 2 × 11 × 11 × 2
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5 Express the following numbers in prime factor form.
a 72 b 24 c 38 d 44
e 124 f 80 g 96 h 16
i 75 j 111 k 64 l 56
6 Express these numbers in prime factor form.
a 600 b 800 c 5000 d 2400
e 1 000 000 f 45 000 g 820 h 690
Example 8
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7 Match the correct composite number (a to d) to its set of prime factors (A to D).
a 120 A 2 × 3 × 52
b 150 B 22 × 32 × 5
c 144 C 24 × 32
d 180 D 2 × 3 × 2 × 5 × 2
8 Find the smallest composite number that has the fi ve smallest prime numbers as factors.
9 a Express 144 and 96 in prime factor form.
b By considering the prime factor form, determine the HCF of 144 and 96.
10 a Express 25 200 and 77 000 in prime factor form.
b By considering the prime factor form, determine the HCF of 25 200 and 77 000.
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11D
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Number and Algebra 563
11 Represent the number 24 with four different factor trees,
each resulting in the same set of prime factors. Note that
simply swapping the order of a pair of factors does not
qualify it as a different form of the factor tree.
12 Only one of the following is the correct set of prime factors
for 424.
A 22 × 32 × 5 B 2 × 32 × 52
C 53 × 8 D 23 × 53
i Justify why you can eliminate alternatives A and B
straight away.
ii Why can option C be discarded as an option?
iii Show that option D is the correct answer.
13 a State the error in each of the following prime factor trees.
i
60 = 2 × 52
60
5 10
2 5
ii
60 = 2 × 5 × 6
60
2 30
6 5
iii
60 = 2 × 3 × 5
60
2 30
2 15
3 5
b What is the correct way to express 60 in prime factor form?
14 Write 15 different (i.e. distinct) factor trees for the number 72.
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Enrichment: Four distinct prime factors
15 There are 17 composite numbers that are smaller than 1000 which have four distinct (i.e. different)
prime factors. For example: 546 = 2 × 3 × 7 × 13.
By considering the prime factor possibilities, find the other 16 composite numbers and express each of
them in prime factor form.
16 A conjecture is a statement that may appear to be true but has not been proved conclusively.
Goldbach’s conjecture states that ‘Every even number greater than 2 is the sum of two prime
numbers.’ For example, 53 = 47 + 5.
Challenge: Try this for every even number from 4 to 50.
17 Use the internet to find the largest-known prime number.
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Chapter 11 564 Introducing indices
Squares, square roots, cubes and cube roots
A square number can be illustrated by considering the area of a square
with a whole number as its side length.
For example:
4 cm
4 cm
Area of square = 4 cm × 4 cm = 16 cm2
Therefore, 16 is a square number.
Another way of representing square numbers is through a square
array of dots. For example:
Number of dots = 3 rows of 3 dots
= 3 × 3 dots
= 32 dots
= 9 dots
Therefore, 9 is a square number.
To produce a square number you must multiply the number
by itself. All square numbers written in index form will have a
power of 2.
Finding a square root of a number is the opposite of squaring a number.
For example: 42 = 16 and therefore 16 = 4.
To fi nd square roots we use our knowledge of square numbers. A calculator is also frequently used to
fi nd square roots.
Geometrically, the square root of a number is the side length of a square whose area is that number.
If 4 × 4 is ‘4 squared’, then 4 × 4 × 4 is ‘4 cubed’.
Note that ‘5 cubed’ is 5 × 5 × 5 = 125 (not 15).
The opposite of this is called a cube root. For example: 2 × 2 × 2 = 8, so 3 8 = 2 .
let’s start: Speed squaring tests
In pairs, test one another’s knowledge of square numbers.
• Ask 10 quick questions, such as ‘3 squared’, ‘5 squared’ etc.
• Have two turns each. Time how long it takes each of you to answer the 10 questions.
• Aim to be quicker on your second attempt.
Write down the fi rst 10 square numbers.
• Begin to memorise these important numbers.
• Time how quickly you can recall the fi rst 10 square numbers without looking at a list of numbers.
• Can you go under 5 seconds?
11E
A block like this shows ‘3 cubed’.
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Number and Algebra 565
■■ Any whole number multiplied by itself produces a square number.
For example: 52 = 5 × 5 = 25. Therefore, 25 is a square number.
– Square numbers are also known as perfect squares.
– The first 12 square numbers are:
Index form 12 22 32 42 52 62 72 82 92 102 112 122
Basic numeral 1 4 9 16 25 36 49 64 81 100 121 144
– All square numbers have an odd number of factors.
– The symbol for squaring is ( )2 . The brackets are optional, but can be very useful when
simplifying more difficult expressions.
■■ The square root of a given number is the positive number that, when multiplied by itself,
produces the given number.
– The symbol for square rooting is .
– Finding a square root of a number is the opposite of squaring a number.
For example: 42 = 16; hence, 16 = 4.
We read this as: ‘4 squared equals 16, therefore, the square root of 16 equals 4.’
– Squaring and square rooting are ‘opposite’ operations.
( x )2 = x also (x)2 = x
– Below is a list of common square roots.
Square root form 1 4 9 16 25 36 49 64 81 100 121 144
Basic numeral 1 2 3 4 5 6 7 8 9 10 11 12
■■ Any whole number when multiplied by itself twice produces a cubic number.
For example: 43 = 4 × 4 × 4 = 64
■■ The cube root of a given number is the number that, when multiplied by itself twice,
produces the given number.
–– The symbol used is 3 .
For example: 3 64 = 4
We read this as ‘The cube root of 64 is 4.’
Key ideas
Example 9 Evaluating squares, square roots, cubes and cube roots
Evaluate:
a 62 b 64 c 1600
d 63 e 3 64
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Chapter 11 566 Introducing indices
Solution ExplAnAtion
a 62 = 36 62 = 6 × 6
b 64 = 8 8 × 8 = 64  64 = 8
c 1600 = 40 40 × 40 = 1600  1600 = 40
d 63 = 6 × 6 × 6 = 216 6 × 6 = 36
36 × 6 = 216
e 3 8 = 2 2 × 2 × 2 = 8
Example 10 Evaluating expressions involving squares, square roots, cubes and
cube roots
Evaluate:
a 33 − 9 +12 b 82 + 62 c 100 28
9
3 −
Solution ExplAnAtion
a 33 − 9 + 12 = 27 – 3 + 1
= 25
33 = 3 × 3 × 3, 9 = 3, 12 = 1 × 1
b 8 6 64 36
100
10
2 + 2 = +
=
=
82 = 8 × 8, 62 = 6 × 6
100 = 10
c
100 28
9
72
9
8
2
3 3
3
− =
=
=
Simplify the fraction fi rst.
1 Draw a square of side length 6 cm. What would be the area of this shape? What special
type of number is your answer?
2 a Write down the fi rst 15 square numbers in index form and as basic numerals.
b Repeat part a for the fi rst 10 cubic numbers.
3 We can confi rm that 9 is a square number by drawing the diagram shown at right.
a Show, using dots, why 6 is not a square number.
b Show, using dots, why 16 is a square number.
Exercise 11E
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Number and Algebra 567
4 Evaluate:
a 62 b 5 squared c (11)3
d 10 to the power of 2 e 203 f 12 × 12
5 Evaluate:
a 25 b square root of 16 c 3 1
d the cube root of 27 e 3 1 000 000
Example 9d
Example 9e
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6 Evaluate:
a 82 b 72 c 12 d 122
e 32 f 152 g 52 h 02
i 112 j 1002 k 172 l 332
7 Evaluate:
a 25 b 9 c 1 d 121
e 0 f 81 g 49 h 16
i 4 j 144 k 400 l 169
8 Using a calculator, evaluate the following.
a 3 3375 b 3 15 625 c 3 9261 d 3 6859
9 Without a calculator, evaluate the following. Then use a calculator to check your answers.
a 32 + 52 − 16 b 4 × 42 c 82 – 02 + 12
d 12 × 22 × 32 e 52 − 32 f 81 − 32
g 62 ÷ 22 × 32 h 9 × 64 ÷ 36 i 122 + 52
j
100 64
9
− k
9 5
9
− l
28 4
28 22
+
+
10 Using a calculator, evaluate the following.
a 256 +152 b 333 +196 c 144 132
3
−
d
27 18 2
3
× ×
e 5 × 42 × 81 f 162 ÷ 42 + 52
Example 9a
Example 9b
Example 9c
Example 10
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11 List all the square numbers between 50 and 101.
12 List all the square numbers between 101 and 200. Hint: There are only four.
13 a Find two square numbers that add to 85.
b Find two square numbers that have a difference of 85.
14 Find three different square numbers that sum to 59.
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Chapter 11 568 Introducing indices
15 a Evaluate 32 × 42.
b Evaluate 122.
c The rule a2 × b2 = (a × b)2 can be used to link 32 × 42 and 122. What are the values of
a and b if 32 × 42 = 122?
d Check this formula using other numbers.
16 a Show that 32 + 42 = 52.
b What does 62 + 82 equal?
c What does 92 + 122 equal?
d What does 302 + 402 equal?
17 a Evaluate 112 and 1112.
b Predict an answer for 11112.
c Evaluate 11112 and test your prediction.
18 Stuart decides there are no odd square numbers. His justifi cation is that ‘because an even number
multiplied by an even number produces an even number, and that an odd number multiplied by an
odd number also produces an even number, then there are no odd square numbers’. Do you agree
with Stuart’s claim? If not, give an example to explain your answer.
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Enrichment: properties of square roots
19 Trial different numbers in the following formulas to determine whether these algebraic statements
involving square roots are true or false.
a a + b = a + b b a − b = a − b
c a × b = a × b d a
b
a
b
=
e a2 = a f a3 = a a
g a2 + b2 = a + b h a2 − b2 = a − b
11E
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number and Algebra 569
the zero index and index laws
Recall that 54 = 5 × 5 × 5 × 5, 53 = 5 × 5 × 5 and 52 = 5 × 5. Therefore, it makes sense that 51 = 5. What
about 5 to the power of 0? Does 50 have a value?
Also, what happens when we multiply 54 and 53 together? Is there any easy way to do that?
let’s start: Patterns of powers
• What comes next in these patterns of equations?
We know that 54 = 5 × 5 × 5 × 5. We also know that 53 = 5 × 5 × 5.
Therefore, 54 × 53 = 5 × 5 × 5 × 5 × 5 × 5 × 5 = 57.
• Can you see a shortcut?
• Can you simplify 84 × 85?
• Does the shortcut work for the expression 64 × 85?
We know that 56 = 5 × 5 × 5 × 5 × 5 × 5.
We also know that 54 = 5 × 5 × 5 × 5.
Therefore, 5 5
5 5 5 5 5 5
5 5 5 5
6 ÷ 4 = 52 × × × × ×
× × ×
= .
• Can you see a shortcut?
• Can you simplify 89 ÷ 85?
• Does the shortcut work for this expression 69 × 85?
We know that 54 = 5 × 5 × 5 × 5.
Therefore, (54)3 = (5 × 5 × 5 × 5) × (5 × 5 × 5 × 5) × (5 × 5 × 5 × 5) = 512.
• Can you see a shortcut?
• Can you simplify (89)5?
11F
54 = 625
53 = 125
52 = 25
51 = 5
5? = ?
÷ 5 ÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
÷ 5
34 = 81
33 = 27
32 = 9
31 = 3
3? = ?
÷ 3 ÷ 3
÷ 3
÷ 3
÷ 3
÷ 3
÷ 3
÷ 3
60 = 1
80 = 1
100 = 1
5000 = 1
a 0 = 1
Any positive number raised to the
power of zero is equal to 1.
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Chapter 11 570 Introducing indices
Example 11 Using the zero index
Simplify each of the following terms.
a 250 b (-25)0 c -(25)0 d (6 × 5)0 e 6 × 50
Solution Explanation
a 250 = 1 Any number raised to the power of 0 is 1.
b (-25)0 = 1 Any number raised to the power of 0 is 1.
c -(25)0 = -1 × (25)0 = -1 × 1 = -1 The negative symbol is not in the brackets, so the zero
index does not apply to it.
d (6 × 5)0 = 300
= 1
Both the 6 and the 5 are in the brackets.
The zero index applies to all the numbers in the brackets.
e 6 × 50 = 6 × 1 = 6 The zero index applies to the 5 but not the 6.
■■ The zero index
–– When a number other than 0 is raised to the power 0, the result is 1.
For example: 50 = 1 and 100 = 1 but 00 ≠ 1.
–– In general, if a is any number other than 0, a0 = 1.
■■ The index law for multiplication
–– When multiplying terms with the same base, add the powers.
For example: 89 × 85 = 814
■■ The index law for division
–– When dividing terms with the same base, subtract the powers.
For example: 89 ÷ 85 = 84
■■ The index law for power of a power
–– When raising a term in index form to a power, retain the base and multiply the indices.
For example: (89)5 = 845
Key ideas
Example 12 Using the index laws for multiplication and division
Simplify, giving your answer in index form.
a 73 × 75 b 1010 × 1010 c 25 × 24 × 23 d 78 ÷ 75 e 78 ÷ 78
Solution Explanation
a 73 × 75 = 78 In a multiplication, if the bases are equal then add the powers.
b 1010 × 1010 = 1020 In a multiplication, if the bases are equal then add the powers.
c 25 × 24 × 23 = 212 In a multiplication, if the bases are equal then add the powers.
d 78 ÷ 75 = 73 In a division, if the bases are equal then subtract the powers.
e 78 ÷ 78 = 70 =1 In a division, if the bases are equal then subtract the powers.
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number and Algebra 571
Example 13 using the index law for the power of a power
Simplify, giving your answer in index form.
a (73)5 b (73)0
Solution ExplAnAtion
a (73)5 = 715 In this situation, multiply the powers.
b (73)0 = 70 = 1 In this situation, multiply the powers.
1 Copy and complete to give an answer in index form. Use cancelling in parts c and d.
a 32 × 34 = 3 × × 3 × × ×
= 3
b 64 × 63 = 6 × × × × 6 × ×
= 6
c 55 ÷ 53 =
5
5
× × × ×
× ×
= 5
d 94 ÷ 92 =
9
9
× × ×
×
= 9
2 Decide if these statements are true or false.
a 5 × 5 × 5 × 5 = 54 b 26 × 22 = 26 + 2 c 72 × 74 = 74 – 2 d 84 ÷ 82 = 84 + 2
3 Write the missing words or numbers in these sentences.
a When raising a term or numbers in index form to another power, ________ the indices.
b Any number (except 0) raised to the power 0 is equal to ___.
4 Write the missing numbers in these tables.
a index form 26 25 24 23 22 21 20
Basic numeral 64 32
b index form 45 44 43 42 41 40
Basic numeral 1024 256
5 Copy and complete this working.
a (42)3 = 4 × × 4 × × 4 ×
= 4
b (123)3 = (12 × × )× (12 × × )× (12 × × )
= 12
Exercise 11F
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Chapter 11 572 Introducing indices
6 Evaluate each of the following.
a 50 b 90 c (−6)0 d (−3)0
e −(40) f
3
4
0 
 

 
g − 
 

 
1
7
0
7 Simplify, giving your answers in index form.
a 24 × 23 b 56 × 53 c 72 × 74 d 89 × 8
8 Apply the index law for power of a power to simplify each of the following. Leave your answers in
index form.
a (32)3 b (43)5 c (35)6 d (75)2
9 Simplify each of the following by combining various index laws.
a 4 × (43)2 b (34)2 × 3 c 78 ÷ (73)2
d (42)3 ÷ 45 e (36)3 ÷ (35)2
Example 11
Example 12
Example 13
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10 Evaluate, without using a calculator.
a 77 ÷ 75 b 106 ÷ 105 c 1311 ÷ 139 d 220 ÷ 217
e 1015 ÷ 1014 f 20030 ÷ 20028 g 7 × 3116 ÷ 3115 h 3 × 50200 ÷ 50198
11 Evaluate these without using a calculator.
a (24)8 ÷ 230 b (103)7 ÷ 1018
c ((−1)11)2 × ((−1)2)11 d −2((−2)3)3 ÷ (−2)8
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11F
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Number and Algebra 573
Enrichment: Changing the base
14 The base of a number in index form can be changed using the index law for power of a power.
For example: 82 = (23)2
= 26
Change the base numbers and simplify the following using the smallest possible base integer.
a 84 b 323 c 93
d 815 e 255 f 24310
g 2569 h 240120 i 100 00010
A research scientist in a microbiology laboratory would use indices to express the numbers of microbes
being studied
12 Explain why the following statements are incorrect and give the correct answer.
a 2 × 50 = 1 b 22 × 22 = 44 c 2 × 36 = 66 d 23 + 23 = 26
13 Write down the next three equations in this pattern, using fractions on the right-hand side.
34 = 81
33 = 27
32 = 9
31 = 3
30 = 1
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Chapter 11 574 Introducing indices
Fibonacci sequences
Leonardo Fibonacci was a famous 13th century mathematician who discovered some very
interesting patterns of numbers that are found in nature.
Fibonacci’s rabbits
These rules determine how fast rabbits can breed in ideal circumstances.
• Generation 1: One pair of newborn rabbits is in a paddock. A pair is one female and one male.
• Generation 2: When it is 2 months old, the female produces another pair of rabbits.
• Generation 3: When it is 3 months old, this same female produces another pair of rabbits.
• Every female rabbit always produces one new pair every month from age 2 months.
a Using the ‘rabbit breeding rules’, complete a drawing of the
fi rst fi ve generations of rabbit pairs. Use it to complete the
table below.
Month 1 2 3 4 5
number of rabbits 2
number of pairs
b Write down the numbers of pairs of rabbits at the end of
each month for 12 months. This is the Fibonacci sequence.
c How many rabbits will there be after 1 year?
d Explain the rule for the Fibonacci sequence.
Fibonacci sequence in plants
a Count the clockwise and anticlockwise spiralling ‘lumps’ of some pineapples
and show how these numbers relate to the Fibonacci sequence.
b Find three examples of fl owers that have two terms of the Fibonacci sequence
as the ratio of the numbers of clockwise and anticlockwise spirals of petals.
c On many plants, the number of petals is a Fibonacci number. Research the
names and images of some of these ‘Fibonacci’ fl owers.
Fibonacci sequence and the golden ratio
a Write down the next 10 terms of the Fibonacci sequence: 1, 1, 2, 3, 5, . . .
b Write down a new set of numbers that is one Fibonacci number divided by its previous
Fibonacci number. Copy and complete this table.
Fibonacci sequence 1 1 2 3 5
Ratio 1 1 2 1.5
c What do you notice about the new sequence (ratio)?
d Research the golden ratio and explain how it links to your new sequence.
1 ÷ 1 2 ÷ 1 3 ÷ 2
cont.
investigation
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number and Algebra 575
1 A number is said to be a ‘perfect number’ if the sum of its factors equals the number. For this
exercise, we must exclude the number itself as one of the factors.
The number 6 is the fi rst perfect number.
Factors of 6 (excluding the numeral 6) are 1, 2 and 3.
The sum of these three factors is 1 + 2 + 3 = 6. Hence, we have a perfect number.
a Find the next perfect number. Hint: It is less than 50.
b The third perfect number is 496. Find all the factors for this number and show that it is a
perfect number.
2 When presented with a diffi cult multiplication question, such as 75 × 24, you can use your
knowledge of factors to help you calculate the answer.
For example: 75 × 24 = 25 × 3 × 6 × 4
Choosing appropriate pairs of factors can suddenly make the whole question easier.
Therefore: 75 × 24 = (25 × 4) × (3 × 6)
= 100 × 18
= 1800
Use factors to help you calculate the following.
a 28 × 50 b 15 × 32 c 36 × 15
d 18 × 35 e 120 × 36 f 44 × 25
3 Instead of carrying out a complex division algorithm, you could convert the divisor into a
smaller pair of factors and complete two simpler division questions to arrive at the correct
answer.
For example: 1458 ÷ 18 = (1458 ÷ 2) ÷ 9
= 729 ÷ 9
= 81
Use factors to help you calculate the following.
a 555 ÷ 15 b 860 ÷ 20 c 3600 ÷ 48
d 1456 ÷ 16 e 6006 ÷ 42 f 2024 ÷ 22
puzzles and challenges
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Chapter 11 576 Introducing indices
4 The easiest way of coming up with a large quantity of prime numbers is to place numbers in a
grid and simply cross off multiples of 2, 3, 4, 5 and so on, until you reach the square root of the
largest number in the grid.
For example: To find all the prime numbers that are less than 100, we could place the numbers
in a 10 × 10 grid and cross off multiples of 2, 3, 4, 5, 6, 7, 8, 9 and 10. We can stop at 10
because 10 is the square root of 100. Actually, because some of these multiples are simply
multiples of other numbers, we can cross out 6, 8, 9 and 10. This means that on our grid of 100
numbers, we only have to cross out the multiples of 2, 3, 5 and 7. Once you have crossed out all
these multiples, the numbers that remain and are not crossed out are prime.
Using the blank sieve (grid) below as an example, work out all the prime numbers less than 100.
01, 02, 03, 04, 05, 06, 07, 08, 09, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90,
91, 92, 93, 94, 95, 96, 97, 98, 99, 100
5 Complete this sequence.
22 = 12 + 3
32 = 22 + 5
42 = 32 + 7
52 = _______
62 = _______
6 Use the digits 1, 9, 7 and 2, in that order, and any operations and brackets you like, to make as
your answers the whole numbers 0 to 10.
Can you do any in more than one way?
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number and Algebra 577
Chapter summary
20: 1, 2, 4, 5, 10, 20
15: 1, 3, 5, 15
Highest common factor
HCF = 5
Factors
5: factors 1 and 5
17: factors 1 and 17
A prime number
only has two
factors: 1 and itself.
Prime numbers
10: factors 1, 2, 5, 10
9: factors 1, 3, 9
A composite number
has more than two
factors.
Composite numbers
The number 1 is
neither a prime
nor a composite
number.
Number
properties
20: 20, 40, 60, 80, ...
15: 15, 30, 45, 60, 75, ...
Lowest common multiple
LCM = 60
Multiples
dividend
divisor
52 = 3 × 17 + 1
= 17
Division
52
3
1
3
remainder
quotient
243 = 3 × 3 × 3 × 3 × 3 = 35
Powers
basic
numeral
expanded
form
index
form
base
number
index
number
Factor trees
90 = 2 × 3 × 3 × 5
90 = 2 × 32 × 5
90
9 10
3 3 2 5
Square number
16 = 4 × 4 = 42
25 = 5 × 5 = 52
Square root
16 = 4
25 = 5
Divisibility tests
2: last digit even or 0
3: sum digits ÷ 3
4: last two digits ÷ 4
5: last digit 0 or 5
6: ÷ by 2 and 3
8: last 3 digits ÷ 8
9: sum digits ÷ 9
10: last digit 0
The zero index
Any positive number raised to the power of zero gives 1.
e.g. 20 = 1
If the bases are the same, add the powers.
e.g. 26 × 24 = 210
If the bases are the same, subtract the powers.
e.g. 26 ÷ 24 = 22
Multiply the powers
e.g. (26)4 = 224
Index law for multiplication
Index law for power of a power
Index law for division
Cube and cube roots
2 × 2 × 2 = 23 = 8
We say, ‘2 cubed is 8’.
Therefore, the cube root
of 8 is 2.
3
e.g. 8 = 2
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Chapter 11 578 Introducing indices
Multiple-choice questions
1 Which number is not a power of 2?
3, 6, 9, 12, 15, 18, 22, 24, 27, 30
A 16 B 22 C 32 D 8 E 2
2 Which group of numbers contains every factor of 60?
A 2, 3, 4, 5, 10, 12, 15, 60 B 2, 3, 4, 5, 10, 12, 15, 20, 30
C 1, 2, 3, 4, 5, 10, 12, 15, 20, 30 D 2, 3, 4, 5, 10, 15, 20, 30, 60
E 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
3 Which of the following numbers is not divisible by only prime numbers, itself and 1?
A 21 B 77 C 110 D 221 E 65
4 Which of the following groups of numbers include one prime and two composite numbers?
A 2, 10, 7 B 54, 7, 11 C 9, 32, 44 D 5, 17, 23 E 18, 3, 12
5 7 × 7 × 7 × 7 × 7 can be simplified to:
A 57 B 75 C 7 × 5 D 75 E 77 777
6 Evaluate 32 + 42 .
A 7 B 5 C 14 D 25 E 6
7 The prime factor form of 48 is:
A 24 × 3 B 22 × 32 C 2 × 33 D 3 × 42 E 23 × 6
8 Evaluate 43 – 3 × (24 – 32).
A 427 B 18 C 43 D 320 E 68
9 Factors of 189 are:
A 3, 7, 9, 18, 21, 27 B 3, 9, 18, 21 C 3, 9, 18
D 3, 7, 9, 17, 21 E 3, 7, 9, 21, 27, 63
10 Which number is not divisible by 3?
A 25 697 403 B 31 975 C 7 297 008
D 28 650 180 E 38 629 634 073
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Number and Algebra 579
Short-answer questions
1 a Find the complete set of factors of 120 and circle those that are composite numbers.
b Determine three numbers between 1000 and 2000 that each have factors 1, 2, 3, 4, 5
and itself.
2 a Write down the first 12 multiples for each of 8 and 7 and circle the odd numbers.
b Which two prime numbers less than 20 have multiples that include both 1365 and 1274?
3 a State whether each of these numbers is a prime or composite number.
21, 30, 11, 16, 7, 3, 2
b How many prime multiples are there of 13?
4 a State the prime factors of 770.
b Determine three composite numbers less than 100, each with only three factors that are all
prime numbers less than 10.
5 Simplify these expressions by writing them in index form.
a 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6
b 5 × 5 × 5 × 5 × 2 × 2 × 2 × 2 × 2
6 Write these numbers as a product of prime numbers. Use a factor tree and then index form.
a 32 b 200 c 225
7 Determine which number to the power of 5 equals each of the following.
a 100 000 b 243 c 1024
8 Evaluate each of the following.
a 52 – 32 b 2 × 42 – 52
c 5 × 34 – 32 + 16 d 122 – (72 – 62)
9 State yes or no. Do the answers to the following divisions contain remainders?
a 32 766 ÷ 4 b 1136 ÷ 8 c 2417 ÷ 3
10 a Carry out divisibility tests on the given number and fill in the table with ticks or crosses. State
the explanation for each result.
Number
Divisible
by 2
Divisible
by 3
Divisible
by 4
Divisible
by 5
Divisible
by 6
Divisible
by 8
Divisible
by 9
Divisible
by 10
84 539 424
b Use divisibility rules to determine a 10-digit number that is divisible by 3, 5, 6 and 9.
c Determine a six-digit number that is divisible by 2, 3, 5, 6, 9 and 10.
11 Evaluate:
a 25 b 2500 c 52 +122
d 42 − 25 + 72 e 16 × 49 ÷ 4 f 102 ÷ 32 + 42
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580 Chapter 11 Introducing indices
12 Write down the smallest number greater than 100 that is:
a divisible by 2 b divisible by 3 c divisible by 4
d divisible by 5 e divisible by 6 f divisible by 8
g divisible by 9 h divisible by 10 i prime
13 Write down the value of:
a cube root of 1728 b 6 cubed
14 Write down the value of:
a 50 b 2 + 50 c (2 + 5)0
d 2 × 50 e (2 × 5)0 f 20 + 50
15 Simplify the following expressions using a power of 5.
a 512 × 53 b 512 ÷ 53
c (512)3 d 5 × 58
Extended-response question
1 For the following questions, write the answers in index notation (i.e. bx ) and simplify where possible.
a A rectangle has breadth 27 cm and length 125 cm. Determine power expressions for its area and
perimeter.
b A square’s side length is equal to 43. Determine three power expressions for each of the area and
perimeter of this square.
c a × a × a × a × c × c
d 43 + 43 + 43 + 43
e
(3 3 3 )
3
x + x + x
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Semester review 2 581
Chapter 7: Time
Multiple-choice questions
1 An ancient building is dated back to 2500 bc. How old does that make it in 2014? (Note: The
year before 1 ad is 1 bc; there is no year 0.)
A 4514 years B 486 years C 10 000 years D 2500 years E 2014 years
2 Anastasia walks to school in 25 minutes and 54 seconds, then home in 37 minutes and
17 seconds. What is Anastasia’s total walking time?
A 62 min 11 s B 1 h 2 min 11 s C 53 min 11 s
D 17 min 37 s E 1 h 3 min 11 s
3 How many hours behind New South Wales is South Australia?
A 1
2
h B 1 h C 1
1
2
h D 2 h E 2
1
2
h
4 6.1 hours is the same as:
A 6 h 1 min B 6 h 10 min C 6 h 12 min
D 6 h 6 min E 6 h 10 min 10 s
5 When it is 3 a.m. in New York, what time is it in Sydney, using Australian standard time?
A 3 a.m. B 12 noon C 5 p.m. D 3 p.m. E 6 p.m.
Short-answer questions
1 Write the following, using the units shown in brackets.
a 2
1
2
days (h) b 180 min (h) c 3.5 min (min and s)
d 6
1
2
h (h and min) e 9:45 p.m. (24-h time) f 1326 (a.m./p.m.)
g 5.75 h (h and min) h 6.32 h (h, min, s)
2 Calculate these time intervals.
a 2:25 a.m. to 3:10 a.m. b 6:18 p.m. to 8:09 p.m.
c 6 h 40 min 10 s to 7 h 51 min 11 s d 2 h 18 min 50 s to 4 h 10 min 40 s
3 a When it is 9 a.m. UTC, what time is it in the following places?
i New South Wales ii Western Australia
iii Iraq iv Central Greenland
v Alaska vi New Zealand
b When it is 4:20 p.m. in New South Wales during daylight saving time, what time is it in the
following places?
i Western Australia ii South Australia
iii Queensland iv Tasmania
Semester review 2
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Semester review 2
582
4 Use the given train timetable for Richmond to Chatswood to answer the following questions.
Station a.m. p.m.
Richmond 6:37 2:56
Seven Hills 7:21 3:40
Parramatta 7:32 3:52
Central 8:07 4:25
Chatswood 8:35 4:53
a How long does it take to travel from:
i Richmond to Parramatta in the morning?
ii Seven Hills to Chatswood in the morning?
iii Parramatta to Chatswood in the afternoon?
iv Richmond to Chatswood in the afternoon?
b Does it take longer to travel from Richmond to Chatswood in the morning or afternoon?
c Domenic travels from Richmond to Parramatta in the morning, then from Parramatta to
Chatswood in the afternoon. What is Domenic’s total travel time?
5 How many hours and minutes are there between 2:30 p.m. Monday and 11:45 a.m. Tuesday?
6 A busy airport operates 18 hours a day. An aircraft lands every 9 minutes. How many aircraft
arrive at the airport each day?
7 The local time in Sydney and Melbourne is 2 hours behind Auckland and 10 hours ahead of
London. When it is 5 p.m. in Auckland, what time is it in London?
Extended-response question
1 Anchen is researching his next big trip. He plans to travel from Sydney to Cairns, to Alice
Springs, to Perth and then return home.
a It is currently 4:30 p.m. at home (Australian standard time). What is the current time in these
places?
i Cairns
ii Alice Springs
iii Perth
b Anchen leaves Sydney at 0635 hours and arrives in Cairns at 0914 hours. What is the duration
of the fl ight?
c After staying a week in Cairns, Anchen leaves on a 2 hour and 30 minute fl ight to Alice
Springs. If he leaves at 1:30 p.m., what will be the time in Alice Springs when he arrives?
Chapter 8: Algebraic techniques 1
Multiple-choice questions
1 12 − x means:
A 12 less than x B x less than 12 C x has the value of 12
D x is less than 12 E x is more than 12
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2 Double the sum of x and y is:
A 2(x + y) B 2x + y C x + 2y D (x + y)2 E x + y + 2
3 Half the product of a and b is:
A 2ab B
a + b
2
C
ab
2
D
1
2
a +
1
2
b E
a
2
+ b
4 4a + 3b + c + 5b − c is the same as:
A 32ab B 4a + 8b + 2c C 8a + 4b D 64abc E 4a + 8b
5 If a = 3 and b = 7, then 3a2 + 2b is equal to:
A 66 B 95 C 23 D 41 E 20
Short-answer questions
1 Consider the expression 5x + 7y + 3x + 9.
a How many terms are in this expression?
b Can the expression be simplifi ed?
c What is the value of the constant term?
d What is the coeffi cient of y?
2 Write an algebraic expression for each of the following.
a the sum of x and 3 b the product of a and 12
c the sum of double x and triple y d w divided by 6
e double x taken from y
3 Find how many:
a cents are in $m b hours are in x days
c millimetres are in p kilometres d days are in y hours
4 If m = 6, fi nd the value of each of the following.
a m + 7 b 2m − 1 c 6m + 3
d 2(m − 3) e
m+ 6
2
f
m
2
+ 4m − 3
5 Evaluate the expression 3(2x + y) when x = 5 and y = 2.
6 Simplify each of the following.
a 6a + 4a b 7x − 3x c 9a + 2a + a
d m + m − m e 6 + 2a + 3a f x + y + 3x + y
7 a Write an expression for the perimeter of rectangle ABCD.
b Write an expression for the area of rectangle ABCD.
D
3
x + 4
A B
C
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Semester review 2
584
8 Find the missing term.
a 3a × ____ = 18abc b 10ab ÷ ____ = 2a
c 2p + 2p + 2p = 6____
9 Expand:
a 2(a + 3) b 12(a − b) c 8(3m + 4)
10 Write the simplest expression for the perimeter of this fi gure.
2xy
Extended-response question
1 A bottle of soft drink costs $3 and a pie costs $2.
a Find the cost of:
i 2 bottles of soft drink and 3 pies
ii x bottles of soft drink and 3 pies
iii x bottles of soft drink and y pies
b If Anh has $50, fi nd his change if he buys x bottles of soft drink and y pies.
Chapter 9: Equations 1
Multiple-choice questions
1 The solution to the equation x – 3 = 7 is:
A 4 B 10 C 9 D 11 E 3
2 The solution to the equation 2(x + 3) = 12 is:
A 4.5 B 2 C 7 D 6 E 3
3 m = 4 is a solution to:
A 3m + 12 = 0 B
m
4
= 16 C 10 – 2m = 2
D m + 4 = 0 E 3m – 6 = 2
4 The solution to 2p – 3 = 7 is:
A p = 4 B p = 5 C p = 2 D p = 10 E p = 3
5 Ying thinks of a number. If he adds 4 to his number and then multiplies the sum by 5, the result is
35. What equation represents this information?
A y + 9 = 35 B 5y – 4 = 35 C 5y + 4 = 35
D 5( y + 4) = 35 E y + 20 = 35
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Semester review 2 585
Short-answer questions
1 Solve:
a x + 9 = 12 b
x
9
= 12 c x – 9 = 12 d 9x = 12
2 Solve:
a 3x + 3 = 9 b
x
2
+ 6 = 12 c 3(m – 1) = 18
3 If y = mx + b, fi nd:
a y when m = 3, x = 4 and b = 8 b b when y = 20, m = 4 and x = 4
c m when y = 36, x = 3 and b = 12
4 If P = S – C, fi nd:
a P when S = 190 and C = 87 b S when P = 47.9 and C = 13.1
c C when P = 384 and S = 709
5 Use your knowledge of geometry and shapes to fi nd the value of x in each of the following.
a
(x + 2) cm
P = 28 cm
b
17 cm
(2x + 5) cm c
2x° 36°
6 The perimeter of this triangle is 85 cm. Write an equation and then solve it
to fi nd the value of x.
Extended-response question
1 The cost of hiring a hall for an event is $200 plus $40 per hour.
a What is the cost of hiring the hall for 3 hours?
b What is the cost of hiring the hall for 5 hours?
c What is the cost of hiring the hall for n hours?
d If the cost of hiring the hall totals $460, for how many hours was it hired?
Chapter 10: Measurement and computation of length, perimeter and area
Multiple-choice questions
1 17 mm is the same as:
A 0.17 m B 0.17 cm C 0.017 m D 170 cm E 1.7 m
2 0.006 L is the same as:
A 6 mL B 36 mL C 10 mL D 60 mL E 600 mL
2x cm
25 cm
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586 Semester review 2
3 Which of the following shapes has the largest perimeter?
A B C
D E
4 The perimeters of the two shapes shown below are equal. The area of the square is:
A 30 m2 B 7.5 m2
C 56.25 m2 D 120 m2
E 60 m2
5 Which is the correct formula for the circumference of a circle?
A C = 2r B C = r 2 C C = r D C = 2d E C = d 2
Short-answer questions
1 Complete these conversions.
a 5 m = ____ cm b 6 km = _____ m c 1800 mm = _____ m
d 1.7 cm = _____ m e 180 cm = _____ m f 5
1
2
km = _____ m
2 Find the perimeter of each of the following.
a
68 cm
b
1.3 m
4.2 m
c
75 cm
1.5 m
d
55 cm
e 60 cm
40 cm
1.2 m
f 7 m
12 m
20 m
3 Find the area of each of the following.
a
1.3 m
b
3 m
8 m c
15 m
8 m
13 m
5 m
12 m
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Semester review 2 587
d 4.5 m
7 m
12 m
e
12 m
12 m
7 m
f
7 m
10 m
12 m
8 m
4 Find the exact perimeter of the following sectors.
a
12 cm
b
6 cm
c
45°
4 cm
5 Find the exact perimeter of this composite fi gure, in which two semicircles
have been removed from a square.
Extended-response question
1 Robert has been given 36 m of fencing with which to build the largest rectangular enclosure that he
can, using whole number side lengths.
a Draw three possible enclosures and calculate the area of each one.
b What are the dimensions of the rectangle that gives the largest possible area?
c If Robert chooses the dimensions in part b and puts a post on each corner, and then posts every
metre along the boundary, how many posts will he need?
d If it takes 15 minutes to dig each hole for each post, how many hours will Robert spend
digging?
Chapter 11: Introducing indices
Multiple-choice questions
1 The fi rst prime number after 90 is:
A 91 B 92 C 97 D 95 E 93
2 The highest common factor (HCF) of 12 and 18 is:
A 6 B 12 C 4 D 2 E 9
3 2 × 2 × 2 × 3 is the same as:
A 6 × 3 B 23 × 3 C 83 D 63 E 43
8 cm
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588 Semester review 2
4 Evaluating 32 − 25 + 3 gives:
A 8 B 5 C 4 D 17 E 7
5 The number 48 in prime factor form is:
A 24 × 5 B 2 × 3 × 5 C 23 × 32 D 24 × 3 E 23 × 3
Short-answer questions
1 List the factors of:
a 15 b 30 c 100
2 List the fi rst fi ve multiples of:
a 3 b 7 c 11
3 List all factors common to 30 and 36.
4 What is the highest factor common to 36 and 40?
5 Find the value of:
a 112 b 62 × 22 c 33 − 23
6 What is the square root of 14 400?
7 Is the expression 32 + 42 = 3 + 4 true or false?
8 Find the smallest number that must be added to 36 791 so that it becomes divisible by:
a 2 b 3 c 4
9 Simplify the following, and express your answer using indices.
a (85)2 b 85 × 82 c 85 ÷ 82
d 85 × 8 e 85 ÷ 8 f 85 × 85
10 Write down the value of each of the following.
a 80 b 8 × 80 c (8 × 8)0
d 8 + 80 e (8 + 8)0 f 80 + 80
Extended-response question
1 Copy and complete the following pattern.
a (−1)2 = −1 × −1 = 1
(−1)3 = −1 × −1 × −1 = −1
(−1)4 =
(−1)5 =
b What is the value of −1 to the power of 279? How do you know?
c Will the value of −3 to the power of 79 be positive or negative? Explain.
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589
ATanbslwe eorfs Contents
Chapter 1
Pre-test
1 a C b A c D d B
2 a 57 b 116 c 2044 d 11 002
3 a 13 b 37 c 999 d 8000
e 26 f 28
4 a 42, 49, 56, 63, 70, 77, 84 b 54, 63, 72, 81, 90, 99, 108
c 66, 77, 88, 99, 110, 121, 132
5 a 2 b 1 c 3 d 12
6 a 14 b 23 c 119 d 150
e 9 f 32 g 79 h 79
i 30 j 63 k 144 l 88
m 5 n 2 o 11 p 11
7 a 37, 58, 59, 62, 73, 159 b 31, 103, 130, 301, 310
c 13 429, 24 319, 24 913, 24 931, 29 143
8 a 0 b 1 c 1 d 2
e 1 f 2 g 2 h 1
Exercise 1A
1 a Babylonian b Roman c Egyptian
2 a i ii iii iv
b i ii iii
c i I ii V iii X iv L v C
3 5 - 1 = 4
4 a i ii iii iv
b i ii iii iv
c i II ii IX iii XXIV iv CLVI
5 a i 33 ii 111 iii 213 iv 241
b i 12 ii 24 iii 71 iv 305
c i 4 ii 8 iii 16 iv 40
6 a XXXVI b c d DCLXXVIII
7 (21)
8 CLXXXVIII (188)
9 (64)
10 a Roman b Babylonian c Roman
11 a IV b IX c XIV d XIX
e XXIX f XLI g XLIX h LXXXIX
i XCIX j CDXLIX k CMXXII l MMMCDI
12 A separate picture is to be used for each 1, 10, 100 etc. The
number 999 uses 27 pictures.
13 a i ii iii ....
iv v vi ....
b third position = 60 × 60 = 3600 c 216 000
14 Answers may vary.
Exercise 1B
1 a hundreds b thousands c tens d ones
2 a 263 b 7421 c 36 015 d 100 001
3 a B b E c D d A
e C f F g G and H
4 a 7 b 70 c 70 d 700
e 700 f 7000 g 700 h 70 000
5 a 20 b 2000 c 200 d 200 000
6 a true b false c true d true
e false f true g false h true
i false j false k false l true
7 a 1 × 10 + 7 × 1 b 2 × 100 + 8 × 10 + 1 × 1
c 9 × 100 + 3 × 10 + 5 × 1 d 2 × 10
e 4 × 1000 + 4 × 100 + 9 × 10 + 1 × 1
f 2 × 1000 + 3 × 1 g 1 × 10 000 + 1 × 1
h 5 × 10 000 + 5 × 1000 + 5 × 100 + 5 × 10 + 5 × 1
8 a 347 b 9416 c 7020
d 600 003 e 4 030 700 f 90 003 020
9 a 44, 45, 54, 55 b 29, 92, 279, 729, 927
c 4, 23, 136, 951
d 345, 354, 435, 453, 534, 543
e 12 345, 31 254, 34 512, 54 321
f 1001, 1010, 1100, 10 001, 10 100
10 a 6 b 6 c 24
11 27
12 a a × 10 + b × 1
b a × 1000 + b × 100 + c × 10 + d × 1
c a × 100 000 + a × 1
13 Position gives the place value and only one digit is needed
for each place. Hindu–Arabic system also gives us a way to
represent the number zero.
14 a You do not need to write the zeros.
b i 41 × 102 ii 37 × 104 iii 2177 × 104
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c i 38 100 ii 7 204 000 iii 1 028 000 000
d i 1 × 106 ii 1 × 109 iii 1 × 1012
iv 1 × 10100 v 1 × 10googol
Exercise 1C
1 a add, plus, sum
b minus, take away, difference
2 a 10 b 69
c 12 d 20
3 a i 8 ii 27 iii 132
b i 6 ii 16 iii 8
4 a true b true c true
d false e true f false
5 a 18 b 19 c 32
d 140 e 21 f 9
6 a 64 b 97 c 579
d 748 e 948 f 5597
g 378 683 h 591 579 i 201 011
7 a 11 b 36 c 112
d 4 e 3111 f 10 001
8 a 24 b 75 c 95
d 133 e 167 f 297
9 a 24 b 26 c 108
d 222 e 317 f 5017
10 a 51 b 128 c 244
d 119 e 242 f 502
11 a 12 b 27 c 107
d 133 e 14 f 90
g 1019 h 0 i 3
12 38 hours
13 107 runs
14 32 cows
15 29 marbles
16 107 cards
17 a i 1
6 5
2 4 3
ii 5
2 4
3 6 1
b i 6
9
1
4 8 3 5
7
2
ii 9
1
6
7 3 5 8
4
2
18 a Because 3 + 9 is more than 10, so you have to carry.
b Because 8 - 6 is easy, but 1 - 6 means you have to carry.
19 a c - b = a b b - a = c
20 a four ways (totals are 9, 10, 11 and 12)
b Answers may vary.
21 29 and 58
22 a 6
7
2
1
5
9
8
3
4
b 10
9
14
15
11
7
8
13
12
c 15
14
19
20
16
12
13
18
17
d 1
12
8
15
6
10
14
7
11
13 3 2
4
9
5
16
Exercise 1D
1 a 17 b 101 c 144 d 110
e 1005 f 143 g 201 h 1105
2 a 8 b 27 c 67 d 84
e 15 f 92 g 29 h 979
3 a 8 b 1 c 1 d 6
e 2 f 1 g 8 h 3
4 a 87 b 99 c 41 d 86
e 226 f 745 g 1923 h 5080
5 a 161 b 225 c 2229 d 1975
6 a 77 b 192 c 418 d 4208
e 1223 f 1982
7 a 31 b 20 c 19 d 58
e 36 f 112 g 79 h 72
8 a 16 b 47 c 485 d 166
9 1854 sheep
10 576 kilometres
11 a 1821 students
b 79 students
12 a 3 8
+ 5 3
9 1
b 1 1 4
+ 7 7
1 9 1
c 6 7
+ 8 4 7
9 1 4
13 a 6 2
- 2 8
3 4
b 2 6 5
- 1 8 4
8 1
c 3 0 9 2
- 9 2 7
2 1 6 5
14 a i 29 ii 37
b yes
c no
d The balance of -19 + 20 is +1, so add 1 to 36.
15 a Answers may vary.
b Different combinations in the middle column can be used to
create the sum.
16 452 and 526
17 Students’ own answers.
18 a
62
61
66
67
63
59
60
65
64
b
101
112
108
115
106
110
114
107
111
113 103 102
104
109
105
116
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Answers
Exercise 1E
1 a 20, 24, 28 b 44, 55, 66 c 68, 85, 102
2 a true b true
c false d true
e true f true
g false h true
i false
3 a 3 b 0
c 5 d 2
4 a 56 b 54
c 48 d 121
e 72 f 35
g 108 h 39
5 a 57 b 174
c 112 d 266
e 105 f 124
g 252 h 159
6 a 96 b 54
c 96 d 72
7 a 66 b 129
c 432 d 165
e 258 f 2849
g 2630 h 31 581
8 a 235 b 4173
c 3825 d 29 190
9 1680 metres
10 $264
11 116 cards
12 no
13 a 3 9
× 7
2 7 3
b 2 5
× 5
1 2 5
c 7 9
× 3
2 3 7
d 1 3 2
× 8
1 0 5 6
e 2 7
× 7
1 8 9
f 3 9
× 9
3 5 1
g 2 3 2
× 5
1 1 6 0
h 3 1 4
× 7
2 1 9 8
14 twelve ways
15 a 3 × 21 b 9 × 52
c 7 × 32 d 5 × 97
e a × 38 f a × 203
16 three ways: (0, 1), (1, 5), (2, 9). You cannot carry a number
to the hundreds column.
17 a Answers may vary; e.g. 2 1 7
× 7
1 5 1 9
b Answers may vary; e.g. 2 9 5
× 3
8 8 5
18 6, 22
Exercise 1F
1 a 2 b 0 c 0 d 4
2 a 100 b 10 c 10 000
3 a incorrect, 104 b incorrect, 546
c correct d incorrect, 2448
4 a 400 b 290 c 1830 d 4600
e 50 000 f 63 000 g 14 410 h 29 100 000
5 a 340 b 1440 c 6440 d 22 500
e 41 400 f 460 000 g 63 400 h 9 387 000
6 a 407 b 1368 c 1890 d 9416
e 2116 f 40 768 g 18 620 h 33 858
7 a 209 b 546 c 555 d 2178
8 $2176
9 $6020
10 86 400 seconds
11 a 2 3
× 1 7
1 6 1
2 3 0
3 9 1
b 1 4 3
× 1 3
4 2 9
1 4 3 0
1 8 5 9
c 4 9
× 3 7
3 4 3
1 4 7 0
1 8 1 3
d 1 2 6
× 2 1
1 2 6
2 5 2 0
2 6 4 6
12 60 480 degrees
13 one number is a 1
14 a 39 984 b 927 908 c 4 752 188 d 146 420 482
15 a 1600 b 780 c 810 d 1000
16 a 84 000 b 3185
17 123, 117
18 a 100, 121, 144, 169, 196, 225, 256, 289
b 961 c 9801
Exercise 1G
1 a 2 b 3 c 7 d 12
2 a 1 b 2 c 2 d 5
3 a 1 b 1 c 5 d 5
4 a 4 b 3 c 6 d 5
e 7 f 9 g 8 h 11
5 a 21 b 19 c 19 d 41
e 29 f 21 g 302 h 98
6 a 22 b 31 c 17 d 7
7 a 26 b 1094 c 0 d 0
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8 a 23 rem. 2 b 13 rem. 1 c 69 rem. 1 d 41 rem. 1
e 543 rem. 1 f 20 333 rem. 2
g 818 rem. 3 h 10 001 rem. 0
9 a 131
2
4
or 131
1
2
b 241
4
7
c 390
5
6
d 11 542
1
8
10 13 packs
11 124 packs
12 a $243 b $27
13 67 posts
14 15 taxis
15 19 trips; any remainder needs 1 more trip
16 2
36
3
9
6
4
12
1
18
17 a 1, 12 b 13, 7 c 4, 5
18 $68
19 a a b 0 c 1
20 8 or 23
21 a = b or a = -b
22 a 33
8
11
b 54
8
17
c 31
1
13
d 108
1
15
e 91
16
23
f 123
26
56
or 123
13
28
23 a 3 rem. 269 b 11 rem. 5 c 18 rem. 625
24 1
40
14
6
28
5
20
2
24
12 8 7
56
3
4
10
Exercise 1H
1 a up b down
c up d up
e down f down
2 a larger b smaller
c smaller d larger
3 a 60 b 30
c 120 d 190
e 200 f 900
g 100 h 600
i 2000
4 a 20 b 30
c 100 d 900
e 6000 f 90 000
g 10 000 h 10
5 a 130 b 80 c 150
d 940 e 100 f 1000
g 1100 h 2600 i 1000
6 a 120 b 160 c 100 d 12
e 40 f 2000 g 4000 h 100
7 a 1200 b 6300
c 20 000 d 8 000 000
e 5 f 16
g 10 h 25
8 Answers will vary.
9 ≈ 2100 scoops
10 ≈ 1200 sheep
11 ≈ 8 people
12 a 200 b 100 000 c 800 d 3 000 000
13 a i larger ii larger
b i larger ii larger
c i smaller ii smaller
d i larger ii larger
14 a i 9 ii 152 iii 10 iv 448
b One number is rounded up and the other is rounded down.
c i 3 ii 3 iii 1 iv 2
d If the numerator is decreased, then the approximation
will be larger. If the denominator is increased then the
approximation will also be smaller. If the opposite occurs the
approximation will be larger.
Exercise 1I
1 a addition b division
c multiplication d multiplication
e division f addition
g division h multiplication
i division j subtraction
k multiplication l division
2 a true b false c false d true
3 a 23 b 21 c 0 d 18
e 32 f 2 g 22 h 22
i 38 j 153 k 28 l 200
4 a 10 b 3 c 2 d 22
e 2 f 9 g 18 h 3
i 10 j 121 k 20 l 1
5 a 48 b 18 c 13 d 28
e 22
6 a 27 b 10 c 8 d 77
e 30 f 21 g 192
7 75 books
8 45 TV sets
9 a (4 + 2) × 3 = 18 b 9 ÷ (12 – 9) = 3
c 2 × (3 + 4) – 5 = 9 d (3 + 2) × (7 – 3) = 20
e (10 – 7) ÷ (21 – 18) = 1
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Answers
f (4 + 10) ÷ (21 ÷ 3) = 2
g [20 – (31 – 19)] × 2 = 16
h 50 ÷ (2 × 5) – 4 = 1
i (25 – 19) × (3 + 7) ÷ 12 + 1 = 6
10 fi rst prize $38, second prize $8
11 a no b yes c no d yes
e yes f no g no h yes
i yes
12 a no b yes
c no d yes
13 a b b 0
c a + 1 d b
14 a multiply by 2 and add 1
b multiply by 3 and subtract 3
c multiply by itself and add 1
Puzzles and challenges
1 The two people pay $24 each, which is $48 in total. Of
that $48 the waiter has $3, leaving a balance of $45 for
the bill.
2 2 9 4
3
6 1 8
7 5
3 Answers may vary; e.g.
2
4
8
3 6 7 1
5
9
or 2
7
5
3 4 9 1
8
6
4 one way is (2 + 7) × 11 + 4 - 3
5
5 4 6 1 3 2 9 8 7
2 8 3 7 5 9 6 1 4
7 9 1 6 8 4 3 2 5
8 2 4 5 1 3 7 6 9
1 3 7 9 2 6 4 5 8
9 6 5 4 7 8 1 3 2
3 7 9 2 6 5 8 4 1
6 1 2 8 4 7 5 9 3
4 5 8 3 9 1 2 7 6
1 7 5 4 6 9 3 8 2
2 9 4 1 3 8 6 5 7
8 6 3 7 5 2 9 1 4
3 4 1 2 7 6 8 9 5
9 2 8 5 4 3 7 6 1
7 5 6 8 9 1 2 4 3
4 3 9 6 2 5 1 7 8
6 1 7 3 8 4 5 2 9
5 8 2 9 1 7 4 3 6
Multiple-choice questions
1 B 2 C
3 E 4 A
5 B 6 A
7 D 8 C
9 B 10 A
Short-answer questions
1 a i ii iii
b i ii .... iii
c i XIV ii XL iii CXLVI
2 a 50 b 5000 c 50 000
3 a 459 b 363 c 95 d 217
4 a 128 b 2324 c 191 d 295
5 a 95 b 132 c 220 d 41
e 33 f 24 g 29 000 h 10 800
i 14 678
6 a 1413 b 351 c 46
5
7
d 7540
1
2
7 a 2 2 3
+ 7 3 8
9 6 1
b 7 2 9
- 4 7 3
2 5 6
c 5 3
× 2 7
3 7 1
1 0 6 0
1 4 3 1
d 1 8 3
9 4 1 1 5
with no
remainder
5)
8 a 70 b 3300 c 1000
9 a 800 b 400 c 5000 d 10
10 a 24 b 4 c 14
d 20 e 0 f 13
Extended-response questions
1 a 646 loads b 9044 kilometres
c $36 430 d $295
2 a 3034 sweets b 249
c liquorice sticks, 6 d yes (124)
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Chapter 2
Pre-test
1 a III b IV c V
d II e I f VI
2 a 90° b 180° c 360° d 270°
e 45° f 315°
3 a 60° b 140° c 125°
4 a no b yes c no
5 a 80° b 150° c 150° d 255°
Exercise 2A
1 a P b N
A
c A
B C
d T
S
e
f
B
C
A
2 a all in a straight line b all meet at the same point
3 a ray b line c segment
4 a line b plane c plane d point
e line f plane g line h point
5 a point T b line CD c angle BAC
d plane e ray PQ f segment ST
6 a ∠BOC b ∠BAC c ∠BEA
d ∠AOC e refl ex ∠BCD
7 a segments AB, BC, AC, BD, CD; angles ABD, CBD, CDB,
BCD
b segments PQ, PR, QR, RS, QS; angles PQS, RQS, QRS, RSQ
8 a C, B and D b A, C and D
9 a no b yes
10 a AB = BC = CD = DA
∠DAB = ∠BCD
∠ABC = ∠ADC
b AB = AD
BC = CD
∠ADC = ∠ABC
11 a 8 b 14
12 10
13 Missing numbers are 0, 1, 3, 6, 10, 15. For 5 points, add 4 to
the previous total; for 6 points, add 5 to the previous total, and
so on.
14 All segments should intersect at the same point; i.e. are
concurrent.
15 a yes b no
16 number of segments =
2
( − 1) n
n
Exercise 2B
1 Answers may vary.
2 a 2 b 3 c 4
3 a 50° b 145° c 90° d 250°
4 a acute, 40° b acute, 55° c right, 90°
d obtuse, 125° e obtuse, 165°
f straight, 180° g refl ex, 230°
h refl ex, 270° i refl ex, 325°
5 a i 20° ii 25° iii 35°
iv 40° v 35°
b ∠AOF (155°)
6 a
40°
b
75°
c
90°
d
135°
e 175° f 205°
g 260° h 270°
i 295° j 352°
7 a 29° b 55° c 35° d 130°
8 Yes, the two smaller angles make up the larger angle.
9 a 180° b 360° c 90° d 270°
e 30° f 120° g 330° h 6°
i 54° j 63° k 255° l 129°
10 a 180° b 90° c 120° d 30°
11 a i 70° ii 70° iii 90° iv 90°
v 80° vi 80°
b no
c Subtract 360° until you have a number that is less
than 180° and then change the sign if it’s negative.
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Answers
12 Use the revolution to get 360° - 60° = 300°.
13 a 115° b 127.5° c 85°
d 77.5° e 122° f 176.5°
Exercise 2C
1 a, b angles should add to 90° c complementary
2 a, b angles should add to 180° c supplementary
3 a, b angles should add to 360° c vertically opposite angles
4 a ∠BOC b ∠AOD and ∠BOC c ∠COD
5 a 60 b 15 c 135 d 70
e 40 f 115 g 37 h 240
i 130
6 a N b N c S d N
e C f C g C h S
7 a EF ⊥ GH b ST ⊥ UV c AX ⊥ NP
8 a 30 b 75 c 60
d 135 e 45 f 130
9 a No, should add to 90. b Yes, they add to 180.
c Yes, they add to 360. d Yes, they are equal.
e No, they should be equal. f No, should add to 360.
10 a 30 b 60 c 60
d 45 e 180 f 36
11 24°
12 a a + b = 90 b a + b + c = 180
c a + b = 270
13 Only one angle – the others are either supplementary or
vertically opposite.
14 a 360° b 72 c 108
d Regular shape a b
triangle
square
pentagon
hexagon
heptagon
octagon
120
90
72
60
(360 ÷ 7)
45
60
90
108
120
(900 ÷ 7)
135
Exercise 2D
1 a 4 b no
2 a 2 b yes
3 a equal b supplementary
c equal d equal
4 a ∠DEH b ∠BEF c ∠DEB d ∠CBG
5 a ∠FEG b ∠DEB c ∠GEB d ∠ABC
6 a ∠CFG b ∠BCF
7 a 130, corresponding b 70, corresponding
c 110, alternate d 120, alternate
e 130, vertically opposite f 67, vertically opposite
g 65, cointerior h 118, cointerior
i 100, corresponding j 117, vertically opposite
k 116, cointerior l 116, alternate
8 a a = 70, b = 70, c = 110
b a = 120, b = 120, c = 60
c a = 98, b = 82, c = 82, d = 82
d a = 90, b = 90, c = 90
e a = 95, b = 85, c = 95
f a = 61, b = 119
9 a No, corresponding angles should be equal.
b Yes, alternate angles are equal.
c Yes, cointerior angles are supplementary.
d Yes, corresponding angles are equal.
e No, alternate angles should be equal.
f No, cointerior angles should be supplementary.
10 a 35 b 41 c 110
d 30 e 60 f 141
11 a 65 b 100 c 62 d 67
e 42 f 57 g 100 h 130
i 59
12 a 12 angles b two angles
13 120
14 a i The angle marked a is alternate to the 20° angle.
ii The angle marked b is alternate to the 45° angle.
b i a = 25, b = 50 ii a = 35, b = 41
iii a = 35, b = 25
15 a a = 120, b = 120 b 60
c opposite angles are equal
16 a Both angles do not add to 180°.
b The cointerior angles do not add to 180°.
c Alternate angles are not equal.
17 a i ∠BDE, alternate ii ∠BED, alternate
b add to 180°
c Three inside angles of a triangle add to 180°, which is
always true.
18 Each triangle adds to 180°, so the total is 360°.
Exercise 2E
1 a a = 65, b = 115 b a = 106, b = 106
c a = 55, b = 55
2 a ∠BED, a = 30 b ∠EBD, a = 70
c ∠ADC, a = 50
3 a 60 b 120 c 115 d 123
e 50 f 80 g 60 h 65
i 45 j 60 k 55 l 335
4 a 130° b 120° c 55°
d 75° e 90° f 75°
5 a 50 b 150 c 60
6 a 1 b 2 c 2
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7 a 30 b 60 c 40
d 30 e 120 f 10
8 a 60 b 45 c 12
9 a 110 b 250 c 40
d 110 e 40 f 300
Exercise 2F
1 a–e
A minor arc
O B
2 a radius b chord c minor arc
d centre e tangent f diameter
3 Your two circles should just touch.
4 ∠BAC and ∠ABC should be equal.
5 ∠AEC should be 90°.
6 ∠BAD should be 60°.
7 ∠AOE and ∠BOE should be equal.
8 a Construct the two circles so that they have the same radius.
b Use two circles of the same size. Point E should be the
midpoint of AB.
9 a Construct a 60° angle (see Question 5) and then bisect this
angle by constructing the angle bisector to form two 30°
angles (see Question 6).
b Construct an angle bisector of one of the 30° angles from
part a.
10 a First, construct a 90° angle by constructing a perpendicular
line and then construct the angle bisector of the 90° angle.
b Construct the angle bisector of one of the 45° angles from
part a.
11 No, the circles must overlap.
12 a Yes, follow the procedure as in Question 7.
b Yes, construct as for an acute or obtuse angle and draw the
angle bisector on the reverse side of the vertex.
13 a
A
C B
b
A
B
c
A B
Exercise 2G
1–9 Answers may vary and can be checked by testing the
properties of the constructions.
Puzzles and challenges
1 77.5°
2 720°
3
4
5 6 6 143.5°
Multiple-choice questions
1 C 2 B 3 D 4 C 5 D
6 E 7 C 8 A 9 E 10 B
Short-answer questions
1 a segment CD b angle AOB c point P
d plane e ray AC f line ST
2 a acute, 35° b obtuse, 115° c reflex, 305°
3 a 180° b 90° c 90° d 150°
4 a 20 b 230 c 35 d 41 e 15
f 38 g 60 h 120 i 30
5 a a° and b° b a° and d° c a° and c° d b° and c°
e c° and d° or b° and d°
6 a Yes, corresponding angles are equal.
b No, alternate angles should be equal.
c No, cointerior angles should be supplementary.
7 a 100 b 95 c 51
d 30 e 130 f 78
8 a 145° b 140° c 100°
9 a The angle should be divided in half.
b The angle should be 90°.
Extended-response questions
1 a i 32° ii 32° iii 148° iv 58°
b i corresponding
ii cointerior iii supplementary
c i 21° ii 159° iii 69°
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Answers
2 a 12 pieces b 30
c i 15 ii 22.5 iii 20 iv 24
Chapter 3
Pre-test
1 a < b < c < d >
2 a 5°C b 2°C c 7°C
3 a 30 b 77 c 39 d 112
4 a 5 b 11 c 7 d 17
5 a 22 b 2 c 52 d 4
e 30 f 18
6 a positive b negative c negative d positive
7 A(1, 1), B(3, 2), C (2, 3)
8
2
3
1
1
D
A
E
B
C
0 2 3 4 5
4
5
y
x
Exercise 3A
1 a -2, 2 b 0, 2 c -7, -5 d -5, -3, 0
2 a -2 b -6 c 3 d 7
e 15 f -21 g -132 h 1071
3 a greater b less c greater d less
4 a
-2 -1 0 1 2
b
-5 -4 -3 -2 -1 0 1
c
-10 -9 -8 -7 -6
d
-32 -31 -30 -29 -28 -27 -26 -25
5 a -2, -1, 0, 1, 2, 3, 4 b -7, -6, -5, -4, -3, -2, -1, 0
c -2, -1, 0, 1 d -4, -3, -2, -1, 0
e -3, -2, -1, 0, 1, 2, 3 f -9, -8, -7, -6, -5, -4
6 a < b > c > d <
e > f < g < h >
i < j > k < l >
7 a 4°C b -1°C c -7°C d -25°C
8 a -10, -6, -3, -1, 0, 2, 4 b -304, -142, -2, 0, 1, 71, 126
9 a 0, -1, -2 b -2, 0, 2 c -5, -10, -15
d -44, -46, -48 e -79, -75, -71
f -101, -201, -301
10 a $5 b -$10
11 a -50 m b -212.5 m c 0 m
12 a 2 b 4 c 4 d 7
e 3 f 3 g 6 h 44
13 a -2 b 1 c -1
d -7 e -51 f 357
Exercise 3B
1 a right b right c left d left
2 a D b A c B d C
3 a 1 b 3 c 2 d 1
e -1 f -3 g -2 h -2
i -4 j -8 k -1 l 2
m 2 n -9 o 6 p -31
4 a -2 b -1 c -8 d -19
e -4 f -10 g -15 h -7
i -41 j -12 k -22 l -47
m -300 n -100 o -93 p -634
5 a 5 b 9 c 5 d 2
e 5 f 7 g 3 h 10
i 5 j 16 k -4 l -5
m -6 n -13 o -30 p -113
6 a 5 b -9 c 1 d -13
e 1 f -22 g -32 h -4
7 a $145 b $55 c $5250
8 a 3°C b -3°C c -46°C
9 69°C
10 a 59 m b 56 m
11 Answers may vary.
12 a i positive ii positive iii negative iv zero
b i no ii yes
13 Other combinations may be possible.
a –, + b +, –, – c +, +, –, +
d –, +, +, +, – e +, +, – f –, +, –, –
Exercise 3C
1 b 6, -3 c -3, 1 d -11, -7 e 3, 5
f 6, -2 g -3, 7 h -11, -7
2 a 4 b subtracting c -5
d subtracting e 2 f adding
3 a false b true c true d false
e false f true g false h false
4 a 1 b 5 c 6 d 2
e -3 f -5 g -2 h -4
i -3 j -22 k -35 l -80
m -10 n -29 o -50 p -112
5 a 5 b 11 c 50 d 90
e -4 f -3 g -5 h -34
i 2 j 1 k 0 l 8
m 28 n 34 o -12 p -76
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6 a -3 b -10 c -4 d 4
e -1 f 4 g -1 h -5
i -4 j 4 k 2 l -24
m -6 n -5 o 2 p 4
7 a 0 b -5 c 8 d 12
e -9 f 5 g -6 h -91
i -15 j 6 k 17 l 11
8 -143 m
9 -$35 000
10 -$30
11 a i $8000 ii -$6000
b $2000
12 a -2
8
-3
0
1
2
5
-6
4
b
-13 -11
-3
-14
-10
-9
-6
-17
-7
13 a 3 + 4 b -2 + (-9) c 5 - (-2)
d 1 - (-2) e a + (-b) f a - (-b)
14 a 4 b -1 c -3
15 a i no ii yes
b i yes ii no
c Yes, if b < a then subtracting b takes the result to a
number bigger than zero.
16 a 5050 b -50 c -100
Exercise 3D
1 a × -2 -1 0 1 2
-2 4 2 0 -2 -4
-1 2 1 0 -1 -2
0 0 0 0 0 0
1 -2 -1 0 1 2
2 -4 -2 0 2 4
b × -4 -2 0 2 4
-4 16 8 0 -8 -16
-2 8 4 0 -4 -8
0 0 0 0 0 0
2 -8 -4 0 4 8
4 -16 -8 0 8 16
2 a 2 b -3 c -4 d -4
3 a positive b positive c negative
d positive e positive f negative
4 a -15 b -10 c -6 d -54
e 32 f 28 g 144 h -99
i -39 j -84 k 38 l -108
m 66 n -45 o 63 p 72
5 a -2 b -12 c -2 d -4
e 3 f 1 g -5 h -19
i -7 j -12 k -68 l 8
m 12 n 13 o -13 p 13
6 a 24 b 15 c -4 d 5
e 1 f -10 g 72 h 18
i 1 j -1 k -69 l -3
7 a -7 b 4 c -4 d 8
e 27 f -140 g 2 h -3
i -3 j -1 k -2 l 40
8 a -3 b -3 c 8 d 31
e 3 f 5 g -30 h -100
9 a 4 b 1 c 81 d 100
e 36 f 64 g 9 h 2.25
10 a (1, 6), (2, 3), (-1, -6), (-2, -3)
b (1, 16), (2, 8), (4, 4), (-1, -16), (-2, -8), (-4, -4)
c (-1, 5), (-5, 1)
d (-1, 24), (-24, 1), (-2, 12), (-12, 2), (-3, 8), (-8, 3),
(-4, 6), (-6, 4)
11 a ×, ÷ b ÷, × c ×, × d ÷, ×
12 a (4, -2), (-4, 2) b (33, -3), (-33, 3)
13 a i -8 ii 64 iii -27 iv 81
b parts ii and iv, even number of negative factors
c parts i and iii, odd number of negative factors
14 a -ab b -ab c ab
15 Answers will vary. You may use a calculator to check that your
answers are correct.
Exercise 3E
1 a division b multiplication c multiplication
d division e addition f subtraction
g multiplication h multiplication i multiplication
2 a true b false c true
d false e false f true
3 a -7 b 7 c 19 d 9
e 16 f 14 g 6 h -32
i -5 j -4 k -18 l -4
m -10 n 4 o 0
4 a -10 b -2 c -6 d 1
e 2 f 9 g 1 h 4
i -14 j -20 k 2 l -5
m 8 n -6 o -12
5 $528
6 -$50
7 a (-2 + 3) × 8 = 8 b -10 ÷ (4 + 1) = -2
c (-1 + 7) 2 - 15 = -3 d (-5 − 1) ÷ (-6) = 1
e (3 − 8) ÷ 5 + 1 = 0 f 50 × (7 − 8) × (-1) = 50
g -2 × (3 − (-7)) - 1 = -21 h (-3 + 9) ÷ (-7 + 5) = -3
i (32 − (−8)) ÷ (-3 + 7) = 10
8 three answers (-10, -21, -31)
9 a no b no c yes
d yes e no f yes
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Answers
10 a true b true c true
d true e false f false
11 a i negative ii negative iii negative
iv positive v negative vi negative
b If there is an even number of negative factors the result will
be positive, if odd then negative
12 a 4 b 4 c -4
d -32 e -32 f 32
g 2 h -2 i -1
13 Kevin should have typed (-3)4 to raise -3 to the power of 4.
-34 is -1 × 34.
Exercise 3F
1 a D b B
c A d C
e E f H
g F h G
2 a 9 b 18
c 15 d 6
e 10 f 2
g 1
3 A ( 2, 1), B (3, -2), C (-1, -4), D (-2, 2), E (4, 3), F (2, -3),
G (-3, -1), H (-4, 4)
4a,b
2
3
1
-2
-3
-4 -3 -2 -1-1
-4
O 1 2 3 4
4
y
x
5 A(3, 0), B (0, -2), C (-1, 0), D (0, 4), E (0, 2), F (1, 0),
G (0, -4), H (-3, 0)
6 a
2
3
1
-2
-3
-4 -3 -2 -1-1
-4
O 1 2 3 4
4
y
x
b They lie in a straight line.
7 a
2
3
1
-2
-3
-4 -3 -2 -1-1
-4
-5
-6
-7
O 1 2 3 4
4
5
y
x
b They lie in a straight line.
8 a triangle b rectangle c trapezium d kite
9 a 4 square units b 6 square units
c 4 square units d 15 square units
10 28 km
11 y = 2
12 y = 3
13 a B b C c B, A, C
14 a i 1 and 4 ii 4 iii 3 and 4 iv 3
b y
x
O
15 a C (2, 4), D (-1, 3) b (2, 4), (-1, 3), (1, -3), (4, -2)
c H (-2, 2)
Puzzles and challenges
1 a -3
-4
1
2
-2
-6
-5
0
-1
b -9
2
-2
5
-4
0
4
-3
1
3 -7 -8
-6
-1
-5
6
c −14
−3
−7
0
−9
−5
−1
−8
−4
−2 −12 −13
−11
−6
−10
1
2 a -81, 243, -729 b 4, -2, 1 c -10, -15, -21
d -8, -13, -21
3 a 0 b -153
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4 a -3 × (4 + -2) = -6
b -2 × 5 × -1 + 11 = 21 or -2 × 5 ÷ -1 + 11 = 21
c (1 - 30 ÷ -6) ÷ -2 = -3 or 1 × 30 ÷ -6 - -2 = -3
5 a 11 and -3 b 21 and -10
6 a -3
2 1
-2 0 -1
b 2
-3 -2
1 -1 0
c 1
0 -2
-3 2 -1
Multiple-choice questions
1 C 2 E 3 B 4 D 5 C
6 A 7 E 8 C 9 B 10 C
Short-answer questions
1 a < b < c > d <
2 a -5 b -2 c -15 d 1
e -2 f -5 g 12 h -18
i -6 j 5 k -11 l 5
3 a -1 b -9 c -1 d 2
e -21 f -2 g -87 h 30
4 a -10 b -21 c 30 d -5
e -3 f 4 g 1 h -8
5 a -2 b -50 c -36 d -1
6 a -37 b 8 c -3
d 1 e 56 f 80
7 a -3 b -7
8 A(3, 0), B (2, 3), C (-1, 2), D (-4, -2), E (0, -3), F (4, -4)
Extended-response questions
1 a 16°C b -31°C c 8°C d 19°C e 27°C
2 rocket
Chapter 4
Pre-test
1 C
2 C
3 B
4 a 3
4
b 1
2
c
2
3
d 4
5
5 a 2
3
4
b 1
1
2
c 9
1
2
d 5
1
4
6 one-quarter of a block
7 a 2, 2
1
2
, 3 b 4
3
,
5
3
,
6
3
c 5
4
,
6
4
,
7
4
d 2
3
,
5
6
, 1
8 a 3 ×
1
2
b 4 ×
3
4
c
5
6
× 1 =
5
6
d
6
8
÷ 1 =
6
8
9 a $7.50 b $40 c 75c d $2
10 a true b true c true d false
Exercise 4A
1 a M b N c F d N
e F f F g M h F
i N j M k N l F
2 a F b N c M d N
e N f N g N h M
i F j M k F l N
3 a 1, 2, 5, 10 b 1, 2, 3, 4, 6, 8, 12, 24
c 1, 17 d 1, 2, 3, 4, 6, 9, 12, 18, 36
e 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
f 1, 2, 3, 6, 7, 14, 21, 42
g 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
h 1, 2, 3, 4, 6, 12
i 1, 2, 4, 7, 14, 28
4 a 5, 10, 15, 20, 25, 30 b 8, 16, 24, 32, 40, 48
c 12, 24, 36, 48, 60, 72 d 7, 14, 21, 28, 35, 42
e 20, 40, 60, 80, 100, 120 f 75, 150, 225, 300, 375, 450
g 15, 30, 45, 60, 75, 90 h 100, 200, 300, 400, 500, 600
i 37, 74, 111, 148, 185, 222
5 a 3, 18 b 5 c 1, 4, 6, 9, 12, 24
d 3, 4, 5, 8, 12, 15, 24, 40, 120
6 a 22 b 162 c 21 d 117
7 a 24 b 1 c 1, 4, 9, 16, 25
8 a 12 × 16 b 21 × 15 c 12 × 15
d 11 × 11 e 12 × 28 or 24 × 14 or 21 × 16
f 19 × 26
9 a 20 min b 5 laps c 4 laps
10 a 25 b 0 c 23
d 2, 7, 12, 17, 37, 47, 62, 87, 137, 287 e 2
11 a false b true c false
d false e true
12 a 840 b 2520
c Answers will vary but should include cancelling out common
factors.
13 a 1, 2, 4, 5, 10, 20, 25, 50, 100
b 1, 2, 4, 5, 10, 20, 25, 50, 100
14 Answers may vary, but they should be multiples of 9.
15 The larger number gives the reply. Any number is a multiple
of its factors. So the answer is ‘yes’.
16 Check the output each time.
Exercise 4B
1 a 1, 2, 4 b 4
2 Factors of 18 are 1, 2, 3, 6, 9 and 18.
Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30.
Therefore, the HCF of 18 and 30 is 6.
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Answers
3 a 24, 48 b 24
4 Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81 and 90.
Multiples of 15 are 15, 30, 45, 60, 75, 90, 105 and 120.
Therefore, the LCM of 9 and 15 is 45.
5 a 1 b 1 c 2
d 3 e 4 f 15
g 50 h 24 i 40
j 25 k 21 l 14
6 a 10 b 3 c 1
d 1 e 8 f 12
7 a 36 b 21 c 60
d 110 e 12 f 10
g 36 h 18 i 60
j 48 k 132 l 105
8 a 30 b 84 c 12
d 45 e 40 f 36
9 a HCF = 5, LCM = 60 b HCF = 12, LCM = 24
c HCF = 7, LCM = 42 d HCF = 9, LCM = 135
10 312
11 9
12 LCM = 780, HCF = 130
13 a 12 min
b Andrew 9 laps, Bryan 12 laps, Chris 6 laps
c 3 times (including the fi nish)
d Answer will vary.
14 a 8, 16 b 24, 32
15 1 and 20; 2 and 20; 4 and 20; 5 and 20; 10 and 20; 4 and 5;
4 and 10.
16 a 2520 b 2520
c Identical answers; 2520 is already divisible by 10, so
adding 10 to list does not alter LCM.
d 27 720 (23 × 5 × 7 × 9 × 11)
Exercise 4C
1 a 9 b 7
2 proper: b, e, f, g
improper: a, c, h, i, k, l
whole numbers: d, j, k
3 A a 4 b 1 c i 4 ii 1 iii
1
4
B a 8 b 7 c i 8 ii 7 iii
7
8
C a 3 b 2 c i 3 ii 2 iii
2
3
D a 12 b 5 c i 12 ii 5 iii
5
12
4 d, g, j, l, m
5 a
0 1 1
7
2
7
3
7
4
7
5
7
6
7
b
0 1 1 2
3
2
3
4
3
5
3
c
0 1 1
6
2
6
3
6
4
6
5
6
d
0 1 1
4
2
4
3
4
5
4
6
4
7
4
2 9
4
10
4
11
4
3
e
−2 −1 0 1 2
11
5
8
5 −
f
-3 -11 -2 -1
4
-10
4
-9
4
-5
4
-7
4
-6
4
-1
4
-2
4
-3
4
0
6 Answers may vary.
a
b
c
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602
7 a 7
5
,
8
5
,
9
5
b 9
8
,
10
8
,
11
8
c 5
3
,
6
3
,
7
3
d
7
7
,
6
7
,
5
7
e
1
2
,
1
2
,
3
2
− − f − − −
2
4
7
4
12
4
, ,
8 a 1
1
2
,
3
2
; 3
1
2
,
7
2
; 5,
10
   2 = = =
b
1
5
;
4
5
; 2
1
5
,
11
5 = = =
c
3
7
; 1
4
7
,
11
7
; 2
2
7
,
16
7 = = =
d 2
3
;
1
2
;
2
3 = − = − =
9 division
10 a
6
11
b
4
8
1
2
= c
7
12
d
5
6 e
3
12
1
4
= f
5
9
11 a
12
43
b
13
15
c
11
12
d
1
12
e
2
11
f
144
475
g 7
20
h
1
4
i
3
7
12 Proper fraction: When you have part of a whole, and therefore
you have a numerator that is smaller than the denominator.
Improper fraction: It is called improper because it is impossible
to break a whole into parts and end up with more than a whole.
An improper fraction is when the numerator is greater than the
denominator.
13 c
14 a i
1
5
ii 50 mL b i 40 mL ii
4
25
c i 32 mL ii
16
125
d i 90 mL ii
9
25
e i 122 mL ii
61
125 f Approximately, yes they will.
Exercise 4D
1 3
6
2
4
11
22
5
10
, , , 2
4
10
,
16
40
,
2
5
,
80
200
3 a 2, 12, 10, 20, 300 b 1, 3, 24, 20, 40
4 a
1
5
b
1
6
c
2
3
d
3
4
5 a 10, 10, 10, 1 b 2, 2, 2, 2
c 4, 4, 4, 7 d 3, 3, 3, 3, 5
6 Answers may vary.
a
2
4
,
3
6
,
4
8
,
10
20
b
2
8
,
3
12
,
4
16
,
10
40
c
4
10
,
6
15
,
8
20
,
10
25
d
6
10
,
12
20
,
24
40
,
48
80
e
4
18
,
6
27
,
8
36
,
20
90
f
6
14
,
9
21
,
12
28
,
30
70
g
10
24
,
50
120
,
500
1200
,
5000
12 000
h
6
22
,
9
33
,
12
44
,
30
110
7 a 9 b 50 c 33 d 56
e 8 f 2 g 12 h 39
i 35 j 200 k 105 l 2
8 a ≠ b = c ≠ d ≠ e =
f = g = h = i =
9 a 3
4
b 2
3
c 1
3
d
4
11
e
2
5
f
1
11
g
1
7
h
1
3
i
7
9
j
3
8
k
5
6
l
5
1
= 5
10 a
7
14
1
2
= b
12
16
3
4
= c
4
42
2
21
= d
7
63
1
9
=
11
2
9
12 a 35 b 45 c 30 d 24
13 Justin 4, Joanna 3, Jack 5
14 a 75 b
1
3 c 150
15 Answers may vary;
6
10 ; The number of answers is infinite
because there are an infinite number of equivalent fractions that
can be created simply by multiplying the numerator and the
denominator by a common value.
16 Answers may vary.
a
b 12 quavers (eighth notes) to a bar
c
d
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Answers
e Each dot increases the length of the note by another 50%.
e.g. • =
3
4
note (value = 3 beats)
= half note + 50%
= half note + quarter note
Exercise 4E
1 a 2 and 3 b 11 and 12 c 36 and 37
2 a 24 b 360 c 60 d 24
3 a b
c
d
4 a 8 b 12 c 28 d 44
e 17 f 7 g 10 h 24
5 a A 7
1
2
, B 10
1
2
b C 1
2
3
, D 2
1
3
, E 4
2
3
c F 23
1
5
, G 24
2
5
, H 25
4
5
, I 26
1
5
d J
1
3
− , K
2
3
− , L 1
2
3
− , M 2
1
3
−
6 a Answers may vary.
b Answers may vary.
7 a
11
5
b
8
5
c
10
3
d
17
3
e
29
7
f
24
7
g 5
2
h
13
2
i
22
5
j
23
2
k
42
5
l
103
10
m
55
9
n
25
9
o
42
8
p
21
8
q
23
12
r
38
11
s
53
12
t
115
12
u
115
20
v
803
100
w
643
10
x
104
5
y
103
25
8 a 1
2
5
b 1
1
3
c 1
2
3
d 1
3
4
e 3
2
3
f 4
1
5
g 2
2
7
h 2
1
2
i 1
5
7
j 3
1
6
k 6
2
3
l 10
1
4
m 4
3
8
n 5
1
5
o 6
6
7
p 13
2
3
q 3
1
12
r 7
4
11
s 9
3
10
t 11
1
7
u 2
31
100
v 33
3
10
w 12
3
11
x 12
5
12
9 a 2
1
2
b 2
4
5
c 1
1
3
d 1
1
3
e 1
1
8
f 3
1
3
g 2
2
3
h 2
2
5
10 a
0 1 2
2
3 4
2
3
5
3 31
3
b
0 1 2 3 4
3
4
12
4 31
2 21
4
c
0 1 2 3 4
4
5
19
5
10
5
14
5 31
5
11 a 2
1
3
, 3
2
3
, 4
b 1
4
7
, 2
1
7
, 3, 3
4
7
, 3
6
7
c 2
2
5
, 4
1
5
, 4
4
5
, 6, 7
1
5
12 a 15 b 1
7
8
c 9 d 1
1
8
13 a 11 b 9 c 4x – 1 d 3y – 1
e mn
14 a i 1
2
3
, 2
1
3
, 3
1
2
ii 1
5
6
b i 2
3
4
, 3
2
4
, 4
2
3
ii 1
11
12
c i 3
4
5
, 4
3
5
, 5
3
4
ii 1
19
20
d 1
29
30
e A mixed number with a whole number part equal to 1.
Denominator equal to product of two largest numbers,
numerator is one less than denominator.
Exercise 4F
1 a
5
7
b
7
3
c
9
11
d
8
5
2 a 10 b 21 c 20 d 30
e 6 f 10 g 12 h 24
i 30 j 12 k 24 l 30
3 a 15 b 20 c 21 d 10
e 24 f 60 g 12 h 12
4 a 6 b 8 c 4
d 6 e 15 f 15
5 a > b > c = d <
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e < f > g < h =
i > j > k < l =
m > n > o < p >
q > r > s > t <
6 a
3
5
, 1
2
5
,
8
5
b
2
9
,
1
3
,
5
9
c
2
5
,
3
4
,
4
5 d
3
5
,
2
3
,
5
6
e 2
1
4
,
5
2
,
11
4
, 3
1
3
f
5
3
,
7
4
,
11
6
,
15
8
g
11
5
,
9
4
, 2
1
2
, 2
3
5
, 2
7
10
h 4
1
6
, 4
10
27
, 4
4
9
, 4
2
3
, 5
7 a 2
1
2
,
3
2
,
5
4
− − − b
1
3
,
1
4
,
1
6
− − −
8 a
1
2
,
1
3
,
1
4
,
1
5
b
3
5
,
3
6
,
3
7
,
3
8
c 7
2
,
7
5
,
7
7
,
7
8
d
1
10
,
1
15
,
1
50
,
1
100
e 10
2
3
, 8
3
5
, 7
1
11
, 5
4
9
f 2
1
3
, 2
1
5
, 2
1
6
, 2
1
9
9
1
4
,
1
8
,
1
11
10 Andrea, Rob, Dean, David
11 a
5
9
,
6
9
b
11
4
,
14
4
c
5
6
,
3
6
d
10
14
,
11
14
12 Answers may vary.
a
13
20
b
1
3
c
5
21
d
3
4
e 2
1
4
f 8
29
40
13 Answers will vary. The first step is to ensure both fractions have
a common denominator. This is done through the technique of
equivalent fractions. Once you have a common denominator,
you simply have to choose a new numerator that fits between
the two existing numerators.
14 a 7, 8 b 16
15 a Thomas b
1
10
th of a donut
16 a i
1
4
ii
3
8
iii
1
2
iv
1
3
b Teacher to check
c Answers will vary
Exercise 4G
1 a
2
10
b
3
100
c
7
1000
2 a 5 b 6 c 7 d 37
3 a
6
10
b
6
100
c
6
1000
d
6
10
e 6 f
6
100
g
6
100
h
6
1000
4 a false b false c true d false
e true f true g false h true
i true j true k true l true
5 a 0.3 b 0.8 c 0.15 d 0.23
e 0.9 f 0.02 g 0.121 h 0.074
6 a 6.4 b 5.7 c 212.3 d 1.16
e 14.83 f 7.51 g 5.07 h 18.612
7 a 7.6 b 12.9 c 33.04
d 26.15 e 8.42 f 99.012
8 a 0.1 b 0.03 c 0.02 d 0.5
e 0.001 f 0.01 g 0.15 h 0.11
9 a 3.05, 3.25, 3.52, 3.55 b 3.06, 3.6, 30.3, 30.6
c 1.718, 1.871, 11.87, 17.81
d 22.69, 22.96, 26.92, 29.26, 29.62
10 a Waugh, Border, Gilchrist, Taylor, Hughes
b first
11 a Day 6 b Day 4 c Days 2, 5 and 6
12 a c.c, c.a, b.c, b.a, a.c, a.b
b c.bc, c.ab, b.ca, b.bb, a.ca, a.bc, a.aa, ba.ca, ab.ab
13 a 0.a b 0.0a c 0.aa d a.a 0a
14 a i 0.1, 1.0 (2 ways)
ii 0.12, 0.21, 1.02, 1.20, 2.01, 2.10, 10.2, 12.0,
20.1, 21.0 (10 ways)
iii 0.123, 0.132, 0.213, 0.231, 0.312, 0.321, 1.023, 1.032,
1.203, 1.230, 1.302, 1.320, 2.013, 2.031, 2.103, 2.130,
2.301, 2.310, 3.012, 3.021, 3.102, 3.120, 3.201, 3.210,
10.23, 10.32, 12.03, 12.30, 13.02, 13.20, 20.13, 20.31,
21.03, 21.30, 23.01, 23.10, 30.12, 30.21, 31.02, 31.20,
32.01, 32.10, 102.3, 103.2, 120.3, 123.0, 130.2, 132.0,
201.3, 203.1, 210.3, 213.0, 230.1, 231.0, 301.2, 302.1,
310.2, 312.0, 320.1, 321.0 (60 ways)
b 408 ways
Exercise 4H
1 a 5.8 b 2 c 80 d 6.78
2 a 5 b 9 c 1 d 4
e 0 f 9 g 6 h 5
3 a i 1 ii 7 iii 4 iv 8
b i 25.8 ii 25.82 iii 25.817 iv 26
4 a 14.8 b 7.4 c 15.6 d 0.9
e 6.9 f 9.9 g 55.6 h 8.0
5 a 3.78 b 11.86 c 5.92 d 0.93
e 123.46 f 300.05 g 3.13 h 9.85
i 56.29 j 7.12 k 29.99 l 0.90
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605
Answers
6 a 15.9 b 7.89 c 236 d 1
e 231.9 f 9.4 g 9.40 h 34.713
7 a 24.0 b 14.90 c 7 d 30.000
8 a 28 b 9 c 12 d 124
e 22 f 118 g 3 h 11
9 a $13 b $31 c $7 d $1567
e $120 f $10 g $1 h $36
10 a $51 b $44 c very accurate
11 a 0 s b 0.4 s c 0.34 s d 52.825 s
12 2.25, 3 decimal places
13 Samara: Round to 2 decimal places = 0.45, then round this to
1 decimal = 0.5. Cassandra: Rounding to 1 decimal place,
critical digit is the second 4, which is less than 5, therefore
rounded to 1 decimal place = 0.4.
Samara has a fl aw of rounding an already rounded number.
Cassandra is correct.
14 Depends on your calculator
15 Depends on your software package
Exercise 4I
1 a 5 b 100 c 75, 7 d 5, 4
2 a 2 b 15, 20 c 10, 4 d 16
3 a false b true c true d false e true f true
4 a
1
2
b 6
2
5
c 10
3
20
d 18
3
25
e 3
1
4
f
1
20
g 9
3
40
h 5
24
125
5 a 0.7 b 0.9 c 0.31 d 0.79
e 1.21 f 3.29 g 0.123 h 0.03
6 a
8
10
= 0.8 b
5
10
= 0.5 c
35
100
= 0.35
d
46
100
= 0.46 e 5
95
100
= 5.95 f 3
25
100
= 3.25
g
25
10
= 2.5 h
375
1000
= 0.375
7 a 0.5 b 0.5 c 0.75 d 0.4
e 0.3 i f 0.375 g 0.416 i h 0.428571
i 0.16i j 0.6 i k 0.142857 i i l 0.5
i
8 a 0, 0.5, 1 b 0, 0.3, 0.6, 0.9 (0.9999999... 1) i i i =
c 0, 0.25, 0.5, 0.75, 1 d 0, 0.2, 0.4, 0.6, 0.8, 1.0
9 a
1
4
, 0.4,
1
2
,
5
8
, 0.75, 0.99 b
1
9
, 0.13,
3
7
, 0.58,
4
5
, 0.84
10 a Tan: =
11
37
= 0.297; Lillian: =
6
21
= 0.285714; hence, Tan is
the better chess player.
b two or more
11 0.11 mm, 11
100
mm
12 0.5, 0.3 i
, 0.25, 0.2, 0.16
i , 0.142857, 0.125, 0.1 i , 0.1
13
1
10
,
1
5
,
1
4
,
2
5
,
1
2
,
3
5
,
3
4
,
4
5
,
9
10
14 a 0.3 i b 0.6 i c 0.666666667
d The calculator is correct. It can show only a certain number
of digits. As the answer is an infi nitely recurring decimal,
the calculator, correctly, needs to round the answer to the
maximum number of digits it can display. Hence, it rounds
the fi nal digit it can display to a 7. Note that the calculator
answer is a rounded answer and not an exact answer in
this case.
15 Answers may vary; 2
4
9
, 2
45
100
, 2
3
7
16 2,
4
7
= 0.571428, repeating pattern
17 Answers will vary.
Exercise 4J
1 C
2 C
3 C
4 a
3
4
= 0.75 = 75% b
9
10
= 0.9 = 90%
c
2
5
= 0.4 = 40% d
4
5
= 0.8 = 80%
5 a 150 b 350 c 75 d 175
e 180 f 220%
6 a
3
10
= 0.3 = 30% b
3
5
= 0.6 = 60%
c 5
4
= 1.25 = 125% d
4
3
1 3 133
1
3
= . = %
i
7 a i
1
2
,
2
4
,
3
6
,
4
8
,
5
10
,
6
12
ii
1
3
,
2
5
,
2
6
,
3
8
,
3
9
,
4
9
,
3
10
,
4
10
,
4
12
,
5
12
b
4
9
8 a 3
4
b
1
2
c
1
4
d 1 e
1
3
f
4
3
1
1
3
=
9 a
1
4
b
1
3
c
1
8
d 0 e
1
2 f
7
12
g
3
4
h
4
5 i
5
8
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606
10 a 0.9 b 0.3 c 0.75
d 0.35 e 0.5 f 0.9
g 0.6 h 1.2 i 2.1
11 a 37.5% b 62.5%
12 a three-quarters b two-thirds
c one-quarter d one-eighth
e three-eighths
13 a 90% b 60%
14 a 1
4
1
4
,
2
6
1
6
,
2
8
2
8
,
1
8
3
8
,
1
10
4
10
,
2
10
3
10
,
1
12
5
12
,
2
12
4
12
,
3
12
3
12
+ + + + + +
+ + +
b 1
3
1
6
,
1
4
2
8
,
1
5
3
10
,
1
3
2
12
,
3
9
1
6
,
3
12
2
8
,
2
6
2
12
,
1
6
4
12
,
3
9
2
12
etc.
+ + + + + +
+ + +
c
1
6
1
6
1
6
,
1
8
1
8
2
8
,
1
10
1
10
3
10
,
1
10
2
10
2
10
,
2
12
2
12
2
12
,
1
12
2
12
3
12
,
1
12
1
12
4
12
+ + + + + + + +
+ + + + + +
d
1
4
1
6
1
12
+ +
e
2
5
15 a This is true because 0.5 is one-half and so is 50%.
b This is false because one-third is
33
99 not
33
100 .
c This is false because
1
5
2
10
20
100
= = = 20%, not 15% .
d This is true because
9
10
90
100
= = 90%
e This is true because
1
2
= 50% and
1
4
= 25%
so
1
8
= 12.5%, which is close to 12%.
f This is true because
1
3
33
1
3
% so
2
3
66
2
3
= = %, which is
greater than 66%.
16 B
30
90
1
3
33
1
3
= = % discount ; A 30%, C 31%, D 25%
17 a 0.5 m b 0.25 m c 0.25 m
d 0.125 m e no
Exercise 4K
1 a 95% b 60% c 75% d 26%
2 C 3 B 4 A
5 a 50 b 50%
c i 5 ii 100 iii 20 iv 1
d i 50% ii 50% iii 50% iv 50%
6 a 100, ÷, 100, . b 35, 100, 35, 0 c out of, 126, 100, 26
7 a 0.32 b 0.27 c 0.68 d 0.54
e 0.06 f 0.09 g 1 h 0.01
i 2.18 j 1.42 k 0.75 l 1.99
8 a 0.225 b 0.175 c 0.3333 d 0.0825
e 1.1235 f 1.888 g 1.50 h 5.20
i 0.0079 j 0.00025 k 0.0104 l 0.0095
9 a 80% b 30% c 45% d 71%
e 41.6% f 37.5% g 250% h 231.4%
i 2.5% j 0.14% k 1270% l 100.4%
10 a 86%, 0.78, 75%, 0.6, 22%, 0.125, 2%, 0.5%
b 50, 7.2, 2.45, 1.8, 124%, 55%, 1.99%, 0.02%
11 a yes b 85%
12 a 25% b 78%
13 24 gigalitres
14 6.7%
15 a ab.cd % b aac % c abdc %
d d.d % e cdb.a % f cc.cddd %
16 a 0.0ab b b.cd c 0.ac
d 0.00da e ab.bb f 0.ddd
17 a no
b Answers may vary. Examples include: percentage score
on a Maths test, percentage of damaged fruit in a crate,
percentage of spectators wearing a hat, percentage of the
day spent sleeping.
c Answers may vary. Examples include: percentage profit,
percentage increase in prices, percentage increase in the
price of a house, percentage score on a Maths test with a
bonus question.
18 a F ÷ A × 100
b F: points scored for the team; A: points scored against the team
c 100%
d 158 points
e yes; Hawthorn 90.60%, Port Adelaide 91.32%
Exercise 4L
1 a 70, 70 b 48, 48 c 60, 60
d 20, 20 e 40, 40 f 63, 63
2 a
1
4
25%,
2
4
50%,
3
4
75%,
4
4
= = = = 100%
b
c
1
3
33
1
3
%,
2
3
66
2
3
%,
3
3
= = = 100%
3 a 86% b 20%
4 a
11
100 b
71
100 c
43
100 d
49
100
e
1
4 f
3
10 g
3
20
h
22
25
1
5
20%,
2
5
40%,
3
5
60%,
4
5
80%,
5
5
= = = = = 100%
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607
Answers
5 a 1
1
5 b 1
4
5 c 2
37
100 d 4
1
100
e 1
3
4 f 1
1
10 g 3
4
25 h 8
2
5
6 a 8% b 15% c 97% d 50%
e 35% f 32% g 86% h 90%
i 112% j 135% k 400% l 160%
7 a 12
1
2
% b 33
1
3
% c 26
2
3
% d 83
1
3
%
e 115% f 420% g 290% h 32
1
2
%
8 a
3
4
b 75% c
1
4
d 25%
9 12
1
2
%
10 70%
11 4%
12 70%, 80%
13 a 18
25
b 72%
14
55
1000
11
200
=
15 a
25
1000
b
825
10000
c
125
1000
d
1
3
Exercise 4M
1 a 10 b 100 c 2 d 1
e 5 f 4
2 a ii b ii c i d i
3 a 70 b 36 c 10 d 27
e 10 f 7 g 150 h 200
i 4 j 48 k 44 l 190
m 22 n 84 o 36 p 63
4 a 96 b 600 c 66 d 100
e 15 f 72 g 73 h 600
5 10% of $200 = $20
20% of $120 = $24
10% of $80 = $8
50% of $60 = $30
20% of $200 = $40
5% of $500 = $25
30% of $310 = $93
10% of $160 = $16
1% of $6000 = $60
50% of $88 = $44
6 a $42 b 24 mm c 9 kg d 90 tonnes
e 8 min f 400 cm g 1.5 g h 3 hectares
i 144 seconds
7 35
8 no (red $45, striped $44)
9 240 10 12
11 a 120 b 420 c 660
12 a $80 b $2080 c $4160
13 a computer games 30 min, drums 24 min, outside 48 min,
reading 12 min
b 5% time remaining c yes, with 1 min to spare
14 80 min
15 a 20 b 90 c 20 d 64
16 Answers will vary.
17 they are the same
18 37
1
2
%
19 a i 1200
ii 24%, 22%, 20%, 18%, 16% iii 2%
b i 30%
ii week 1: 30%, 2400 pieces; week 2: 25%, 2000 pieces;
week 3: 20%, 1600 pieces; week 4: 15%, 1200 pieces;
week 5: 10%, 800 pieces
iii
Week Cumulative % Pieces completed
Week 1 30% 2400
Week 2 55% 4400
Week 3 75% 6000
Week 4 90% 7200
Week 5 100% 8000
Exercise 4N
1 a 4 b 4 c
1
2
d 50% e
1
2
f 50%
2 a 10 b
1
5
c
4
5
d 20% e 80%
3 a
3
10
, 30% b
3
5
, 60% c
1
5
, 20% d
3
4
, 75%
e
1
20
, 5% f
1
2
, 50% g
1
4
, 25% h
3
20
, 15%
4 a
3
5
, 60% b
1
2
, 50% c
1
4
, 25% d
2
5
, 40%
e
3
4
, 75% f
4
5
, 80%
5 a
1
10
b 10%
6 a 1
5
b 20% c 4
5
d 80%
7 a
3
4
b 75% c
1
4
d 25%
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8 95%
9 80%
10 a
1
25
, 4% b
2
5
, 40% c
1
20
, 5% d
1
100
, 1%
e
1
50
, 2% f
1
40
, 2.5%
11 a
1
5
b 40%
12 a 5% b
1
5
13 Ross 75%, Maleisha 72%
14 hatch 15%, 4WD 16%; hence, 4WD has larger price reduction.
15 yellow 20%, blue 19%; hence, blue has least percentage of
sugar.
16 a i
+
b
a b
ii
+
a
a b
b i
100
+
b
a b
ii
100
+
a
a b
17 a
+
a
a b
b
100
+
b
a b
18 a 3 b 1
30
c 1
6
d
17
30
e 20% f 96.67%
Puzzles and challenges
1 Delhi, Agra and Jaipur
2 Answers will vary.
3 a
1
9 b
9
1
c 14 d
8
9
e
1
2
,
2
4
,
3
6
,
4
8
f
3
4
,
6
8 g 16 h eight ways
i three ways j 9
4 20% of 50 is not the same as 20% of 60.
5 nine times bigger; 800%
6 3
Multiple-choice questions
1 B 2 C 3 A 4 C 5 D
6 C 7 E 8 D 9 A 10 C
Short-answer questions
1
1
14
,
1
10
,
1
8
,
1
6
,
1
5
,
1
4
,
1
3
,
1
2
2 3
5
6
10
9
15
12
20
15
25
= = = = etc.
Multiplying by the same number in the numerator and
denominator is equivalent to multiplying by 1, and so does
not change the value of the fraction.
3 a
3
5
b
2
7
c
5
7
4 a 1
1
2 b 1
3
4 c 1
2
3 d 3
1
2
5 a
2
7
4
7
< b
3
8
1
8
> c 1
2
3
1
3
5
> d 3
1
9
29
9
<
6 a
5
7
b
5
8
7 a 10 b 21 c 24
8 a 10 b 21 c 24
9 a 2
1
5
,
9
5
, 1
3
5
b
9
4
,
11
6
,
14
8
,
5
3
c 5
2
3
5
7
18
5
1
3
48
9
5
2
9
, , = ,
10 a
1
2
b
5
6
c 1
1
2
d
1
2
e
1
2
f
2
5
11 Percentage form Fraction
36%
9
25
220% 2
1
5
5%
5
100
140% 1
2
5
44%
11
25
18%
9
50
12 a ii b i
13 a
3
5
, 60% b
1
5
, 20%
c
1
16
, 6.25% d
3
10
, 30%
14 a 3 hundredths =
3
100
b 3 thousandths = 3
1000
c 3 units = 3
15 a 8.1 = 810% b 0.81 = 81%
c 8.01 = 801% d 0.801 = 80.1%
16 45.265
45.266
45.267
45.268
45.270
45.271
45.272
45.273
45.274
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Answers
17 a 1, 2, 3, 4, 6, 8, 12, 24
b 1, 2, 4, 8, 16, 32
c 1, 2, 3, 4, 6, 9, 12, 18, 36
d 1, 2, 4, 8, 16, 32, 64
e 1, 2, 4, 5, 10, 20, 25, 50, 100
f 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
g 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
h 1, 3, 5, 15, 25, 75
18 a 8 b 4 c 72 d 25
19 a 12 b 10 c 6 d 6
e 60 f 4 g 15 h 6
i 8 j 20 k 10 l 40
Extended-response questions
1
Simplifi ed fraction Decimal Percentage
7
20 0.35 35%
3
5 0.6 60%
1
1
10 1.1 110%
2
3
4
2.75 275%
7
1
2
7.5 750%
5
3
1.6
i
166.6
i
5
6
0.83
i
83.3
i
3
8
0.375 37.5
2 Science
3 a i
1
4
ii 0.25
b i 15% ii
15
100
3
20
=
c i $20 ii $15
d i 30% ii
3
10 iii 0.3 iv $30
e i $50 ii
1
2 , 0.5, 50%
Chapter 5
Pre-test
1 a
1
2 b
2
3 c
3
4 d
1
4
2 a 5 b 3 c
2
5
3 a 0.5 b 0.4 c 0.2 d 0.2
4 C, A, B, D
5 Answers may vary.
6
1
2
= 0.5 = 50%,
1
3
= 0.3
i = 33
1
3
%,
1
4
= 0.25 = 25%,
1
5
= 0.2 = 20%,
1
10
= 0.1 = 10%
Exercise 5A
1 a D b A c B d C
2 a certain b even chance c unlikely d impossible
3 a true b false c true
d false e true f false
4 a D b C c A d B
5 a i true ii false iii false iv true
v true
b i spinner landing on yellow
ii spinner not landing on red
iii spinner landing on green, blue or red
iv spinner landing on blue or on red
6 a spinner 3 b spinner 2 c spinner 1
7 Spinner 3 contains two red sections. Together, they are twice
the size of the blue section. In spinner 2, the red and blue
sections are the same size.
8 Answers will vary.
9 a Blue, red and green equally likely.
b Red and green both have an even chance.
c Green and blue equally likely, red and blue are not equally
likely.
d Blue is certain.
e Blue, red and green all possible, but no two colours are
equally likely.
f Red and blue both have an even chance.
10 a
1
2
5
4
3
6
b All sectors have the same size; i.e. 60°.
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c
1
2
5
4
3
6
d by replacing the 5 with a 6 (so that there are two faces with 6).
e with 52 equal segments
11 a spinner 1:
1
4
, spinner 2:
1
3
, spinner 3:
1
9
b
1
2
c i
red
ii
blue
green
d i 50% ii 0% iii 0%, 50% iv 50%
e If the two fractions are equal, the two events are
equally likely.
f The proportion of the spinner’s area cannot exceed 100% (or 1)
and must be greater than 0%.
Exercise 5B
1 a C b A c D d B
2 a sample space b 0 c certain d more
e even chance
3 a {1, 2, 3, 4, 5, 6} b {1, 3, 5} c
1
2
4 a 3: red, green, blue b
1
3
c
2
3
d
2
3
e 0
5 a {1, 2, 3, 4, 5, 6, 7} b
1
7
c 0
d
2
7
e
3
7
f
4
7
g Number chosen is less than 10; other solutions possible.
6 a {M, A, T, H, S} b 0.2 c 0.8 d 60%
7 a
1
11
b
2
11
c
5
11
d
9
11
e
4
11
f Choosing a letter in the word ROPE; other solutions possible.
8 a 30% b 50% c 80%
9 a {2, 3, 4, 5, 6, 7, 8, 9} b 0.5
c i 0.375 ii 0.375 iii 0
d Possible spinner shown:
2
5
31
10 a i 12 red, 6 blue, 4 yellow, 2 green ii
1
12
b i 18 red, 9 blue, 6 yellow, 3 green ii
1
12
c 12
d No, because it is always 1 –
1
2
–
1
4
–
1
6
.
11 a a = 1, b =
1
2
, c = 1, d =
1
3
, e = 1, f =
2
5
, g =
3
5
, h = 1
b
10
13
c
3
7
12 a yes, 19
210
b 210 c 840
13 a
red
green blue
b Cannot be done because adds to more than 1.
c Cannot be done because adds to less than 1.
d
blue
red
green
Other answers possible
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Answers
Exercise 5C
1 a
1
10
b
2
5
c
3
10
d false
2 a
53
100
b
47
100
c
1
2
d tossing a tail
3 a 50 b
2
5
c
1
10
d
1
2
e No, just that nobody did it within the group surveyed.
4 a 500 b 1750 c i 7 tails ii more
5 a 100 b 300 c yes (but this is very unlikely)
d from 2 rolls
6 a–e Answers will vary. f true
7 a
1
4
b
3
100
c
31
100
d 60
e white, blue, purple or green
8 a 5 b 40 c 70 d
26
35
e
4
7
f 126 g 1, 2, 3 or 4 children
9 a 2 red, 3 green and 5 blue
b i yes ii yes iii yes iv yes v no
10 a C b D c B d A
11 a Answers will vary. b Answers will vary.
c No technique for fi nding theoretical probability has been
taught yet.
d less accurate
e more accurate
12 a False; there is no guarantee it will occur exactly half of the
time.
b False; e.g. in two rolls, a die might land 3 one time. The
theoretical probability is not
1
2
, though.
c False; perhaps the event did not happen yet but it could.
d True; if it is theoretically impossible it cannot happen in an
experiment.
e False; experiment might have been lucky.
f True; if it is certain, then it must happen in an experiment.
13 a red: 25%, green: 42.2%, blue: 32.8%
b Fifth set is furthest from the fi nal estimate.
c have 4 green sectors, 3 blue and 3 red sectors
d red: 90°, green: 150°, blue: 120°
Exercise 5D
1 a event; outcome
b equally likely outcomes; mutually exclusive events
c probability
d at least e certain f certain g less than
2 a 80% b 0% c 80% d 20%
e 80% f 40%
3 a
2
3
b 0 c
1
3
d
1
6
e 2
3
f 5
6
g 5
6
h
1
3
i
1
6
j 5
6
k
2
3
4 a 0 b
12
51
4
17
= c 39
51
13
17
=
5 a
1
7
b
6
7
6 Answers will vary.
7 a There are 10: AB, AC, AD, AE, BC, BD, BE, CD, CE, CE
b 10 c 45
d i 3 838 380 ii 7 059 052
Exercise 5E
1 a Like
bananas
Dislike
bananas Total
Like apples 30 15 45
Dislike apples 10 20 30
Total 40 35 75
b 30 c 20 d 75
e mutually exclusive
2 a i 2 ii 4 iii 1 iv 3
b No, there are two people who own a dog AND a cat.
3 a 8 b
5 10 7
8
cricket soccer
c 15 d 22 e 12
4 a 15
b Employed Unemployed Total
University
degree 10 3 13
No university
degree 5 2 7
Total 15 5 20
c The 10, 13, 15 and 20 would all increase by 1.
5 a 26 b 12 c 11
d i
2
13
ii
7
26
iii
7
13
iv
15
26
e Being a boy or being a girl; paying own bill or not paying
own bill.
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6 a
17 8 2
13
car home
b i
1
5
ii
17
40
iii
1
4
c
8
25
d
4
5
7 a 12 b
2
15
c
13
15
d
24
30
or
4
5
e The ‘inclusive or’ includes the people who like one or the
other or both.
10 2 14
4
juice soft drink
The ‘exclusive or’ does not include the 2 people who like
BOTH juice AND soft drink.
10 2 14
4
juice soft drink
8 a Sports Not sports Total
Automatic 2 13 15
Not automatic 8 17 25
Total 10 30 40
b
1
5
c
13
40
d
2
15
9 a
1
5
b 45 c
2
9
d
1
3
10 a 4; if only the totals are missing, they can be calculated from
the first four spaces.
b No, not if the 5 spots are in the last row/column.
c Filling it requires a negative number, which is impossible.
11
B Not B Total
A x w x + w
Not A y z y + z
Total x + y w + z w + x + y + z
12 a 120 b 60
c i
59
120
ii
19
40
iii
1
2
iv
29
120
v 21
60
7
20
= vi
17
60
d i
10
19
ii
5
19
e i
29
59
ii
34
59
f They are male, under 40 and not using a trolley.
g More likely to be female
30
59
29
59
>  
 
.
h More likely to use trolley
30
57
27
57
>  
 
.
Exercise 5F
1 a 10 b H2, H4, T2, T4 c
2
5
d T1, T3, T5 e
3
10
2 a
1
4
b HH, TT c
1
2
3 a 1 2 3 4 5 6
H H1 H2 H3 H4 H5 H6
T T1 T2 T3 T4 T5 T6
b 12 c
1
12
d
1
4
4 a R I D E
L LR LI LD LE
I IR II ID IE
N NR NI ND NE
E ER EI ED EE
b 16 c
1
16
d
1
16
e
1
4
f
1
4
g
1
8
5 a
R P B
R RR RP RB
P PR PP PB
G GR GP GB
B BR BP BB
b 1
12
c
1
12
d
1
12
e
1
6
f
1
4
6 a
1
12
b
1
6
c
1
12
d
1
2
7 a
Sum 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
b
1
9
c
1
2
d 7 e 2 and 12
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Answers
8 a
Y W B B B
W WY WW WB WB WB
O OY OW OB OB OB
O OY OW OB OB OB
b
2
15
c
1
15
d
2
15
e
1
5
f
1
15
9 a
1
8
b
1
3
10 63
11 a
1
4
b
1
52
c
1
26
d i
1
16
ii
1
4
iii
1
2704
e The second card would then depend on the fi rst card (e.g. if
a red card was selected, it would be a little less likely that
the next card would be red).
12 a i
1
4
ii
1
2
b i
1
16
ii
1
4
iii
1
16
iv
1
8
c OYYBBB (or some rearrangement of those letters)
d i
1
9
ii
7
18
iii
1
12
iv
1
6
v
11
36
vi
3
4
e i
1
24
ii
1
8
iii
1
12
iv
1
4
v
1
4
vi
3
4
Puzzles and challenges
1 0.25
2 a MOON b OFF c DING
d PROBABILITY e STUMBLE f TRY
3 32
4
5
9
5 0.000977
6 1 red, 3 blue, 4 yellow
7 PROBABLE
8 Reykjavik
Multiple-choice questions
1 E 2 C 3 C 4 B 5 B
6 C 7 C 8 C 9 C 10 C
Short-answer questions
1 a 1 b
1
8
c
19
20
d
3
4
e 0
2 a {1, 2, 3, 4, 5, 6} b {heads, tails}
c {D, E, S, I, G, N} d {blue, yellow, green}
3 a 9
b i
5
9
ii
1
3
iii 0 iv
1
3
v
4
9
4 a
1
26
b
2
13
c
1
52
d
3
13
5 a 42% b 50%
6 a
1
2
b
1
4
c 25 d 250
Extended-response questions
1 a 50 b 25 c 24% d 1%
e
12
25
f
24
25
2 a
Uses public
transport
Does not
use public
transport Total
Owns a car 20 80 100
Does not
own a car 65 35 100
Total 85 115 200
b 200 c
1
2
d
1
10
e
4
17
f
1
5
g i More public transport users expected.
ii People less likely to use public transport in regional area.
3 a 1, it has the most occurrences.
b 3 and 10, as they have the closest outcomes.
c
3
8
d
1
3
5
10
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Chapter 6
Pre-test
1 a 0.1 b 0.3 c 1.7
d 0.01 e 0.001 f 4.7
2 a 0.5 b 0.25 c 0.75
3 a $0.70 b $0.85 c $1.00
d $0.05 e $1.05 f $0.03
4 a 50 b 25 c 75 d 250
5 a $2.70 b $12.60 c $22.50 d $1.89
6 25c
7 $15.50
8 a $85 b $0.10 c $2.70
d $0.70 e $11.20 f $0.24
9 a $87.60 b $38 c $13.20
10 $104.65
11 a 523 b 839 c 1352 d 1237
Exercise 6A
1 C
2 B
3 a 27.97 b 25.94 c 247.4 d 58.31
4 a 16.06 b 21.33 c 343.75
d 37.567 e 21.592 f 340.0606
5 a 12.1 b 114.13 c 6.33 d 70.79
6 a 12.3 b 131.4 c 22.23
d 13.457 e 43.27 f 4947.341
7 a 62.223 b 50.203
8 a 7, 8 b 6, 5, 1, 4 c 0, 1, 0, 2 d 7, 5, 1, 6, 1
9 186.19
10 $54.30
11 49.11 mm
12 + 0.01 0.05 0.38 1.42
0.3 0.31 0.35 0.68 1.72
0.75 0.76 0.80 1.13 2.17
1.20 1.21 1.25 1.58 2.62
1.61 1.62 1.66 1.99 3.03
13 $2036.10
14 a 8.038 b 0.182 c 2.438 d 1.308
15 a Answers will vary; e.g. 3.57 + 4.15 + 3.44
b Answers may vary; e.g. 1.35 + 2.87 + 6.94 = 11.16
16 Always end up with $10.89, unless the starting value has the
same fi rst and last digit, in which case we end up with zero. The
8 forms as it comes from adding two 9s. This produces a 1 to
be carried over, which results in the answer being a $10 answer,
rather than a $9 answer.
Exercise 6B
1 a denominator
b denominator, numerators
c denominators, lowest common denominator
d check, simplifi ed
2 a 5 b 5 c 4, 3, 7 d 8, 15, 23, 3
3 a 15 b 20 c 6 d 6
e 8 f 10 g 77 h 9
i 24 j 18 k 30 l 48
4 a ✓ b ✘ c ✘ d ✓
e ✓ f ✘ g ✓ h ✘
i ✓ j ✘ k ✓ l ✘
5 a
5
8
b
5
7
c
4
5
d
9
11
e
7
8
f
7
12
g
7
15
h
5
9
i 1
2
7
j 1
3
10
k 1
4
5
l 1
4
19
6 a
3
4
b
14
15
c
2
3
d
7
12
e
13
20
f
19
20 g
13
21
h
23
40
i 1
13
30 j 1
9
28 k 1
13
33 l 1
5
12
7 a 3
4
5
b 7
3
7 c 12
3
4 d 5
5
9
e 10
1
3 f 21
1
6
g 19
3
11 h 12
2
5
8 a 4
5
12
b 7
7
30 c 12
1
6
d 13
9
28
e 15
1
10
f 19
1
9
g 25
21
44
h 15
5
24
9 a 23
31
35 b 79
31
36 c 200
5
24 d 259
47
100
10 a
14
15
b
1
15
11 8
3
20
km
12 a
3
4
b 400 c
1
4
, 250 pieces
13 a
47
60 b
13
60
c 39
14 a 5, 2 b 2, 4, 8 c 5, 1
d 5, 1, 15; other answers possible.
15 Jim 36
60
 
 
, Vesna 37
60
 
 
, Juliet 38
60
 
 
, Mikhail 39
60
 
 
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Answers
16 a maximum:
6
1
5
2
4
3
9
5
6
+ + =
minimum:
1
4
2
5
3
6
1
3
20
+ + =
b maximum:
8
1
7
2
6
3
5
4
14
3
4
+ + + =
minimum:
1
5
2
6
3
7
4
8
1
97
210
+ + + =
c Maximum: Largest numbers as numerators, smallest numbers
as denominators; combine largest numerator available with
smallest denominator available to produce the fractions.
Minimum: Smallest numbers as numerators, largest numbers
as denominators; combine smallest numerator available with
smallest denominator available to produce the fractions.
Exercise 6C
1 a denominators b multiply
c simplify d multiply
2 a 12 b 10 c 15 d 18
e 24 f 60 g 56 h 63
3 a 1 b 3 c 4, 3, 1 d 12, 10, 2
4 a ✓ b ✘ c ✘ d ✓
e ✘ f ✓ g ✓ h ✘
i ✓ j ✓ k ✘ l ✘
5 a
2
7
b
3
11 c
7
18 d
1
3
e 0 f
4
9
g
3
19 h
8
23
i
31
100 j
24
50
12
25
= k
16
25
l
4
10
2
5
=
6 a
5
12
b
1
10
c
1
10
d
9
28
e
1
6
f
23
36
g
13
33
h
2
15
i
1
8
j
3
20
k
1
36
l
1
9
7 a 1
3
5
b 8
3
7
c 1
1
7
d 3
2
9
e 2
5
12
f 3
5
28
g 4
7
18
h 7
1
20
8 a 2
2
3 b 4
3
5
c 4
2
3
d 4
8
9
e 4
2
3
f
37
45
g 9
37
44
h 2
29
60
9 a 6
2
5 b 17
7
12
c 1
58
63 d 207
51
55
10
11
20
11
3
5
12
1
6
13 $7
3
4
, $7.75
14 a ice-cream
3
4
, chocolate
3
4
, sponge
7
8
b ice-cream
1
4
, chocolate
1
4
, sponge
1
8
c 2
3
8
d
5
8
15 a 3, 4 b 3, 1 c 1, 2 d 4, 3
16 a
1
4
b 4 years c 10 years
17 a
8
15
: ‘Converting to an improper fraction’ is quick and
effi cient for this question.
b 3
36
55
: ‘Borrowing a whole number’ keeps the numbers
smaller and easier to deal with for this question.
Exercise 6D
1 a 0, 1 b 1 c whole number, proper fraction
2 You get a smaller answer because you are multiplying by a
number that is less than 1.
3 a
b
c
1
12
4 a
b
c
d
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e
f
5 D
6 a
3
20
b
2
21
c
10
21
d
8
45
e
2
5
f
1
7
g
1
4
h
5
11
i
5
22
j
1
3
k
6
11
l
4
11
m
6
35
n
3
10
o
2
7
p
1
6
7 a 6 b 9 c 16 d 15
e 12 f 4 g 80 h 33
8 a 5
5
6
b 1
31
35
c 3
3
10
d 4
7
8
e 5
1
3
f 7 g 6 h 3
1
3
9 a 3
11
15
b 1
25
63
c 7
4
5
d 24
10 a 3
1
5
b
3
16
c 4 d 33
e
2
7
f 3
3
8
g
4
15
h 6
11 a 3
1
5 b
3
16
c 4 d 33
e
2
7 f 3
3
8 g
4
15 h 6
12 a
3
5
b 48 boys, 72 girls
13 11
2
3
L
14 7 cups of self-raising flour, 3 cups of cream
15 7 games
16 a
7
12
× b
7
12
× c
1
12
×
d ✓ e ✓ f
1
12
×
17 D; e.g.
2
7
3
5
6
35
× = . The two numerators will always multiply
to give a smaller number than the two larger denominators.
Hence, the product of two proper fractions will always be a
proper fraction.
18 Answers may vary.
a
2
5
3
2
× b
5
4
3
5
× c
3
7
2
6
×
19 a
2
7
,
3
8
b–c Answers will vary.
Exercise 6E
1 a 00 b 000 c 00 d 00000
2 a 000 b 00 c 0 d 00000
3 a i right 2 places ii left 1 place
iii right 6 places iv no change
v left 3 places vi right 3 places
vii right 1 place viii left 7 places
b right 1 place
4 a 48.7 b 352.83 c 4222.7 d 1430.4
e 5699.23 f 125.963 g 12 700 h 154 230
i 3400 j 2132 k 86 710 000 l 516 000
5 a 4.27 b 35.31 c 2.4422 d 56.893
e 12.13518 f 9.32611 g 0.029 h 0.001362
i 0.00054 j 0.367 k 0.000002 l 0.0100004
6 a 2291.3 b 31.67 c 0.49
d 0.222 e 63 489 000 f 0.0010032
7 a 15 600 b 43 000 c 225.1 d 0.016
e 213 400 f 21.34 g 0.007 h 9 900 000
i 0.0034
8 a 158.4 b 3.36 c 85.4 d 7054
e 71.06 f 7.5 g 2.037 h 21.7
9 $147
10 a 1 200 000 mL
b 12 000
11 3000c, $30
12 $21 400
13 225 kg
14 Answers may vary.
Starting
number Answer
Possible two-step
operations
12.357 1235.7 × 1000, ÷ 10
34.0045 0.0340045 ÷ 100, ÷ 10
0.003601 360.1 × 100, × 1000
bac.dfg ba.cdfg ÷ 100, × 10
d.swkk dswkk × 100 000, ÷ 10
fwy f.wy ÷ 1000, × 10
15 ÷ 1 000 000
16 a i 5 × 1013 ii 4.2 × 107 iii 1.23 × 1016
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Answers
b i 2 × 1019 ii 1.08 × 1021
c –d Answers will vary.
e 3.5 × 10–11
f i 1 × 10–6 ii 9 × 10–10 iii 7.653 × 10–12
Exercise 6F
1 a 1 b 2 c 3 d 3 e 5
f 2 g 4 h 3 i 9
2 a 19.2 b 1.92 c 0.192
3 It helps you check the position of the decimal point in the answer.
4 a The decimal point is the actual ‘dot’; decimal places are the
numbers after the decimal point.
b 1 decimal point, 4 decimal places
5 in the question; decimal places
6 a 20.84 b 26.6 c 183.44 d 100.8
e 218.46 f 15.516 g 23.12 h 12.42
i 5.44 j 311.112 k 0.000966 l 1.32131
7 a 100.8 b 483 c 25 400 d 9800
e 14 400 f 364 550 g 0.68 h 371
i 90.12
8 a $31.50, $32 b $22.65, $23 c $74.80, $75
d $17.40, $17 e $145.20, $145 f $37 440, $37 440
g $88.92, $89 h $4.41, $4 i $18.0625, $18
9 29.47 m 10 3.56 kg
11 165.85 km
12 a 67.2 m b $198.24
13 $1531.25
14 a 738.4 km b yes c 1.57 L left in the tank
15 Answers may vary; 0.25, 0.26
16 Answers may vary; 0.0043
17 a 38.76 b 73.6 c 0.75 d 42, 0.42
Exercise 6G
1 A
2 a
5
11
5
3
× b
1
3
5
1
× c
7
10
17
12
× d
8
3
1
3
×
3 a
5
2
4
3
,
5
2
3
4
÷ × b 24
16
5
,
24
1
5
16
÷ ×
c
47
11
21
4
,
47
11
4
21
÷ × d
8
3
80
7
,
8
3
7
80
÷ ×
4 a less b more c more
d more e less f less
5 a
7
5
b
5
3
c
9
2
d
8
1
e
3
7 f
5
23 g
6
11
h
3
26
i
1
12
j
1
101
k 9 l 1
6 a
3
8
b
5
33
c
2
5
d
5
7
e
3
4
f 1
1
3
g 1
3
5
h
3
14
7 a 20 b 21 c 100 d 120
e 30 f 40 g 4 h 6
2
3
8 a
5
7
b
4
5
c
11
14
d
3
4
e 1
11
16 f 1
3
11
g 3
1
3
h
3
4
9 a
3
40
b 30 c 1
13
35
d
1
3
e 28 f
4
15
g 6
4
7
h 2
1
10
10 a
3
40
b 30 c 1
13
35 d
1
3
e 28 f
4
15 g 6
4
7
h 2
1
10
11
3
4
12 10
13
9
10
m
14 8
15 22 km
16
1
2
of 8 and
1
2
1
8
÷ = 4, 12 ÷ 4 and 12 ×
1
4
= 3,
10 ×
1
2
and 10 ÷ 2 = 5, 3 ÷
1
2
and 3 × 2 = 6
17 a
9
20 b
45
64
c
5
16
d
1
5
18 a 240 km b 160 km
Exercise 6H
1 B
2 directly above the decimal point in the dividend
3 a 60 b 60 c 60 d 60
e An identical change has occurred in both the dividend and
the divisor.
4 a 32.456, 3 b 12 043.2, 12
c 34.5, 1 d 1 234 120, 4
5 a 4.2 b 6.1 c 21.34 d 0.7055
e 1.571 f 0.308 g 3.526 h 124.3
i 0.0024 j 117.105 k 0.6834 l 0.0025625
6 a 30.7 b 77.5 c 26.8 d 8.5
e 44.4 f 645.3 g 0.08 h 0.050425
i 980 j 800.6 k 0.79 l 2 161 000
8 a 1.1807 b 8.267 c 0.0123748
d 0.00423 e 0.096487 f 0.0007825
9 a 11.83 kg b $30.46 c 304.33
d 239.17 g e 965.05 L f $581.72
10 a 20.84 b 93.36 c 10.93
d 18.49 e 67.875 f 158.35
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11 a 8, 9 b 9 c 5, 5, 1 d 0, 7, 7
12 $1.59/L
13 238 frames
14 26.67, 26 can be fi lled
15 a $66.35 b $21.90
16 apples $3.25/kg; bananas $3.10/kg; hence, bananas are better
value.
17 a 24.5 m/s b 88.2 km/h c yes
18 a 24.53 b 19.7 c 2453
d 1.97 e 2.453 f 197
19 a 0.5 b 0.3 c 0.01 d 1.1 e 4.8
Exercise 6I
1
−4 −3 −2 −1 0 1 2 3 4
−3 4
5 −21
2 11
2 − 7
3 − 1
4
1
4
2 a
1
4
b
1
3
− c
3
5
d
2
7
−
3 a positive b negative c negative d positive
4 a
4
7
− b
1
5
c
7
9
− d 5
1
3
−
e
1
3
− f
2
5
− g
3
2
h
7
11
5 a
1
12
− b
13
35
− c 1
1
10
d
8
9
e
1
4
− f
1
8
g
3
20
− h
4
15
−
6 a
12
35
− b
16
55
− c
4
15
d
5
6
e
4
21
− f
1
8
− g
3
7
h 1
1
5
−
7 a
20
21
− b
9
20
− c
8
15
d 1
1
3
e
2
7
− f
3
20
− g
3
4 h 2
2
5
8 a 6 b 5
8
39
− c
3
11 d
85
203
9
5
3
, 1
1
2
,
3
4
,
1
2
,
1
5
,
1
16
,
3
4
, 3
1
10
− − − − −
10 16
3
4
°C
11 a Mon = 1
2
3
− , Tue = 1
2
− , Wed = 2
1
4
− , Thur =
1
4
hours
b 4
1
6
− hours c 12
1
6
hours
12 1
7
20
metres
13 a > b < c > d <
e > f > g < h >
14 a negative b negative
c negative d positive
15 a
6
539
− b −2
491
605
c
35
81
− d
1
343
16 a < b > c > d <
17 a i 1
1
2
ii
1
15
iii
21
25
− iv 4
23
40
−
b Answers may vary:
5
8
,
3
8
,
2
8
,
1
8
,
11
8
− − − −
c Answers may vary:
7
4
,
5
4
,
3
4
,
1
4
,
1
4
− − − −
Puzzles and challenges
1 Continent of Atlantis
2 Height of rungs: 0.25 m, 0.40 m, 0.54 m, 0.68 m, 0.83 m,
0.97 m, 1.12 m, 1.26 m, 1.40 m, and 1.55 m
3 a A = 1, B = 7, C = 8 b A = 7, B = 5, C = 1, D = 9
c A = 4, B = 2 d A = 3, B = 7
Multiple-choice questions
1 D 2 B 3 C 4 E 5 A
6 D 7 D 8 B 9 A 10 C
Short-answer questions
1 a 0.44, 0.4, 0.04 b 2.16, 2.026, 2.016
c 0.98, 0.932, 0.895
2 a 8.1 b 0.81 c 8.01 d 0.801
3 a 3 hundredths =
3
100
b 3 thousandths = 3
1000
c 3 ones = 3
4 a false b false c true
d true e false f true
5 a 947 b 9470 c 94 700
6 a 423.5 b 15.89 c 7.3 d 70.000
e 2.8 f 0.67 g 0.455
h 0.01234567901234567901234567901234567901234568
7 a 9.53 b 4.137 c 43.35
d 240.49857 e 83.497 f 205.22
8 a true b false c false d false e true
9 a 5 b 5 c 4 d 2
10 a 137 b 790 c 22.51 d 0.096208
e 696.956 f 360.5 g 563 489.3
11 a 19.2 b 63.99 c 19.32
d 0.95 e 1.52 f 6
g 16 h 3 i 3.109
i i
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Answers
12
Decimal Fraction Percentage
0.45
45
100
9
20
= 45%
0.7
70
100
7
10
= 70%
0.32
32
100
8
25
= 32%
0.06
6
100
3
50
= 6%
0.79
79
100
79%
1.05
105
100
21
20
= 105%
0.35
35
100
7
20
= 35%
0.65
65
100
13
20
= 65%
0.125
125
1000
1
8
= 12.5%
Extended-response questions
1 a Jessica $12.57; Jaczinda $13.31; hence, Jaczinda earns
higher pay rate by 74c per hour.
b $43.47, $43.45 to the nearest 5 cents
c $40.56
d $48.34
2 a $28.39 b $17 035.20
3 32 min and 26 s, on average
Semester review 1
Computation with positive integers
Multiple-choice questions
1 E 2 B 3 D 4 C 5 A
Short-answer questions
1 a b LXXIV c
2 a 67 849 b 700 850
3 a 99 323 b 6867 c 441
d 196 000 e 1644 f 764
3
4
4 a false b true c true
5 36
6 a 30 b 56 c 48 d 160
e 16 f 42
7 a false b true c false d true
e true f true
8 18 × (7 + 3)
9 9 times
10 a 3 859 640 b 3 860 000 c 4 000 000
Extended-response question
1 a 28 b $700 c $1000 d 12 h
Angle relationships
Multiple-choice questions
1 A 2 B 3 B 4 B 5 D
Short-answer questions
1 a AC || FD or EB || DC
b BF and BD
c AC, BF, BD at point B or CD, ED, BD at point D or BF,
BE, BD at B
d A and B e E
2 a 30° b 80° c 150°
3 25°
4 78°
5 a a = 140 b a = 50 c a =140
d a = 65 e a = 62 f a = 56
6 a = 100, b = 80, c = 100, d = 80, e = 100, f = 80, g = 100
7 Because the cointerior angles add to more than 180°.
8 Check with your teacher.
Extended-response question
1 a i x = 56 ii y = 95 iii z = 29
b x + y + z = 180
Computation with positive and
negative integers
Multiple-choice questions
1 C 2 B 3 D 4 E 5 A
Short-answer questions
1 a < b < c =
2 a -13 b -84 c -108 d -21 e 84 f 0
3 a -24 b 72 c 144 d 21 e 30 f -31
4 a negative b positive c negative
5 a -15 b -8 c -4
6 a -10 b 10 c -20 d -96 e 52 f -1
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Extended-response question
1 a D b A, B, O and G ; all lie on the x-axis.
c F d i 2 units ii 5 units
e trapezium f 8 square units
g X(4, 2) h DECIDE
Understanding fractions, decimals and
percentages
Multiple-choice questions
1 B 2 C 3 E 4 D 5 A
Short-answer questions
1
3
10
,
1
3
,
2
5
,
1
2
2
17
3
3 a
11
12
b 2
2
3
c 6
1
4
d
1
5
e 4 f
1
2
4
3
20
5 $120
6 $60
7 a true b true c true d false
8 67%
Extended-response question
1 a 6 b
8
9
c 9
d second dose on Sunday week
Probability
Multiple-choice questions
1 C 2 D 3 C 4 C 5 B
Short-answer questions
1 a A b B c B, C, D, A
2 a
1
2
b
1
2
c 1
4
d
1
13
e
1
13
f
1
26
3 a
1
6
b
1
15
c
4
15
d
17
30
e
5
6
4 B, C, A, D, E
Extended-response question
1 a
1
4
b
1
4
c
1
2
d
1
52
e
2
13
f
1
13
g 4
13
h
4
13
Computation with decimals and fractions
Multiple-choice questions
1 E 2 C 3 D
4 A 5 B
Short-answer questions
1 a 0.2 b 0.13 c 1.7
2 a 6 units b
4
1000
c 136.1
3 a 18 b 18.4 c 18.40
4 a 4.07 b 269.33 c 19.01
d 0.24 e 0.09 f 60
5 a 0.833 b 3 c 36
6 a 4.5387 b 0.045387 c 1.23
7 a 36 490 b 0.018 c 3886
8 a
4
5
b 1.1 c
2
3
9 a true b false c false
d true e false f true
Extended-response question
1 a $80.58 b $80.60 c $80.85 d 25c
e No, it will be 11c cheaper if rounding occurs at the
beginning.
Chapter 7
Pre-test
1 a 24 b 60 c 60
d 7 e 12 f 365
2 a Friday b Monday c Wednesday d Sunday
3 a 5 p.m. b 1:45 a.m. c 4:37 a.m. d 8:49 p.m.
e 4 p.m. f 3:45 p.m. g 10:36 a.m. h 2:14 p.m.
4 a 1 min b 2 h c 7 weeks d 360 min
5 a 3 h 55 min b 235 min
6 a 3.00 b 2.30 c 5.45
Exercise 7A
1 a February
b April, June, September, November
c January, March, May, July, August, October, December
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Answers
2 a F b D c A
d E e B f C
3 a M b D c D d M
4 a 120 s b 3 min c 2 h d 240 min
e 72 h f 2 days g 5 weeks h 280 days
5 a 180 min b 630 s c 4 min
d 1.5 h e 144 h f 3 days
g 168 h h 1440 min i 4 h
j 2 weeks k 20 160 min l 86 400 s
m 210 min n 15 s o 1.5 days
p 4.5 h q 1.25 min r 2 h
6 a 1330 h b 2015 h c 1023 h
d 2359 h e 6:30 a.m. f 1 p.m.
g 2:29 p.m. h 7:38 p.m. i 11:51 p.m.
7 a 1.5 h b 4.75 h c 7.25 h d 3.3
i
h
8 a 2 h 30 min b 4 h 15 min c 1 h 20 min
d 6 h 30 min e 3 h 45 min f 9 h 15 min
9 a 2 p.m. b 5 a.m. c 1200 hours d 1800 hours
10 a 7 h 7′ 12″ b 2 h 16′ 48″
c 3 h 3′ d 8 h 55′ 48″
11 a 6:30 a.m. b 2:25 p.m. c 7:27 p.m.
12 52.14 weeks
13 Friday
14 a 0.2 of 1 hour is one-fi fth of 1 hour = 12 min.
b 10 min =
1
6 of 1 hour and
1
6 = 0.16
i
15 a 3600 b 1440 c 3600 d 1440
16 a 2 h 24 min b 7 h 10 min 48 s c 9 h 55 min 12 s
17 Answers will vary.
Exercise 7B
1 a false b true c true
d false e true
2 a 6 h 30 min b 10 h 45 min
c 16 h 20 min d 4 h 30 min
3 a 4 h b 6 h c 3 h 30 min
d 8 h 45 min e 5 h 15 min f 7 h 30 min
4 a 2 h 5 min b 2 h 10 min
c 4 h 5 min d 2 h 11 min
e 1 h 50 min f 3 h 40 min
g 47 min h 2 h 54 min
i 2 h 46 min j 3 h 46 min
5 a 6:30 p.m. b 9 a.m. c 6:30 p.m.
d 4:30 p.m. e 5:30 p.m. f 11:40 a.m.
6 a 2 h 10 min 10 s b 3 h 15 min 6 s
c 1 h 4 min 14 s d 54 min 55 s
e 1 h 53 min 53 s f 5 h 44 min 28 s
7 a 5 h 45 min b 8 h
c 9 h 6 min d 4 h 30 min
e 3 h 50 min 28 s f 13 h 20 min 30 s
8 42 min 41 s
9 a i 16 min ii 27 min iii 24 min iv 40 min
b afternoon c 40 min
10 56 million years
11 17 min 28 s
12 7 h 28 min
13 23 h 15 min
14 1 km = 1000 m and 1 h = 3600 s, so multiply to go from hours
to seconds and divide to go from metres to kilometres.
15 So that morning and afternoon times are not confused
16 $20 857.14
17 a $900 b $90 c $1.50 d 2.5c
18 a 97.5° b 182.5° or 177.5° c 108.5°
19 1
5
7
h or 1 h 42′ 51″
Exercise 7C
1 SA, Vic., Tas., NSW, ACT
2 a i 10 ii 9.5 iii 8 iv 7
v 8 vi 2
b i 0 ii 2 iii 5 iv 5
3 Sunday
4 a 10 a.m. b 9:30 a.m. c 10 a.m.
d 9:30 a.m. e 8 a.m. f 10 a.m.
5 a 6 p.m. b 6:30 p.m. c 6:30 p.m.
6 a 11 a.m. b 12 noon c 8 p.m. d 7:30 p.m.
e 7 a.m. f 5 a.m. g 1 a.m. h 10 a.m.
7 a 5:30 a.m. b 6:30 a.m. c 6:30 a.m. d 1:30 p.m.
e 2:30 p.m. f 2:30 a.m. g 3 p.m. h 5:30 p.m.
8 a 6 h b 2.5 h c 8 h
d 6 h e 7 h
9 midnight
10 7:35 p.m.
11 6:30 a.m.
12 a 8 a.m. 29 March
b 10 p.m. 28 March
c 3 a.m. 29 March
13 a 8:30 a.m. b 9:30 a.m. c 9 a.m. d 6:30 a.m.
14 turn back 1 hour
15 During daylight saving time, it is UTC plus 11 h.
16 3 a.m. 21 April
17 11:30 p.m. 25 October
18 a SA is ahead of Qld in daylight saving time. Qld does not use
daylight saving time.
b Broken Hill uses Central Standard Time, as per SA.
c Lord Howe Island is UTC + 10:30 or 11:00 during daylight
saving
d Central Western Time and The Indian Pacifi c Train
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Puzzles and challenges
1 12 noon
2 3
3
7 or 3 min 26 s
3 Wednesday
4 3 min 10 s
5 75 s
Multiple-choice questions
1 A 2 D 3 E 4 D
5 B 6 C 7 A 8 D
9 E 10 C
Short-answer questions
1 a 90 min b 2 min c 2 days
d 21 days e 1440 min f 0.5 h
2 a 0400 b 1530 c 1919
d 6:35 a.m. e 12:51 p.m. f 11:28 p.m.
3 a 3 h 30 min b 4 h 20 min c 6 h 15 min d 1 h 45 min
4 a 3 h 36′ b 6 h 55′ 12″ c 11 h 26′ 24″
5 1 h 33 min
6 a 5 h 53 min b 4 h 5 min 55 s
c 1 h 59 min 1 s d 3 h 38 min
7 a i 25 min ii 34 min iii 20 min iv 44 min
b morning c 45 min
8 a i 8:15 a.m. ii 6:15 a.m.
b i 1:36 p.m. ii 3:36 p.m.
9 1:40 p.m.
Extended-response question
1 a i 9 p.m. ii 2 p.m. iii 4 p.m.
b i 7:30 a.m. ii 1:30 p.m. iii 1:30 a.m.
c i Tuesday ii Monday
d 1 p.m.
Chapter 8
Pre-test
1 a 11 b 5 c 5 d 21
2 a 8 b 36 c 40 d 10
3 a 13 b 21 c 4 d 5
4 a 6 × 3 b 10 ÷ 2 c 12 + 3 d 10 - 9
5 a Output = 3 × input
b Output = input + 5
c Output = input + 6
d Output = 2 × input – 1
6 a 11 b 6 c 40 d 2
7 a 20 b 44 c 22 d 100
8 a 20 b 6 c 2 d 6
9 a 1 b 9 c 3 d 45
e 25 f 5 g 0 h 25
Exercise 8A
1 a 4x, 3y, 24z, 7 b 7 c 4 d z
2 a x + 4 b x – 4 c 4x
d
4
x
e 4 – x f
4
x
3 a i 2 ii 17
b i 3 ii 15
c i 3 ii 21
d i 4 ii 2
e i 2 ii 1
f i 4 ii 12
4 a x + 1 b 5 + k c 2u d 4y e
2
p
f
3
q
g r – 12 h 9n i 10 – t j
8
y
5 a 2(x + 5) b 3a + 4 c 8k – 3 d 8(k – 3)
e 6(x + y ) f
7
2
x
g
2
+ 2 p
h 12 – xy
6 a The product of 7 and x.
b The sum of a and b.
c The sum of x and 4 is doubled.
d a is tripled and subtracted from 5.
7 a 70 b 10n
8 a $8x b $(x + 3) c $8(x + 3)
9 a 1000x b 100x c 100 000x
10 a
$
4
A
b
$A
n
c
$A − 20
n
11 ‘One-quarter of the sum of a and b.’ Other answers possible.
12 a true b true c false
d true e false
13 a false b false c true d true
14 c ÷ 2, c - 4, c + 1, 2c, 3c, 3c + 5, 4c - 2, c × c
15 ‘The sum of twice the value of x is taken from 3’ becomes
3 - 2x. Other answers possible.
Exercise 8B
1 a 14 b 1 c 10 d 8
2 17
3 15
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Answers
4 3
5 a 17 b 20 c 72 d 12
6 a 8 b 10 c 9 d 14 e 12
f 3 g 19 h 9 i 7 j 20
k 3 l 1 m 47 n 20 o 35
p 86 q 8 r 6
7 a 8 b 4 c 5 d 9
e 4 f 45 g 5 h 24
i 50 j 8 k 5 l 20
8 a 27 b 22 c 41
d 8 e 10 f 70
9 n 1 2 3 4 5 6
n + 4 5 6 7 8 9 10
x 1 2 3 4 5 6
12 – x 11 10 9 8 7 6
b 1 2 3 4 5 6
2(b – 1) 0 2 4 6 8 10
q 1 2 3 4 5 6
10q – q 9 18 27 36 45 54
10 a 75 b 45 c 54 d 11
e 12 f 5 g 33 h 19
11 5
12 x is between 4 and 33 inclusive.
13 x 5 9 12 1 6 7
x + 6 11 15 18 7 12 13
4 x 20 36 48 4 24 28
14 a 1 and 24, 2 and 12, 3 and 8, 4 and 6
b Infi nitely many answers; e.g. x =
1
5
, y = 120.
15 Because 5 × (a + a ) is 5 × 2 × a, which is 10a. Every multiple
of 10 ends with 0.
16 a x 5 10 7 9 5 8
y 3 4 2 5 2 0
x + y 8 14 9 14 7 8
x – y 2 6 5 4 3 8
xy 15 40 14 45 10 0
b 2 and 2, 3 and 1
1
2
, 6 and 1
1
5
; other answers possible.
Exercise 8C
1 a
x = 0 x = 1 x = 2 x = 3
2 x + 2 2 4 6 8
(x + 1) × 2 2 4 6 8
b equivalent
2 a
x = 0 x = 1 x = 2 x = 3
5 x + 3 3 8 13 18
6 x + 3 3 9 15 21
b no
3 6 x + 5 4 x + 5 + 2x
x = 1 11 11
x = 2 17 17
x = 3 23 23
x = 4 29 29
They are equivalent because they are always equal.
4 a 2x + 4 and x + 4 + x b 5a and 4a + a
c 2k + 2 and 2(k + 1) d b + b and 4b – 2b
5 a C b A c E
d F e B f D
6 2x + 2 + 2x, 2(2x + 1); other answers possible.
7 2(l + b ); other answers possible.
8 9a + 4b ; other answers possible (must have 2 terms).
9 6
10 If x = 8, all four expressions have different values.
11 b x × y and y × x c a × (b + c) and a × b + a × c
d a – (b + c) and a – b – c e a – (b – c) and a – b + c
f a ÷ b ÷ c and a ÷ (b × c)
12 a 4 × ( a + 2) 8 + 4 a
x = 1 12 12
x = 2 16 16
x = 3 20 20
x = 4 24 24
b 5(2 + a) c 24 + 6a
13 2a + a + 5b, 3a + 12b – 7b; others answers possible.
14 a Yes; for any value of x expressions A and B are equal, and
expression B and C are equal, so expressions A and C are
equal.
b No; e.g. if expression A is 7x, expression B is 3x and
expression C is 5x + 2x.
Exercise 8D
1 a x and y b a, b and c c k d p and q
2 a like b like terms
c terms d 15a; others answers possible.
e equivalent f 5x + 4; others answers possible.
3 a N b L c L d N
e N f L g N h L
i L j L k L l N
4 a 2a b 5x c 7b d 8d
e 12u f 12ab g 11ab h xy
5 a 3a + 5b b 7a + 9b c x + 6y d 7a + 2
e 7 + 7b f 6k – 2 g 5f + 12 h 4a + 6b – 4
i 6x + 4y j 8a + 4b + 3 k 7h + 4 l 14x + 30y
m 2x + 9y + 10 n 8a + 13 o 12b
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p 9ab + 4 q 6xy + 5x r 7cd – 3d + 2c
s 9uv + 7v t 11pq + 2p – q u 6ab + 36
6 a $27n b $31n c $58n
7 a 3a + 4 b 19
8 a 4x b 7x c 11x d 3x
9 a 12xy b 6ab + 5 c 10ab
d 6xy + 3 e 5xy + 14 f 10cde
g 6xy + 6x + 4 h 9ab + 9 i 7xy – 2y
10 a 3x + 2x 5x
x = 1 5 5
x = 2 10 10
x = 3 15 15
b For example, if x = 5 and y = 10, then 3x + 2y = 35 but
5xy = 250.
11 a 5x + 4 – 2x 3x + 4
x = 1 7 7
x = 2 10 10
x = 3 13 13
b For example, if x = 10, 5x + 4 – 2x = 34 but 7x + 4 = 74.
c For example, if x = 1, 5x + 4 – 2x = 7 but 7x – 4 = 3.
12 a 2a + a + 3b, b + 3a + 2b; other answers possible.
b 12
Exercise 8E
1 a both are 21 b both are 35 c both are 56 d yes
2 a true b true c false d true e false
3 a
2
3 b
2
3
c
2
3
4 a C b E c B d A e D
5 a 2x b 5p c 8ab d 6a
e 10ab f 10b g 28xz h 36abc
i 42ab j 42abc k 54abd l 84abc
6 a 36a b 63d c 8e d 15a
e 12ab f 63eg g 8abc h 28adf
i 12abc j 8abc k 60defg l 24abcd
7 a w 2 b a 2 c 3d 2
d 2k 2 e 7p 2 f 3q 2
g 12x 2 h 15z 2 i 36r 2
8 a 5
x
b 2
z
c 12
a
d 5
b
e
2
x f
5
d
g
x
y h
a
b
i
4 1
5
x +
j
2
5
x + y
k
2
1
+
+
x
y l
5
3
−
+
x
b
m 2x +
5
y
n 2 + x + y o
5
3
x − + b p 4b
q 0 r 11b s 7b t -4c
u -7c
9 a
2
5
b
5
9
c
9
4
a
d
2
5
b
e
2
x
f
3
4
x
g
2
3
h
3
4
i 2a j 3 k 2y l
3
y
10 a 3k b 6x c 12xy
11 a 4x grams b nx grams c 2nx grams
12 a 2 × 3 × a × b b 3 × 7 × x × y
c 22 × e × f × g d 3 × 11 × q × q × r
13 a $20 b
$
5
C
14 a 6p b 3 × 2p also simplifies to 6p, so they are equivalent.
15 a
6
2a b 12a
c
12
3
a
a
simplifies to 4. It has four times the area.
d area is multiplied by 9
16 a 3 × a × a × b × b b 7x 2y 3
c 1200a 3b 2c 2 d 24a 3b 2c 2
Exercise 8F
1 a 3a + 6
b x + y + x + y = 2x + 2y
c p + 1 + p + 1 + p + 1 + p + 1 = 4p + 4
d 4a + 2b + 4a + 2b + 4a + 2b = 12a + 6b
2 a 4x b 12 c 4x + 12
3 a 60, 3, 63
b 30, 4, 210, 28, 238
c 20, 1, 100, 5, 95
4 a 4(x + 3) 4x + 12
x = 1 16 16
x = 2 20 20
x = 3 24 24
x = 4 28 28
b equivalent expressions
5 a 12(x + 4) and 12x + 48
b 8(z + 9) and 8z + 72
c (3 + a)(3 + b) and 9 + 3a + 3b + ab
6 a 6y + 48 b 7l + 28 c 8s + 56 d 8 + 4a
e 7x + 35 f 18 + 3a g 81 – 9x h 5j – 20
i 8y – 64 j 8e – 56 k 6e – 18 l 80 – 10y
7 a 60g – 70 b 15e + 40 c 35w + 50 d 10u + 25
e 56x – 14 f 27v – 12 g 7q – 49 h 20c – 4v
i 4u + 12 j 48l + 48 k 5k – 50 l 9o + 63
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Answers
8 a 6it – 6iv b 2dv + 2dm
c 10cw – 5ct d 6es + 6ep
e dx + 9ds f 10ax + 15av
g 5jr + 35jp h in + 4iw
i 8ds – 24dt j 2fu + fv
k 14kv + 35ky l 4em + 40ey
9 a 5(x + 3) = 5x + 15 b 2(b + 6) = 2b + 12
c 3(z – 4) = 3z – 12 d 7(10 – y ) = 70 – 7y
10 a b + g + 1 b 2(b + g + 1) = 2b + 2g + 2
11 3(4x + 8y ) and 2(6x + 12y ); others answers possible.
12 2l + 2b
13 x 3
5 5x 15
14 2
6
a
3 3a
b ab 2b
15 a 4 ways, including 1(10x + 20y)
b infi nitely many ways
16 a Rosemary likes Maths and Rosemary likes English.
b Priscilla eats fruit and Priscilla eats vegetables.
c Bailey likes the opera and Lucia likes the opera.
d Frank plays video games and Igor plays video games.
e Pyodir likes chocolate and Pyodir likes tennis and Astrid
likes chocolate and Astrid likes tennis.
Exercise 8G
1 a 35 b 20
2 a 24 cm b 40 cm
3 a 3x b 36
4 $2n
5 a $10x b $15x c $kx
6 a 180 km b 30 km c 70n km
7 a $200 b $680 c $(50 + 80x)
8 a B b A c D
d E e C
9 a
No. of
hours ( t ) 1 2 3 4 5
Total cost ($) 150 250 350 450 550
b 100t + 50 c $3050
10 a $25 b 10x + 5 c $75
11 a 33 b g = 8, b = 5
c g = 3 and b = 2, g = 1 and b = 14, g = 0 and b = 20
12 a 90c b $6.30 c 0.3 + 0.6t
13 a 5b + 2c + 6d b 5b + 4c + 3d c $8
14 a cn b 3cn c 6cn
15 a 10 + 30x b 2 h 20 min
c Because you don’t pay the booking fee twice.
d $32
e It will get closer to $30.
16 a 0.2 + 0.6t b 0.8 + 0.4t c Emma’s d 3 min
e Answers will vary.
Exercise 8H
1 C 2 B 3 no, 20
4 a 17 b 23 c -8 d 8
e 6 f 11 g -7 h 6
i 3 j -9 k -9 l 6
5 a -8 b -2 c 2 d -13
e -25 f 17 g 36 h 18
6 a 38 b 5 c 4 d -13
e -4 f -16
7 a 10 b -1 c -24 d -6
8 a 6 m2 b 16 m2
9 a 32 metres per second
b 14 metres per second
c 2 metres per second
10 a i 16 ii -2
b A negative width is not possible.
11 Answer may vary.
12 Answers may vary.
13 a a - a = 0 and -b + b = 0 b = 1 a
a
c a – a = 0 d
ab
b
cancels to simply give a.
14 a i 5°C ii -15°C iii -25°C
b F
C = + 9
5
32
Exercise 8I
1 a 8, 11, 14, 17, 20 b 32, 31, 30, 29, 28
c 2, -2, -6, -10, -14 d 123, 130, 137, 144, 151
2 a 3, 6, 12, 24, 48 b 5, 20, 80, 320, 1280
c 240, 120, 60, 30, 15 d 625, 125, 25, 5, 1
3 a ratio of 3 b difference subtracting 2
c difference adding 11 d neither
e ratio of
1
2
f neither
g neither h difference subtracting 3
4 a 23, 28, 33 b 44, 54, 64 c 14, 11, 8
d 114, 116, 118 e 27, 18, 9
f 0, -1, -2 g 505, 606, 707 h -7, -13, -19
5 a 32, 64, 128 b 80, 160, 320
c 12, 6, 3 d 45, 15, 5
e 176, 352, 704 f 70 000, 700 000, 7 000 000
g 16, 8, 4 h 76, 38, 19
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6 a 50, 32, 26 b 25, 45, 55 c 32, 64, 256 d 9, 15, 21
e 55, 44, 33 f 333, 111 g 70, 98, 154 h 126, 378, 3402
7 a 17, 23, 30 b 16, 22, 29 c 36, 49, 64 d 17, 12, 6
e 17, 19, 23 f 47, 95, 191 g 5, 7, 6 h 32, 40, 38
8 a 49, 64, 81; square numbers
b 21, 34, 55; Fibonacci
c 216, 343, 512; cubes (i.e. powers of 3)
d 19, 23, 29; primes
e 16, 18, 20; composite
f 161, 171, 181; palindromes
9 a 115, 121, 128 b 24, 48, 16
c 42, 41, 123 d 9, 6, 13
10 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 (total = 55)
11 a 39 b 57
c 110 d 192
12 1, 0, 0, 1, 2
13 difference is 0, ratio is 1, Jemima
14 a 55 b 100 c 2485 d 258
15 a 3 b 10 c 45 d 276
e n × (n – 1) ÷ 2
16 a 135 b 624 c 945
Exercise 8J
1 a
b
c
d
e
2
3 a
and
b
and
c
and
d
and
e
and
f
and
4 a i
ii 4, 7, 10, 13, 16
iii 4 sticks are required to start, then 3 are added
b i
ii 3, 5, 7, 9, 11
iii 3 sticks are required at the start, then 2 are added
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Answers
c i
ii 6, 11, 16, 21, 26
iii 6 sticks are required to start, then 5 are added
d i
ii 7, 12, 17, 22, 27
iii 7 sticks are required to start, then 5 are added
e i
ii 6, 11, 16, 21, 26
iii 6 sticks are required to start, then 5 are added
f i
ii 4, 7, 10, 13, 16
iii 4 sticks are required to start, then 3 are added
5 a
b No. of crosses 1 2 3 4 5
No. of sticks required 4 8 12 16 20
c Number of sticks = 4 × number of crosses
d 80 sticks
6 a
b
No. of fence sections 1 2 3 4 5
No. of planks required 4 7 10 13 16
c Number of planks = 3 × number of fence sections + 1
d 61 planks
7 a No. of tables 1 2 3 4 5
No. of students 5 8 11 14 17
b Number of students = 3 × number of tables + 2
c 23 students
d 21 tables
8 a Spa length 1 2 3 4 5 6
No. of tiles 8 10 12 14 16 18
b Number of tiles = 2 × spa length + 6
c 36 tiles
d 12 units
9 Check with your teacher.
10 A
11 A
12 Answers may vary.
a
b
c
d
e
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f
13 a 5 b 41 c 4g + 1 d 16
14 a m + n
b number of pieces for each new panel
c
15 b
No. of straight cuts 1 2 3 4 5 6 7
No. of sections created 2 4 7 11 16 22 29
c 56 d 211
Exercise 8K
1 a true b false c true d true
2 C
3 B
4 a A b D c B d C
5 a
Input 4 5 6 7 10
Output 7 8 9 10 13
b Input 5 1 3 21 0
Output 10 2 6 42 0
c Input 11 18 9 44 100
Output 3 10 1 36 92
d Input 5 15 55 0 100
Output 1 3 11 0 20
6 a
Input 1 2 3 4 5
Output 7 17 27 37 47
b Input 6 8 10 12 14
Output 7 8 9 10 11
c Input 5 12 2 9 0
Output 16 37 7 28 1
d Input 3 10 11 7 50
Output 2 16 18 10 96
7 a Output = input + 1 b Output = 4 × input
c Output = input + 11 d Output = input ÷ 6
8 Output = 3 × input
9 Output = input - 9
10 a
Input 3 6 8 12 2
Output 7 34 62 142 2
b
Input 6 12 1 3 8
Output 5 3 25 9 4
c
Input 5 12 2 9 0
Output 30 156 6 90 0
d
Input 3 10 11 7 50
Output 15 190 231 91 4950
11 a Input c d 2p b 2 www
Output c + 6 d + 6 2p + 6 b 2 + 6 www + 6
b Input t k p 2 2f ab
Output 3t – 2 3k – 2 3p 2 – 2 6f – 2 3ab – 2
12 Input b d e g 2 mn x c 0 1
Output bc cd ec g 2c cmn xc c 2 0 c
Ouput = c × input
13 a Output = 2 × input + 1; Output = 3 × input – 2
b infinite
14 a i Output = 2 × input – 3
ii Output = 4 × input + 1
iii Output = 5 × input – 1
iv Output = input ÷ 6 + 2
v Output = 10 × input + 3
vi Output = 4 × input – 4
b–c Answers will vary.
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Answers
Exercise 8L
1
2
3
1
O 1 2 3 4 5 6
4
5
6
y
x
2
2
3
1
O 1 2 3 4
4
y
x
3 C
4 C
5 a x-axis b y-axis
c origin d fi rst
e y-coordinate f x, y, x, y
6
2
3
1
1 2 3 4 5
4
5
y
x
E
G
O
B
H
D A
C
F
7 a A (1, 4), B (2, 1), C (5, 3), D (2, 6), E (4, 0), F (6, 5), G (0, 3),
H (4, 4)
b M (1, 2), N (3, 2), P (5, 1), Q (2, 5), R (2, 0), S (6, 6), T (0, 6),
U (5, 4)
8 a Input (x ) Output (y )
0
1
2
3
2
3
4
5
b
2
3
1
O 1 2 3 4 5
4
5
y
Input
x
Output
9 a Input (x ) Output (y )
1
2
3
4
0
1
2
3
b
2
3
1
O 1 2 3 4 5
4
5
y
Input
x
Output
10 a Input (x ) Output (y )
0
1
2
3
0
2
4
6
b
2
3
1
O 1 2 3 4 5
4
5
6
y
Input
x
Output
11
2
3
1
O 1 2 3 4 5
4
5
y
x
12
2
3
1
O 1 2 3 4 5 6 7 8
4
5
6
7
8
y
x
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13 a
2
3
1
O 1 2 3 4 5 6 7 8 9 10
4
5
6
7
8
9
10
y
x
b (4, 0), (4, 3), (6, 3), (6, 0), but answers may vary.
c Answers may vary. d Answers may vary.
14 a B (1, 4), C (4, 4), D (4, 1)
b
2
3
1
O 1 2 3 4
4
y
x
15 a HELP b (4, 4), (5, 1), (3, 1), (3, 4), (5, 1), (5, 4)
c key under pot plant
d 21510032513451001154004451255143
16 a D (4, 5) b D (4, 1) c D (0, 0) d D (1, 5)
17 a Output = input + 2 b Output = input × 3
c Output = input ÷ 2
18 a (3, 6) or any with x = 3
b any point on a vertical line with x = 3
c Answers may vary. If you are given the base of an isosceles
triangle, you can have an infi nite number of possible
triangles simply by varying the height of the triangle.
d (3, 5)
19 a M(3, 2) b (1, 2)
c Find the average of the x values, then the average of the
y values.
d (4, 2) e (2.5, 3.5) f (-0.5, -0.5) g (5, 3)
Puzzles and challenges
1 a a = 4, b = 12, c = 16, d = 8, e = 36
b a = 6, b = 3, c = 5, d = 10, e = 15
2 x 2 2 3 0 5
y 7 6 3 12 1
3 x 6 6 9 0 15
x + 2 y 16 14 9 24 7
xy 14 12 9 0 5
3 5050
4 a = 1, c = 4, d = 2, e = 1, f = 3, g = 4, h = 1, i = 3,
j = 4, k = 2
5 a A = 4, B = 9, C = 2 b D = 8, E = 6, F = 2, G = 5
6 Because 2(x + 5) - 12 - x + 2 simplifi es to x.
Multiple-choice questions
1 B 2 A 3 C 4 D 5 D
6 C 7 A 8 E 9 A 10 A
Short-answer questions
1 a 5a, 3b, 7c, 12 b 12
2 a u + 7 b 3k c 7 +
2
r
d h – 10 e xy f 12 – x
3 a 15 b 24 c 2 d 32
4 a -15 b -2 c -16 d -3
5 a 16 b 200 c 5 d 20
6 a E b N c E d N
7 a L b N c L d L
e N f L g N h L
8 a 7x + 3 b 11p c 7a + 14b + 4
d 3m + 17mn + 2n e 1 + 7c + 4h – 3o
f 4u + 3v + 2uv
9 a 12ab b 6xyz c 36fgh d 64klm
10 a
3
2
b
3
5
c
3
a
d
4
3
x
z
11 a 3x + 6 b 4p – 12 c 14a + 21 d 24k + 36l
12 2(6b + 9c ), 6(2b + 3c ); other answers possible.
13 9t kg 14 g + b 15 3x
Extended-response questions
1 a i $24.50 ii $45.50 iii $213.50
b 3.5 + 2.1k c $87.50
d If k = 40, 2.1 + 3.5k = $142.10, not $87.50.
e 6 + 1.2k
2 a 26 b $78 c 4x + 2y + 10
d 12x + 6y + 30
e x (x + 5) + 3y ; other answers possible.
Chapter 9
Pre-test
1 a 7 b 20 c 16 d 5
2 a false b true c true d false
3 a 7 b 5 c 15 d 24
4 a 3 b 27 c 2 d 6
5 a k + 5 b 2p c 7y d
2
q
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Answers
6 a 5x b 28b c 5a + 7b d 7 + 10a
7 a a = 5, b = 15, c = 4, d = 7, e = 60
b a = 0, b = 2, c = 12, d = 10, e = 41
c a = 2, b = 5, c = 7, d = 1, e = 13
d a = 3, b = 2, c = 1, d = 0, e = 4
8 a ÷ b - c × d +
Exercise 9A
1 a E b N c E d E e E
f N g E h E i N
2 a true b false c false
3 a 19 b 19 c true
4 true
5 a true b false c true d true
e false f false g true h false
i true j true k false l true
m false n false o true
6 a false b true c false d true
7 a true b true c false d true
8 a true b false c true d true
e true f false g false h true
i true
9 a 3 + x = 10 b 5k = 1005 c a + b = 22 d 2d = 78
e 8x = 56 f 3p = 21 g
4
t
= 12 h q + p = q × p
10 a 6c = 546 b 5x = 37.5 c 12a + 3b = 28
d f + 10 = 27 e j + 10 + m + 10 = 80
11 m = 3
12 k = 2, k = 6
13 x = 1 and y = 5, x = 2 and y = 4, x = 3 and y = 3, x = 4 and
y = 2, x = 5 and y = 1
14 a S b S c A d A
e N f N g A h S
i A j S k N l N
15 a 6 = 2 × 3; other solutions possible.
b 5 - 4 = 1; other solutions possible.
c 10 ÷ 2 = 7 - 2; other solutions possible.
d 4 - 2 = 10 ÷ 5; other solutions possible.
16 a 12 b 4
c Yes; if 5 is changed to 1, then 1 + 2 = 3 is a true equation.
d Yes; if + is changed to -, then 5 - 3 = 2 is a true equation.
Exercise 9B
1 a true b false c true d false
2 a 12 b 17 c 13 d 6
3 a 3 b 6 c 10 d 4
e 6 f 70 g 20 h 19
4 a x b c c b d d
5 a y = 8 b l = 2 c l = 6 d d = 2
e l = 12 f a = 6 g s = 12 h x = 8
i e = 8 j r = 10 k s = 8 l z = 3
6 a p = 3 b p = 4 c q = 3 d v = 5
e b = 1 f u = 4 g g = 3 h e = 3
i d = 4 j d = 6 k m = 4 l o = 3
7 a x = 3 b x = 7 c x = 5
d x = 4 e x = 1 f x = 5
g x = ±2 h x = ±10 i x = ±6
8 a 11 b 12 c 16
d 33 e 30 f 2
9 a 10x = 180 b x = 18
10 a 2w = 70 b w = 35
11 a 4.5x = 13.5 b x = 3
12 a y + 12 = 3y b y = 6
13 a x = 9 b 2x + 1 = 181 so x = 90
14 a x = 14, so 14, 15 and 16 are the numbers.
b LHS is 3x + 3 or 3(x + 1), which will always be a multiple of 3.
15 a x = 2 and y = 6; other solutions possible.
b x = 12 and y = 10; other solutions possible.
c x = 1 and y = 1; other solutions possible.
d x = 12 and y = 0.5; other solutions possible.
e x = 10 and y = 0; other solutions possible.
f x = 2 and y = 2; other solutions possible.
Exercise 9C
1 a 10d + 15 = 30 b 7e + 10 = 41
c 2a + 10 = 22 d x + 10 = 22
2 a C b D c E d B e A
3 a 6 + x = 11 b 6x = 14 c 3 = 2q d 6 + a = 10
e 12 + b = 15 f 0 = 3b + 2 g 4 = 7 + a
h 12x + 2 = 8 i 7p = 12
4 a subtracting 2 b adding 2 c dividing by 10
d multiplying by 2 e dividing by 3
f adding 3 g dividing by 4 h adding 4
5 a 3(x + 4) = 27 b 18 = 3(a + 5) c 3d = 18
d 3(11 + a) = 36 e 3(3y + 2) = 33 f 3(2x + 4) = 30
6 a E (+ 2) b A (+ 4) c D (÷ 2) d B (- 2)
e C (× 2)
7 a × 3 then + 2 b × 10 then - 3 c ÷ 5 then - 2
d - 10 then ÷ 3
8 a 2q = 2 b 10x = 7 c 3 + x = 20 d x = 60
9 a 3x + 2 = 14
3x = 12
x = 4
10x = 40
10x + 1 = 41
- 2 - 2
÷ 3 ÷ 3
+ 1 + 1
× 10 × 10
b 5x - 3 = 32
5x = 35
x = 7
x + 2 = 9
+ 3 + 3
÷ 5 ÷ 5
+ 2 + 2
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c (x ÷ 2) + 4 = 9
x ÷ 2 = 5
x = 10
x + 8 = 18
- 4 - 4
× 2 × 2
÷ 2 ÷ 2
+8 + 8
(x + 8) ÷ 2 = 9
10 a i 0 = 0 ii 0 = 0 iii 0 = 0
b Regardless of original equation, will always result in 0 = 0.
11 a x = 3
7x = 21
+ 2 + 2
× 7 × 7
7x + 2 = 23
b x = 3
2x = 6
+ 8 + 8
× 2 × 2
8 + 2x = 14
12 a True; you can + 3 to both sides and then - 3 to get the
original equation again.
b True; simply perform the opposite operations in the reverse
order, so + 4 becomes - 4.
c True; use the operations that take equation 1 to equation 2
and then the operations that take equation 2 to equation 3.
d false; e.g. equation 1: x = 4, equation 2: x = 5,
equation 3: 2x = 8.
Exercise 9D
1 a true b false c true
d false e false
2 a 6 b x = 6
3 g = 2
4 a -5 b ÷10 c ×4 d +12
5 a m = 9 b g = 11 c s = 9 d i = 10
e t = 2 f q = 3 g y = 12 h s = 12
i j = 4 j l = 4 k v = 2 l y = 12
m k = 5 n y = 9 o z = 7 p t = 10
q b = 12 r p = 11 s a = 8 t n = 3
6 a 7a + 3 = 38
7a = 35
a = 5
- 3 - 3
÷ 7 ÷ 7
b 4b - 10 = 14
4b = 24
b = 6
+ 10 + 10
÷ 4 ÷ 4
c 2(q + 6) = 20
q + 6 = 10
q = 4
÷ 2 ÷ 2
- 6 - 6
d
5 = 10
x
- 3 + 3 - 3
2 = 10
x
20 = x
× 10 × 10
7 a subtract 3 b add 7 c divide by 5 d divide by 2
8 a f = 11 b k = 4 c x = 9 d a = 9
e k = 8 f a = 6 g n = 11 h n = 8
i g = 4 j q = 11 k z = 10 l p = 1
m d = 4 n t = 8 o u = 5 p c = 1
q q = 11 r y = 12 s q = 2 t u = 10
9 a x =
3
4
b k =
24
5
c w =
4
3
d x =
4
3
e x =
5
8
f x =
1
3
10 a r = −7 b x = −3 c t = −16 d y = −24
e x = −5 f k = −9 g x = −6 h x = −4
i x =
3
2
−
11 a x + 5 = 12 → x = 7 b 2y = 10 → y = 5
c 2b + 6 = 44 → b = 19 d 3(k – 7) = 18 → k = 13
e
4
b
+ 3 = 6 → b = 12 f
2
k
– 10 = 1 → k = 22
12 a 12n + 50 = 410 b n = 30 h
13 a 12 + 5x b 12 + 5x = 14.5
c x = 0.5, so pens cost 50 cents.
d 12 + 5 (0.5) = 14.5
14 a 3b = 15 → b = 5 b 4x = 12 → x = 3
c 2(10 + x ) = 28 → x = 4 d 4b = 28 → b = 7
15 a x = 6 b x = 8 c x = 5
16 Examples include: x + 1 = 3, 7x = 14, 21 – x = 19,
4
x
= x,
2
x
= 1.
17 a 2x + 5 = 13
2x = 8
x = 4
5x = 20
- 5 -5
÷ 2 ÷ 2
× 5 × 5
b (10 + 2x = 20
2x = 10
x = 5
x - 3 = 2
-10 -10
÷ 2 ÷2
× 2 × 2
- 3 -3
2(x - 3) = 4
c yes d yes
18 a First step, 4x + 2 is not completely divided by 4.
b Second step, LHS divided by 3, RHS has 3 subtracted.
c First step, RHS has 5 added not subtracted.
d First step, LHS has 11a subtracted, not 12.
19 a x = 4 b x = 1 c l = 3 d t = 1
e s = 5 f b = 19 g j = 2 h d = 1
Exercise 9E
1 a true b false c true
d false e true f false
2 a 7 b 9 c false
3 a b = 44 b d = 15 c h = 28 d p = 26
4 a B b C c A d D
5 a m = 12 b c = 18 c s = 16 d r = 10
e u = 20 f y = 18 g x = 4 h a = 16
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Answers
i h = 10 j j = 15 k v = 9 l q = 8
6 a h = 9 b y = 6 c j = 3 d b = 4
e u = 3 f t = 9 g w = 6 h r = 4
i q = 9 j s = 3 k l = 8 l z = 7
m v = 11 n f = 9 o x = 2 p d = 5
q n = 5 r m = 11 s p = 8 t a = 9
7 a y = -1 b a = -10 c x = -10 d x = -48
e u = -30 f y = -10 g u = -4 h d = -5
8 a
2
t
= 9 → t = 18 b
3
q
= 14 → q = 42
c
2
5
r
= 6 → r = 15 d 4
2
3 − = q → q = 10
e
3
4
2 + = x
→ x = 5 f
4
y
+ 3 = 5 → y = 8
9 a
5
b
= 31.50 b b = 157.5 c $157.50
10 a
2
x
+ 5 b
2
x
+ 5 = 11 → x = $12 c $6
11 a 6x = 3; other solutions possible.
b 7x = 5
c yes; e.g. 2x + 1 = 0
12 a The different order in which 3 is added and the result is
multiplied by 5.
b multiply by 5
c subtract 3
d No, the difference between them is always 2.4 for any value
of x.
13 a i multiply by 2 ii divide by
1
2
b x = 26 for both of them.
c Makes the fi rst step a division (by a fraction) rather than
multiplication.
14 a x =
7
2
b x =
17
6
c x =
19
6
d x =
7
6
15 Answers will vary. Substitute x = 12 into your equation to
ensure that it is a valid solution.
Exercise 9F
1 C
2 a 6 b 4 c 2 d 12
3 a true b false c true
d true e false f false
4 a 2x + 2 b 10b + 15 c 6a - 8 d 35a + 5
e 12x + 16 f 24 - 9y g 48a + 36 h 2u - 8
5 a 4a + 2 b 5 + 3x c 3b - 4 d 3a + 12
e 6x + 3 f k + 6 g 2b + 6 h 5k + 1
6 a s = 6 b l = 1 c p = 9 d y = 0
e q = 1 f p = 12 g m = 5 h b = 6
i p = 3 j p = 7 k y = 9 l r = 8
7 a d = 3 b x = 2 c x = 5 d e = 1
e a = 1 f r = 3 g u = 1 h q = 11
8 a s = 1 b i = 1 c c = 5 d v = 8
e k = 1 f q = 3 g y = 4 h f = 3
i t = 2
9 a u = -5 b k = -3 c p = -1 d q = -2
e u = -1 f x = -3 g p = -4 h r = -10
10 a i 2(5 + x ) = 14 ii x = 2
b i 3(q - 3) = 30 ii q = 13
c i 2(2x + 3) = 46 ii x = 10
d i 2(y + 4) - y = 17 ii y = 9
11 a LHS simplifi es to 10, but 10 = 7 is never true.
b LHS simplifi es to 15, not 4.
c LHS simplifi es to 6, not 12.
12 a LHS = 9, RHS = 9, therefore true.
b LHS = 9, RHS = 9, therefore true.
c LHS simplifi es to 9.
d For example, 2(x + 5) - 3 - 2x = 7. Others possible.
13 a s = 6 b l = 1 c p = 9 d y = 0
e q = 1 f p = 12 g m = 5 h b = 6
i p = 3 j p = 7 k y = 9 l r = 8
14 a j = 2 b a = 1 c a = 3 d a = 8
e c = 4 f d = 8 g x = 1 h x = 3
i x = 0
Exercise 9G
1 a N b R c R
d N e R f R
2 a 15 b 36 c 8 d 2
3 a h = 7 b h = 9 c m = 8 d m = 10
4 a y = 23 b x = 4 c x = 7
5 a A = 7 b q = 4 c t = 0
6 a G = 27 b x = 1 c y = 5
7 a 20 = b × 4 → b = 5
b i 25 = 5h → h = 5 ii square
8 a P = 16 b h = 3 c 24 units squared
9 a F = 68 b C = 10 c 12°C d 15°C
10 a
3
5
= −
d
S b
b
5( 32)
9
= −
C
F
c x
Q = − 36
4
11 a Check by substituting values back into equation.
b If D = 20, C should equal 60 not 50, as in row 2.
c Check by substituting values back into equation.
d For example, C = 2t − 10, C =
20
Dt
+ 20; other solutions
possible.
12 a Abbotsford Apes
b 8 goals
c S = 9q + 6g + b
d 0 goals and 0 behinds, 2 goals and 12 behinds, 3 goals and
9 behinds, 7 goals and 7 behinds.
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Exercise 9H
1 a A b C c C
2 a x = 6 b a = 2 c k = 9
3 a Let c = car’s cost b c + 2000 = 40 000
c c = 38 000 d $38 000
4 a Let p = cost of one pen b 12p = 15.6
c p = 1.3 d $1.30
5 a Let h = number of hours worked
b 17h + 65 = 643
c h = 34 d 34 hours
6 a 24l = 720 b l = 30 c 30 m d 108 m
7 2(2x + 3) = 34 → x = 7
8 x = 4
9 1.5 h
10 14 years old
11 2x + 154 = 180 → x = 13
12 3y = 90 → y = 30
13 Examples include: h = 3, b = 6 or h = 12, b = 2.4; other
solutions possible.
14 x = 3.5, y = 2
15 a possible (p = 12)
b Not possible because solution is not a whole number.
c possible (p = 8)
16 a + 2b = 180, so a = 180 - 2b = 2(90 - b ) is always even.
17 a i x = 60 ii One angle is -10°, which is impossible.
b i 60 - x + 70 + x + 50 is always 180, regardless of the
value of x.
ii any value less than 60 and greater than -70
c Answers will vary.
Puzzles and challenges
1 a 26 b 29 c 368 d 31
1
3 e 36
2 7 and 13
3 a = 5, b = 2, c = 12
4 a a = 22.5, b = 37.5 b a = 10, b = 23
c a = 36, b = 108 d a = 60, b = 40
5 A corny joke.
6 26 sheep, 15 ducks
Multiple-choice questions
1 C 2 A 3 D 4 B 5 E
6 C 7 A 8 A 9 B 10 E
Short-answer questions
1 a false b true c true
d false e true f false
2 a 2 + u = 22 b 5k = 41 c 3z = 36 d a + b = 15
3 a x = 3 b x = 6 c y = 1
d y = 9 e a = 2 f a = 10
4 a 2x = 8 b 7a = 28 c 15 = 3r d 16 = 8p
5 a x = 3 b r = 45 c p = 9 d b = 6
e x = 9 f r = 4 g q = 2 h u = 8
6 a u = 8 b p = 3 c x = 4
d y = 8 e y = 4 f x = 15
7 a 6 + 4p b 12x + 48 c 7a + 35 d 18x + 9
8 a x = 8 b x = 8 c x = 7
d y = 10 e z = 5 f q = 3
9 a no b Solution is x =
7
11
, which is not whole.
10 a No; LHS simplifi es to 10.
b i Is a solution. ii Is a solution.
11 a A = 450 cm2 b h = 10 cm
12 a 36 b 6 c 3
13 a 5 b 6 c 20 years d 6 cm
14 a y = 35 b y = 30
Extended-response questions
1 a 75 cents b C = 15 + 2t c $1.75
d 12 seconds e 81 seconds
f 2.5 min in total (50 seconds for the fi rst call, then
100 seconds)
2 a $500
b 30 hours at $x/hour, and 10 hours at $(x + 2)/hour.
c $660
d x = 15
e x = 21, so Gemma earned $630 from Monday to Friday.
Chapter 10
Pre-test
1 a 40 mm b 60 mm c 8 mm
2 a mm, cm, m, km b mg, g, kg, t
c s, min, h, day, year
3 a 10 b 100 c 1000
d 60 e 60 f 1000
4 a 2000 b 2 c 56 d 2500
e 180 f 7200 g 1.4 h 0.027
5 a 55 m b 27 cm c 28 cm
6 a 32 b 7 c 8
7 a kilometres b kilograms
c minutes d centimetres
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Answers
Exercise 10A
1 a i 1 foot = 12 inches = 16 digits = 4 palms
ii 1 mile = 1000 paces
b i 1 foot = 12 inches
ii 3 feet = 1 yard
iii 1 mile = 1760 yards
c i 1 m = 100 cm ii 1 cm = 10 mm iii 1 km = 1000 m
2 digit, inch, palm, foot, cubit, pace, mile
3 inch, foot, yard, rod, chain, furlong, mile
4 millimetre, centimetre, metre, kilometre
5 a 5 b 5000 c 4
d 20 e 16 f 80
6 a 12 b 3 c 36
d 1760 e 22 f 40
7 a 10 b 100 c 1000
d 1000 e 100 000 f 1 000 000
8 a kilometres b millimetres
c metres d metres
e centimetres f kilometres
9 a metres b millimetres
c kilometres d kilometres
e metres f centimetres
10 1 pace
11 5000
12 440
13 a 2 mm b 5 mm c 2 cm
d 5 cm e 8 cm
14 a 6 cm b 5 cm c 25 cm
15 a
mm cm m km
mm 1
1
10
1
1000
1
1000 000
cm 10 1
1
100
1
100 000
m 1000 100 1
1
1000
km 1 000 000 100 000 1000 1
b inch feet yard mile
inch 1
1
12
1
36
1
63 360
feet 12 1
1
3
1
5280
yard 36 3 1
1
1760
mile 63 360 5280 1760 1
c digit palm feet pace mile
digit 1
1
4
1
16
1
80
1
80 000
palm 4 1
1
4
1
20
1
20 000
feet 16 4 1
1
5
1
5000
pace 80 20 5 1
1
1000
mile 80 000 20 000 5000 1000 1
16 4 rods = 1 chain, but conversion to other units is less simple.
17 All conversions involve a power of 10.
18 Answers may vary.
Exercise 10B
1 a 100 b 1000 c divide
d multiply e multiply f divide
2 a 1000 b 1000 c 100 000 d 1 000 000
3 a right b left
4 a 50 mm b 200 cm c 3500 m d 2610 cm
e 4 cm f 5 m g 4.2 km h 47.2 cm
i 684 cm j 20 m k 926.1 cm l 4.23 km
5 a 2.5 cm b 82 mm c 2.5 m
d 730 cm e 6200 m f 25.732 km
6 a 3000 mm b 600 000 cm c 2400 mm
d 4000 cm e 0.47 km f 913 m
g 0.216 km h 0.0005 m
7 a 2 cm b 5 cm c 1.5 cm d 3.2 cm
e 3 cm f 3 cm g 1.2 cm h 2.8 cm
8 a 2.7 m b 0.4 km
9 a 8.5 km b 310 cm c 19 cm
10 a 38 cm, 0.5 m, 540 mm
b 160 cm, 2100 mm, 0.02 km, 25 m
c 142 mm, 20 cm, 0.003 km, 3.1 m
d 10 mm, 0.1 m, 0.001 km, 1000 cm
11 125 cm
12 Bigan tower
13 a $8200 b $6.56 c 41c
14 625 years 15 50 000 years
16 0.08 mm 17 2500 s
18
×1000000
×100000
×1000
÷1000000
÷100000
÷1000
km
m
cm
mm
÷1000 ×1000
÷100
÷10
×100
×10
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19 So that only one unit is used and mm deliver a high degree
of accuracy.
20 a i 1 million ii 10 000 iii 1000
iv 1 billion (1 000 000 000)
b 0.312 mm
c The distance you travel in 1 year at light speed
Exercise 10C
1 a 10 cm b 12 cm
2 a 6 cm b 12 cm c 5.2 cm d 6.4 cm
3 a 15 cm b 37 m c 30 km
d 2.4 m e 26 cm f 10 cm
4 a 42 cm b 34 m c 36 km
5 a 8.4 cm b 14 m c 46.5 mm
6 $21 400
7 a 516 ft b 157.38 m
8 a 70 mm b 72 cm
9 a 40.7 cm b 130.2 cm c 294 cm
10 400 m
11 a 5 cm b 5 m c 7 km
12 4, including a square
13 a P = a + 2b b P = 4a c P = 2a + 2b
d P = 4a e P = 2a + 2b f P = 2a + b + c
14 a P = 2a + 2b + 2c or 2(a + b + c)
b P = 2a + 2b + 2c or 2(a + b + c)
15 a
4
P
b
( 2 )
2
P − a
16 a 160 cm b 216 cm c 40 cm
d 4a + 32 or 4(a + 8) cm
Exercise 10D
1 a 15.71 b 40.84 c 18.85 d 232.48
2 a 3.1 b 3.14 c 3.142
3 a diameter b radius c circumference
d segment e chord f sector
g tangent
4 Answer is close to pi.
5 a false b true c false d true
e true f false g false
6 a 12.57 mm b 113.10 m c 245.04 cm d 12.57 m
e 21.99 km f 15.71 cm
7 a 300 cm b 60 m c 18 km
8 a 44 mm b 132 cm c 220 m
9 11.0 m
10 12 566 m
11 Svenya and Andre
12 B
13 3 cm
14 a i 2 m ii 5 cm iii 15 km
b i 2 mm ii 3 m iii 0.5 cm
15 d = 2r, so 2pr is the same as pd.
16 a 4p mm b 36p m c 78p cm d 4p m
e 7p km f 5p cm
17 a i
2
=

r
C
ii =

d
C
b i 2.23 m ii 6.37 cm
Exercise 10E
1 a
1
4
, quadrant b
1
2 , semicircle
2 a
1
2
b
1
4
c
1
6
d
1
3
e
3
4
f
5
8
3 a square, semicircle b quadrant, rectangle
c sector, triangle
4 a 8.38 cm b 5.59 m c 12.22 mm
d 1.96 m e 136.14 cm f 1.15 km
5 a 14.3 cm b 35.7 m c 51.4 mm
d 36.0 km e 13.1 m f 14.0 cm
6 a 18.7 m b 11.1 cm c 101.1 cm
d 35.4 km e 67.1 mm f 45.1 cm
7 657 cm
8 a 26.88 m b 52.36 m c 6.24 m
9 a 25.13 cm b 56.55 m c 35.71 m
10 The four arcs make one full circle (2πr ) and the four radii
make 4r.
11 The perimeter includes one large semicircle (
1
2 × 2π r = π r )
and the two smaller semicircles, which make one full smaller
circle (π × r ). So, the total is 2π r.
12 a 7.6 m b 34.4 m c 9.5 km
13 a 24 + 3π cm b 9 + 2.5 π cm
14 a 16π cm b 18π m c 5π + 20 cm
Exercise 10F
1 a 8 b 4 cm and 2 cm c 8 cm2
2 a 9 b 3 cm and 3 cm c 9 cm2
3 a 5 square units b 8 square units
c 96 square units
4 a cm2 b m2 c ha
d km2 e ha f mm2
5 a 6 cm2 b 3 cm2 c 2 cm2
d 5 cm2 e 4.5 cm2 f 9 cm2
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Answers
6 a 200 cm2 b 22 mm2 c 7 cm2 d 25 m2
e 1.44 mm2 f 6.25 mm2 g 1.36 m2 h 0.81 cm2
i 179.52 km2
7 a 2 cm b 5 m c 12 km
8 a 2 ha b 10 ha c 0.5 ha
9 5000 m2
10 2500 cm2
11 20 000 cm2
12 a 25 cm2 b 12 cm c 4 units
13 $2100
14 5 L
15 8 ha
16 a i 10 cm ii 9 mm
b Divide the area by the given length.
17 half of a rectangle with area 4 cm2
18 a 121 cm2 b 4
2  
 
P
19 Area is quadrupled (× 4).
20 a i 100 ii 10 000 iii 1 000 000
b ×1000000
÷1000000
km2 m2 cm2 mm2
×10000
÷10000
×100
÷100
c i 200 mm2 ii 100 000 cm2 iii 3 500 000 m2
iv 3 cm2 v 2.16 m2 vi 4.2 km2
vii 5000 mm2 viii 100 ha ix 0.4 ha
Exercise 10G
1 a 20 cm b 7 m c 2 m d 6.3 cm
2 a 6 m b 11 mm c 1.9 m d 3.2 mm
3 a 10 b 56 c 12.5
4 a 16 m2 b 30 cm2 c 160 m2 d 1.2 m2
e 3 mm2 f 24.5 km2 g 1.3 cm2 h 20 m2
i 4.25 m2
5 a 2 cm2 b 6 cm2 c 3 cm2 d 3 cm2
6 480 m2
7 4800 m2
8 6.8 cm2
9 160 m2
10 $6300
11 a 5 cm b 4.4 mm
12 Yes, the base and height for each triangle are equal.
13 No, the base and height are always the same.
14
2 b =
A
h
15 a 7 cm2 b 8 cm2 c 11 cm2 d 11 cm2
Exercise 10H
1 a A = bh
= 5 × 7
= 35
b A = bh
= 20 × 3
= 60
c A = bh
= 8 × 2.5
= 20
2 a b = 6 cm, h = 2 cm
b b = 10 m, h = 4 m
c b = 5 m, h = 7 m
d b = 5.8 cm, h = 6.1 cm
e b = 5 cm, h = 1.5 cm
f b = 1.8 m, h = 0.9 m
3 a 40 m2 b 28 m2 c 36 km2 d 17.5 m2
e 30 m2 f 6.3 cm2 g 1.8 cm2 h 14 cm2
i 176 mm2
4 a 6 cm2 b 4 cm2 c 15 cm2 d 8 cm2
5 54 m2
6 a 2 m b 7 cm c 0.5 mm
7 a 10 cm b 5 m c 2 km
8 $1200
9 a 1800 cm2 b 4200 cm2
10 Because height must be less than 5 cm.
11 half; Area (parallelogram) = bh and Area (triangle) =
1
2
bh
12 Area = twice triangle area
= 2 ×
1
2
bh
= bh
13 $4 500 000
Exercise 10I
1 a b c
2 a addition b subtraction c subtraction
3 a A = s 2 + lb
= 12 + 3 × 1
= 1 + 3
= 4 cm2
b A = lb -
1
2 bh
= 13 × 8 -
1
2 × 5 × 4
= 104 - 10
= 94 m2
4 a 33 m2 b 600 mm2 c 25 m2
d 21 m2 e 171 cm2 f 45 km2
5 a 39 m2 b 95.5 cm2 c 26 m2
d 78.5 m2 e 260 cm2 f 4 m2
6 a 62 cm2 b 16 m2 c 252 cm2
7 a 80 cm2 b 7 m2 c 14.5 m2 d 11.75 cm2
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8 10.08 m2
9 yes, with $100 to spare
10 Subtraction may involve only two simple shapes.
11 a No; bases could vary depending on the position of the top
side.
b Yes, 40 m2; take out the rectangle and join the triangles to
give a base of 10 − 6 = 4.
12 a See given diagram in textbook.
b i 1
16
,
1
32
,
1
64
,
1
128
,
1
256
,
1
512
,
1
1024
,
1
2048
,
1
4096
,
1
8192
ii no
c 1 square unit d Total must equal 1.
Exercise 10J
1 a 1 kg, 1000 g b 1000 mg, 1 g
2 a C b F c A
d D e B f E
3 a B b A c D d C
4 a 2000 kg b 70 000 g
c 2400 mg d 2.3 g
e 4.620 g f 21.6 t
g 470 kg h 0.312 kg
i 0.027 g j 750 kg
k 125 g l 0.0105 kg
m 210 t n 470 kg
o 592 g p 80 g
5 a 4 kg b 12 g c 65 t
6 a 12°C b 37°C c 17°C
d 225°C e 1.7 °C f 31.5°C
7 a 60 kg b 60 000 g c 60 000 000 mg
8 a 3000 g b 3 kg
9 33°C
10 147°C
11 a 8 kg b 8.16 kg
12 a 400 mg, 370 g, 2.5 kg, 0.1 t
b 290 000 mg, 0.000 32 t, 0.41 kg, 710 g
13 a 4th day b 25°C c 26°C
14 50 days
15 yes, by 215 kg
16 a 45.3 t b yes, by 2.7 t
17 a 1 g b 1 t c 1000 t
18 a 12 kg b 1000 kg c 360 000 kg
19 a i 10°C ii 27°C iii 727°C
b i 273 K ii 313 K iii 0 K
20 a 180°F b
5
9 c
9
5
d i 0°C ii 20°C iii 60°C iv 105°C
e
9
5
F = + 32
C
Puzzles and challenges
1 Both lines are the same length.
2 Mark a length of 5 m, then use the 3 m stick to reduce this to
2 m. Place the 3 m stick on the 2 m length to show a remainder
of 1 m.
3 a 3 cm2 b 20 cm2
4 500 L
5 10.5 cm2
6 48 cm2
Multiple-choice questions
1 D 2 E 3 C 4 A 5 E
6 B 7 E 8 B 9 D 10 D
Short-answer questions
1 a i 16 ii 1000
b i 3 ii 12
c i 10 ii 100 000
2 a 50 mm b 2 m c 3700 m d 4.21 km
e 7100 g f 24.9 g g 28.49 t h 9000 g
i 4 L j 29.903 kL k 400 kL l 1000 mL
m 1440 min n 60 min o 3.5 days p 9000 s
3 a 2.5 cm b 2.3 cm c 4.25 kg d 5 L
4 a 16 m b 20.6 cm c 23 m
d 34 km e 3.2 mm f 24 m
5 a 24.01 cm2 b 14 km2 c 67.5 m2 d 12 cm2
e 14 m2 f 5 cm2 g 22 m2 h 21 mm2
i 291 cm2 j 52 m2 k 14 m2 l 0.9 km2
6 a (3p + 18) cm b (3p + 16) cm
7 (15p + 10) cm
Extended-response questions
1 436.64 m2
2 a 97.3 m b 5 min 30 s
c 270 m2 d $2040
Chapter 11
Pre-test
1 a 48 b 72 c 45 d 91 e 180 f 121
2 13 × 7 and 11 × 11
3 a 2 b 8 c 4 d 5
e 6 f 1 g 10 h 6
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Answers
4 a yes b no c no d yes
5 a N b P c P d N
6 a true b true c true d true
7 a 9 b 25 c 36 d 196 e 121 f 100
8 a 9 b 13 c 15
9 a 32 b 81 c 125 d 1
10 a 4 b 7 × 7 c 6 d 11 × 11 = 121
Exercise 11A
1 a not even
b digits do not sum to a multiple of 3
c 26 is not divisible by 4
d last digit is not 0 or 5
e not divisible by 3 (sum of digits is not divisible by 3)
f 125 is not divisible by 8 and it is not even
g sum of digits is not divisible by 9
h last digit is not 0
2 a 2 b 2 c 0 d 0
3 3, 6 and 9
4 2, 5 and 10
5 i a yes b no c yes
d yes e no f yes
g no h yes i no
j yes k no l yes
m yes n no o no
p yes q no r no
ii Using a calculator
6 a 10, 15, 20, 25, 30 b 12, 15, 18, 21, 24
c 10, 12, 14, 16, 18 d 12, 18, 24, 30, 36
e 16, 24, 32, 40, 48 f 18, 27, 36, 45, 54
g 10, 20, 30, 40, 50 h 12, 16, 20, 24, 28
Number
Divisible
by 2
Divisible
by 3
Divisible
by 4
Divisible
by 5
Divisible
by 6
Divisible
by 8
Divisible
by 9
Divisible
by 10
243 567 ✘ ✓ ✘ ✘ ✘ ✘ ✓ ✘
28 080 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
189 000 ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓
1 308 150 ✓ ✓ ✘ ✓ ✓ ✘ ✓ ✓
1 062 347 ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✘
7
12 980
13 966
14 a yes
b Multiples of 3; adding a multiple of 3 does not change the
result of the divisibility test for 3.
c 18 d, e Answers will vary.
15 a 0, 4, 8 b 2, 6
16 36
17 a 11 b 9 c 7 d 5 e 3
18 a 11, 22, 33, 44, 55, 66, 77, 88, 99
b 0
c 110, 121, 132, 143, 154, ..., 209, ... , 308, 319, ...
d equals the centre digit or 11 + the centre digit
e difference is 0 or 11
f Sum the odd- and even-placed digits. If the difference
between these two sums is 0 or is divisible by 11, then the
number is divisible by 11.
g i yes ii yes iii no iv yes
v yes vi yes
Exercise 11B
1 no
2 yes
3 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
4 4, 6, 8, 9, 10, 12, 14, 15, 16, 18
5 101
6 211
7 a C b P c C d P
e C f C g P h P
i C j C k C l P
m P n P o C p P
8 a no b $14.25
9 2, 4, 8, 11, 22, 44
10 Start by working out how many numbers are divisible by 5.
This answer is found by dividing 250 by 5, which equals 50. As
there are 50 multiples of 5 in 250, there are 200 other numbers
between 1 and 250 that are not multiples of 5 and, therefore,
these 200 numbers are not divisible by 5.
11 15
8 a 2, 3, 7 b 3, 13 c 2, 3, 5
d 5 e 2, 7 f 2, 3
9 a 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49
b 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69
c 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99
10 a 5, 11 b 7, 13 c 11, 13
d 11, 17 e 5, 73 f 7, 19
11 14
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12 5 and 7, 11 and 13, 17 and 19, as well as other pairs
13 17 and 71, as well as other pairs
14 (5, 7, 11), as well as other er groups
15 32 = 29 + 3, 34 = 29 + 5, 36 = 29 + 7, 38 = 31 + 7, 40 = 37
+ 3, 42 = 31 + 11, 44 = 41 + 3, 46 = 41 + 5, 48 = 41 + 7
16 2 + 3 = 5 and 2 + 5 = 7. All primes other than 2 are odd and
two odds sum to give an even number that is not a prime. So
any pair that sums to a prime must contain an even prime,
which is 2.
17 Check your spreadsheet using smaller primes.
Exercise 11C
1 E
2 D
3
Value
Base
number
Index
number
Basic
numeral
23 2 3 8
52 5 2 25
104 10 4 10 000
27 2 7 128
112 1 12 1
121 12 1 12
05 0 5 0
4 a 33 b 25 c 154 d 104 e 62
f 203 g 16 h 43 i 1002
5 a 32 × 52 b 22 × 73 c 92 × 122 d 53 × 82
e 33 × 63 f 74 × 132 g 43 × 71 × 131 h 93 × 102
i 23 × 32 × 52
6 26 × 35 × 54
7 a 2 × 2 × 2 × 2 b 17 × 17
c 9 × 9 × 9 d 3 × 3 × 3 × 3 × 3 × 3 × 3
e 14 × 14 × 14 × 14 f 8 × 8 × 8 × 8 × 8 × 8 × 8 × 8
g 10 × 10 × 10 × 10 × 10 h 54 × 54 × 54
8 a 3 × 3 × 3 × 3 × 3 × 2 × 2 × 2
b 4 × 4 × 4 × 3 × 3 × 3 × 3
c 7 × 7 × 5 × 5 × 5
d 4 × 4 × 4 × 4 × 4 × 4 × 9 × 9 × 9
e 5 × 7 × 7 × 7 × 7
f 2 × 2 × 3 × 3 × 3 × 4
g 11 × 11 × 11 × 11 × 11 × 9 × 9
h 20 × 20 × 20 × 30 × 30
9 a 32 b 64 c 1000 d 72
e 10 000 f 1000 g 64 h 121
10 a 25 b 1 c 10 d 64
e 128 f 8 g 22 h 900 i 8
11 a 56 b 810 c 15 750
d 7343 e 92 096 f 29 701
12 a 4 b 2 c 3 d 6
e 3 f 2 g 2 h 4
13 a < b > c = d <
e > f > g < h <
14 125
15 a 166 b 55 min
c 305 175 781 d 70 min
e 70 000 000 000 000 000
f Answers will vary.
16 a + b × c -
d ÷ e × f -
17 x y
18 a = 2, b = 4
19 a 1, 2, 6, 24, 120, 720
b i 24 × 32 × 5 × 7 ii 27 × 32 × 5 × 7
iii 27 × 34 × 5 × 7 iv 28 × 34 × 52 × 7
c 0 d 0
e It is the index number on the base 5.
f e.g. 23! × 4!
Exercise 11D
1 composite: 15, 8, 9, 27, 4, 12; prime: 13, 7, 5, 23, 11, 2
2 a 5, 4 b 6, 2 c 10, 2, 5
3 a 3, 3, 2, 5
b 7, 2, 2, 2
c 5, 11, 2, 2
4 a 23 × 32
b 34 × 52
c 22 × 3 × 72
d 22 × 32 × 112
5 a 23 × 32 b 23 × 3 c 2 × 19 d 22 × 11
e 22 × 31 f 24 × 5 g 25 × 3 h 24
i 3 × 52 j 3 × 37 k 26 l 23 × 7
6 a 23 × 3 × 52 b 25 × 52 c 23 × 54
d 25 × 3 × 52 e 26 × 56 f 23 × 32 × 54
g 22 × 5 × 41 h 2 × 3 × 5 × 23
7 a D b A c C d B
8 2310
9 a 144 = 24 × 32, 96 = 25 × 3
b HCF = 24 × 3 = 48
10 a 25 200 = 24 × 32 × 52 × 7, 77 000 = 23 × 53 × 7 × 11
b HCF = 23 × 52 × 7 = 1400
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Answers
11 24
4 6
2 2 2 3
24
2 12
2 6
2 3
24
2 12
3 4
2 2
24
3 8
4 2
2 2
12 i 424 cannot have a factor of 5.
ii 8 is not a prime number.
iii 424
4 106
2 2 2 53
∴ 424 = 23 × 53
13 a i 5 × 10 ≠ 60 ii 6 is not a prime number.
iii a 2 has been left off
b 60 = 22 × 3 × 5
14 72
2 36
2 18
2 9
3 3
72
2 36
2 18
3 6
2 3
72
2 36
3 12
2 6
2 3
72
2 36
3 12
3 4
2 2
72
3 24
6 4
2 3 2 2
72
3 24
2 12
4 3
2 2
72
3 24
3 8
4 2
2 2
72
3 24
2 12
6 2
2 3
72
2 36
4 9
2 2 3 3
72
2 36
6 6
2 3 2 3
72
4 18
6 3
2 3
2 2
72
6 12
4 3
2 2
2 3
72
4 18
9 2
3 3
2 2
72
6 12
6 2
2 3
2 3
72
8 9
3 3
2 2
4 2
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15 2 × 3 × 5 × 7 = 210 2 × 3 × 7 × 11 = 462
2 × 3 × 5 × 11 = 330 2 × 3 × 7 × 13 = 546
2 × 3 × 5 × 13 = 390 2 × 3 × 7 × 17 = 714
2 × 3 × 5 × 17 = 510 2 × 3 × 7 × 19 = 798
2 × 3 × 5 × 19 = 570 2 × 3 × 7 × 23 = 966
2 × 3 × 5 × 23 = 690 2 × 5 × 7 × 11 = 770
2 × 3 × 5 × 29 = 870 2 × 5 × 7 × 13 = 910
2 × 3 × 5 × 31 = 930 2 × 3 × 11 × 13 = 858
16 Check with your teacher.
17 Check with your teacher.
Exercise 11E
1 36 cm2, a square number
2 a 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25, 62 = 36, 72 = 49,
82 = 64, 92 = 81, 102 = 100, 112 = 121, 122 = 144,
132 = 169, 142 = 196, 152 = 225
b 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
3 a A square is not possible. b draw a 4 by 4 square
4 a 36 b 25 c 1331
d 100 e 8000 f 144
5 a 5 b 4 c 1
d 3 e 100
6 a 64 b 49 c 1 d 144
e 9 f 225 g 25 h 0
i 121 j 10 000 k 289 l 1089
7 a 5 b 3 c 1 d 11
e 0 f 9 g 7 h 4
i 2 j 12 k 20 l 13
8 a 15 b 25 c 21 d 19
9 a 30 b 64 c 65 d 36
e 4 f 0 g 81 h 4
i 13 j 2 k
2
3
l
4
5
10 a 241 b 23 c 2
d 18 e 720 f 41
11 64, 81, 100
12 121, 144, 169, 196
13 a 4 and 81
b 36 and 121
14 1, 9 and 49
15 a 144 b 144 c a = 3, b = 4
d Answers may vary; e.g. 42 × 52 and 202
16 a 32 + 42 = 9 + 16 = 25 b 102 c 152 d 502
17 a 121 and 12 321
b 1 234 321 c 1 234 321
18 no, 92 = 81
19 a false b false c true
d true e true (if a ≥ 0) f false
g false h false
Exercise 11F
1 a 3 × 3 × 3 × 3 × 3 × 3 = 36
b 6 × 6 × 6 × 6 × 6 × 6 × 6 = 67
c
5 5 5 5 5
5 5 5
52 × × × ×
× ×
= d
9 9 9 9
9 9
92 × × ×
×
=
2 a true b true c false d false
3 a multiply b 1
4 a 16, 8, 4, 2, 1 b 64, 16, 4, 1
5 a 4 × 4 × 4 × 4 × 4 × 4 = 46
b 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 × 12 = 129
6 a 1 b 1 c 1 d 1
e -1 f 1 g 1
7 a 27 b 59 c 76 d 810
8 a 36 b 415 c 330 d 710
9 a 47 b 39 c 72 d 4 e 38
10 a 72 = 49 b 10 c 132 = 169
d 23 = 8 e 101 f 2002 = 40 000
g 7 × 31 = 217 h 3 × 502 = 7500
11 a 4 b 1000 c 1 d 4
12 a The zero index applies only to the 5, not the 2. Correct
answer is 2 × 1 = 2.
b When doing multiplication, the indices are added, not the
bases. Correct answer is 24.
c 2 × 36 = 2 × 3 × 3 × 3 × 3 × 3 × 3 but 66 = 6 × 6 × 6 × 6 ×
6 × 6. These are not equal.
d The rule that says ‘add the powers’ applies only when
multiplying terms, not adding.
13 3
1
3
, 3
1
9
−1 = −2 = , 3−3 = 27
14 a 212 b 215 c 36
d 320 e 510 f 350
g 272 h 780 i 1050
Puzzles and challenges
1 a 28 (1, 2, 4, 7, 14) b 1, 2, 4, 8, 16, 31, 62, 124, 248
2 a 1400 b 480 c 540
d 630 e 4320 f 1100
3 a 37 b 43 c 75
d 91 e 143 f 92
4 01, 02, 03, 04, 05, 06, 07, 08, 09, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30,
31, 32, 33, 34, 35, 36, 37, 38, 39, 40,
41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88, 89, 90,
91, 92, 93, 94, 95, 96, 97, 98, 99, 100
ISBN: 9781107626973
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Answers
5 52 = 42 + 9, 62 = 52 + 11
6 Answers may vary. Check that the answer is the whole
number that you are looking for.
Multiple-choice questions
1 B 2 E 3 C 4 E 5 B
6 B 7 A 8 C 9 E 10 B
Short-answer questions
1 a 1, 2, 3, 4 , 5, 6 , 8 , 10 , 12 , 15 , 20 , 24 , 30 ,
40 , 60 , 120
b 1200, 1440, 1800
2 a 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96
7 , 14, 21 , 28, 35 , 42, 49 , 56, 63 , 70, 77 , 84
b 7, 13
3 a composite 21, 30, 16; prime 11, 7, 3, 2
b only one, 13
4 a 2, 5, 7, 11
b 30, 42, 70
5 a 68 b 54 × 25
6 a 25 b 23 × 52 c 32 × 52
7 a 10 b 3 c 4
8 a 16 b 7 c 397 d 131
9 a yes b no c yes
10 a
Number
Divisible
by 2
Divisible
by 3
Divisible
by 4
Divisible
by 5
84 539 424 ✓ ✓ ✓ ✘
Number
Divisible
by 6
Divisible
by 8
Divisible
by 9
Divisible
by 10
84 539 424 ✓ ✓ ✘ ✘
Explanation:
84 539 424 is an even number, therefore is divisible by 2.
84 539 424 has a digit sum of 39, therefore is divisible
by 3, but not by 9.
84 539 424 is divisible by 2 and 3, therefore is
divisible by 6.
The last two digits are 24, which is divisible by 4.
The last three digits are 424, which is divisible by 8.
The last digit is a 4, therefore not divisible by 5 or 10.
b Many possible answers.
c Any six-digit number ending in 0 with sum of digits equals
any multiple of 9.
11 a 5 b 50 c 13 d 18
e 14 f 20
12 a 102 b 102 c 104
d 105 e 102 f 104
g 108 h 110 i 101
13 a 12 b 216
14 a 1 b 3 c 1
d 2 e 1 f 2
15 a 515 b 59 c 536 d 59
Extended-response question
1 a area = 33 × 53 cm2 perimeter = (2 × 33 + 2 × 53 ) cm
b area = 163 or 46 or 212 perimeter = 162 or 44 or 28
c a 4 × c 2 d 162 or 44 or 28 e 3x
Semester review 2
Time
Multiple-choice questions
1 A 2 E 3 A
4 D 5 E
Short-answer questions
1 a 60 h b 3 h c 3 min 30 s
d 6 h 30 min e 2145 f 1:26 p.m.
g 5 h 45 min h 6 h 19 min 12 s
2 a 45 min b 1 h 51 min
c 1 h 11 min 1 s d 1 h 51 min 50 s
3 a i 7 p.m. ii 5 p.m.
iii 12 noon iv 6 a.m.
v 12 midnight vi 9 p.m.
b i 1:20 p.m. ii 3:50 p.m.
iii 3:20 p.m. iv 4:20 p.m.
4 a i 55 min ii 1 h 14 min
iii 1 h 1 min iv 1 h 57 min
b morning c 1 h 56 min
5 21 h 15 min
6 120
7 5 a.m.
Extended-response question
1 a i 4:30 p.m. ii 4 p.m. iii 2:30 p.m.
b 2 h 39 min c 3:30 p.m.
Algebraic techniques 1
Multiple-choice questions
1 B 2 A 3 C 4 E 5 D
ISBN: 9781107626973
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Short-answer questions
1 a 4 b yes c 9 d 7
2 a x + 3 b 12a c 2x + 3y
d
6
w
e y – 2x
3 a 100m b 24x c 1 000 000p d
24
y
4 a 13 b 11 c 39 d 6 e 6 f 24
5 36
6 a 10a b 4x c 12a
d m e 6 + 5a f 4x + 2y
7 a 2x + 14 b 3(x + 4) = 3x + 12
8 a 6bc b 5b c p
9 a 2a + 6 b 12a – 12b c 24m + 32
10 12xy
Extended-response question
1 a i $12 ii $(3x + 6) iii $(3x + 2y )
b $50 – 3x – 2y
Equations 1
Multiple-choice questions
1 B 2 E 3 C 4 B 5 D
Short-answer questions
1 a x = 3 b x = 108 c x = 21 d
4
3
x =
2 a x = 2 b x = 12 c m = 7
3 a y = 20 b b = 4 c m = 8
4 a P = 103 b S = 61 c C = 325
5 a x = 5 b x = 6 c x = 18
6 4x + 25 = 85; x = 15
Extended-response question
1 a $320 b $400 c $(200 + 40n) d 6
1
2
h
Measurement and computation of length,
perimeter and area
Multiple-choice questions
1 C 2 A 3 E 4 C 5 A
Short-answer questions
1 a 500 b 6000 c 1.8 d 0.017 e 1.8 f 5500
2 a 272 cm b 11 m c 3 m
d 220 cm e 3.4 m f 92 m
3 a 1.69 m2 b 24 m2 c 60 m2
d 75 m2 e 114 m2 f 171 m2
4 a (6p + 12) cm b (3p + 12) cm c (p + 8) cm
5 (8p + 16) cm
Extended-response question
1 a Many answers possible; e.g. 8 m × 10 m, 4 m × 14 m,
15 m × 3 m
b 9 m by 9 m (area = 81 m2)
c 36 posts
d 9 h
Introducing indices
Multiple-choice questions
1 C 2 A 3 B 4 E 5 D
Short-answer questions
1 a 1, 3, 5, 15 b 1, 2, 3, 5, 6, 10, 15, 30
c 1, 2, 4, 5, 10, 20, 25, 50, 100
2 a 3, 6, 9, 12, 15 b 7, 14, 21, 28, 35
c 11, 22, 33, 44, 55
3 1, 2, 3 and 6
4 4
5 a 121 b 144 c 25
6 120
7 false
8 a 1 b 1 c 1
9 a 810 b 87 c 83
d 86 e 84 f 810
10 a 1 b 8 c 1
d 9 e 1 f 2
Extended-response question
1 a (-1)4 = 1; (-1)5 = -1
b -1 because 279 is an odd number.
c Will be negative because 79 is an odd number.
ISBN: 9781107626973
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ITnadbelex of Contents
abacus 44
acute angle 60
adjacent angles 66
algebraic expression 361, 363, 390
algorithm 19
alternate angles 71
amicable numbers 139
angle 56, 60
angle bisector 84
approximation 36
arc 83
arc length 503
area (composite fi gures) 525
area (parallelograms) 521
area (rectangle) 510
area (square) 510
area (triangles) 515
area conversions 514
associative law 14, 23, 31
axes 122
axis 122
Babylonian numbers 6
best buys 318
bisect 83
cancelling 286
Cartesian plane 122, 417, 418
chance 222
chance experiment 221, 238
chord 83, 499
circumference 498, 499
coeffi cient 363
collinear points 55
commutative law 14, 23, 31
complement 222
complementary angles 66
complementary events 222
composite fi gure 504, 525
composite number 552
composite shape 505, 525
compound event 222
concurrent lines 55
constant term 363
constructions 83, 87
coointerior angles 71
coordinates 122, 123, 418
cooresponding angles 71
cube root 564
daylight saving 352
decimal point 168
denominator 144, 162, 272
diameter 83, 499
dimensions 493
distributive law 24, 385, 461
dividend 31, 547
divisibility tests 548
divisible 547
divisor 31, 547
dynamic geometry software 87
Egyptian fractions 133, 208
Egyptian numbers 5
equally likely outcomes 222
equation 433
equivalent fractions 151
even chance 223
event 222
exclusive ‘or’ 239
expanded form 10
expanding 385
expansion 385
expected number of occurrences 233
experiment 221
experimental probability 222, 233
exponent 556
expression 363, 435
factor tree 560
factorials 559
factors 135, 547
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favourable outcomes 222
Fibonacci numbers 574
formula 465
fraction wall 182
fractions 144
GMT 349
Greenwich Mean Time (GMT) 349
HCF 140
highest common factor (HCF) 140
Hindu-Arabic numbers 10
imperial system 483
improper fraction 145, 156
inclusive ‘or’ 239
index 556
index form 555, 556
index laws 570
indices 556
integers 101
International Date Line 352
intersection 55
interval 55
LCD 163, 272
LCM 141
like terms 377, 461
likelihood 222
line 55
line segment 55
long division 35
lowest common denominator (LCD) 163, 272
lowest common multiple (LCM) 141
major arc 83
mass 531
metric system 484
minor arc 83
mixed numerals 156
multiples 135
mutually exclusive events 222, 243
negative numbers 101
non-mutually exclusive events 222
number line 101
number plane 101, 418
numerator 144, 162
obtuse angle 60
order of operations 40, 118, 119, 294
origin 418
outcome 222
palindromic numbers 552
parallel 56
parallel lines 72
parentheses 40
partitioning 14
percentages 182, 188, 193, 198, 203
perfect numbers 575
perfect squares 565
perimeter 493
perimeter of a sector 503
perpendicular 56
perpendicular lines 66
pi 498
place value 10, 168
plane 55
point 55, 122
power of zero 570
powers 555, 556
powers of ten 292
prime decomposition 560
prime factors 560
prime number 552, 576
probability 221
pronumeral 363
proper fraction 144
proportion 204
protractor 61
quadrant 504
quotient 31, 547
radius 83, 499
ratio 203
ray 56
reciprocal 304
recurring decimals 176
reflex angle 60
remainder 31, 547
repeating decimals 176
revolution 60
right angle 60
Roman measurement system 484
Roman numbers 5, 6
rounding 36
rounding decimals 172
sample space 222
scientific notation 296
ISBN: 9781107626973
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Index
sector 83, 499
segment (circle) 499
segment (line) 55
semicircle 504
sequence 398
simplifi ed fractions 151
single-step experiment 228
solution 435
spatial pattern 403
spreadsheet 126, 255
square numbers 24, 564
square root 564
standard form 296
straight angle 60
subject 465
substitution 363, 395, 412
supplementary angles 66
table of values 411, 412
tangent 83, 499
temperature 531
term 363, 398
theoretical probabilty 222
time zones 349
transversal 71
trials 221
two-step experiments 250
two-way tables 243
units of area 509
units of length 497
variable 363
Venn diagrams 243
vertex 56
vertically opposite angles
66, 71
x-axis 122, 418
y-axis 122, 418
zero index 570
ISBN: 9781107626973
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ISBN: 9781107626973
Photocopying is restricted under law and this material must not be transferred to another party
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