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Here we will learn about scale drawings, including creating scale drawings, using scale factors, and word problems.

There are also scale diagrams and drawingsworksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you're still stuck.

What is a scale drawing?

A scale drawing is an enlargement of an object.

An enlargement changes the size of an object by multiplying each of the lengths by a scale factor to make it larger or smaller.

The scale of a drawing is usually stated as a ratio.

E.g.

1cm:5m

This is said as " 1 centimetres to 5 metres" and means every 1 centimetre on the diagram represents 5 metres in real life.

In order to interpret and produce scale drawings we need to know the scale factor and the actual lengths of the object.

E.g.

Below is a scale drawing of a pool with a scale of \bf{1cm:2m} or 1:200.

This means that every centimetre on the diagram represents 2 metres in real life.

So the 6cm width of the diagram represents a 12m width on the real pool.

The real pool has been been enlarged by a scale factor of \frac{1}{200} to give the scale drawing.

The majority of scale drawing questions will involve polygons or 2D drawings such as floor plans or using a map although there is overlap with topics such as enlargement, construction of triangles, and loci.

How to calculate the actual / real life distance from a scale

In order to calculate the actual/real life distance from a scale:

1. State the scale of the enlargement as a ratio in the form \bf{1:n} .

2. Multiply \bf{n} by the length given from the scale drawing.

3. Write the units.

Explain how to calculate the actual / real life distance from a scale

Scale drawing worksheet

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Scale drawing worksheet

Get your free scale drawing worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Related lessons on scale

Scale drawing is part of our series of lessons to support revision on scale. You may find it helpful to start with the main scale lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Example 1: map

A map has a scale of 1cm:2km. Find the actual distance represented by 5cm on the map.

State the scale of the enlargement as a ratio in the form \bf{1:n} .

1cm:2km

2 Multiply \bf{n} by the length given from the scale drawing.

2 \times 5=10

3 Write the units.

10km.

How to calculate the scale drawing distance from a scale

In order to calculate the scale drawing distance from a scale:

1. State the scale of the enlargement as a ratio in the form \bf{1:n} .

2. Divide the real life distance by the scale ratio.

3. Write the units.

Calculate scale drawing distance examples

Common misconceptions

Multiplying the real life distance by the ratio scale

The real life distance is multiplied by the ratio scale, giving an incorrect distance on the plan.

Dividing the model / plan / map distance by the ratio scale

The distance on the model / plan / map is divided by the ratio scale meaning that the real life distance is incorrect.

Ratio scale not simplified

If given the ratio 2cm:5km, it is easier to calculate when the ratio is in the form 1:n and so we must find an equivalent ratio before using the scale.

Here the ratio would be 1cm:2.5km so 1cm on the map would be equal to 2.5km in real life.

Incorrect units in the solution

The units for the model are mixed up with the units for the real life distance. For example if we were calculating the distance of 10cm on a map with the scale ratio of 1cm:5km, \ 10 \times 5=50cm is stated whereas the correct solution would be 10 \times 5=50km.

Sometimes the units need to be converted so that you are working with the same units, making calculations much easier to comprehend. Make sure you know how to convert between different metric units. For example, the map scale is given as 1:25 \ 000 which means that 1cm on the map is equivalent to 25 \ 000cm in real life.

If the answer is asked to be written in kilometres, the real life value in centimetres must be divided by 100 \ 000 to get the same measurement in kilometres. 1:25 \ 000 = 1cm:0.25km

Practice scale drawing questions

Scale drawing GCSE questions

1. A map has a scale of 1cm:0.5km. A dog walker walks around 18km per day. Calculate how far they will have walked on a map.

(2 marks)

18 \div 0.5

(1)

=36cm

(1)

2. (a) The distance between two cities in an American State is 146.8km. The distance between the two cities on a map is 29.36cm. Calculate the scale ratio of the map to the actual distance in the form 1cm:nkm.

(b) The river that travels through the state is measured on the plan to be 45cm. Calculate the actual length of the river. State the units in your solution.

(4 marks)

(a)

146.8 \div 29.36=

(1)

1cm:5km

(1)

(b)

45 \times 5=225

(1)

km

(1)

3. Below is the map of a desert island.

Using the scale provided, calculate the distance between the Palm Trees and the Waterfall.

(3 marks)

Distance [6.3-6.7cm]

(1)

Their 6.5 \times 50

(1)

325m

(1)

Learning checklist

You have now learned how to:

Multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1,000 in converting between units such as kilometres and metres

Solve problems involving similar shapes where the scale factor is known or can be found

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One to one maths interventions built for KS4 success Weekly online one to one GCSE maths revision lessons now available Learn more Here we will learn about scale drawings, including creating scale drawings, using scale factors, and word problems. There are also scale diagrams and drawingsworksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you're still stuck.

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Practice 8th grade math for free online with rates and ratios to compare values exercises, including find value given ratio, find value given rate

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Here you will learn about ratios, including how to write a ratio, simplifying ratios, unit rate math and how to solve problems involving ratios and rates.

Students will first learn about ratios as part of ratios and proportions in 6 th grade and 7th grade.

What is a ratio?

A ratio is a multiplicative relationship between two or more quantities.

Ratios are written in the form a : b, which is read as "a to b", where a and b are normally integers, fractions, or decimals.

The order of the quantities in the ratio is important.

For example,

If there are 10 boys in a class and 15 girls, the ratio of boys to girls is 10 : 15 which is read as "10 to 15." This is an example of a part to part ratio. You could also say the ratio of total students to girls is 15 : 25. This is an example of a part to whole ratio.

Step-by-step guide: How to write a ratio

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Since a ratio represents a relationship, there is always more than one way to show it.

This includes unit rate math - which creates equivalent ratios where one part of the ratio is 1.

You can use unit rates to compare different quantities.

For example,

A grocery store sells a bag of 6 bananas for \$ 2.34 and a bag of 4 bananas for \$ 1.44.

Which bag has the better unit price?

Unit price means the price per 1 unit. In this case, the units are bananas. Divide each ratio to find the price for 1 banana.

The bag of 4 bananas is \$ 0.36 per banana, which is cheaper than the bag of 6 bananas which is \$ 0.39 per banana.

Step-by-step guide: Unit rate math

Unit rates are not the only types of equivalent ratios. When simplifying fractions, use the common factors to divide all the numbers in a ratio until they cannot be divided further to write the ratio in lowest terms.

For example,

The ratio of red counters to blue counters is 16 : 12.

You can simplify the ratio to lowest terms by finding the greatest common factor \textbf{(GCF)} of each of the numbers in the ratio.

Factors of 16 \text{:} \, 1, 2, 4, 8, 16

Factor of 12 \text{:} \, 1, 2, 3, 4, 6, 12

The greatest common factor is 4. To simplify the ratio, you divide both sides by 4.

Step-by-step guide: Simplifying ratios

Another way to write ratios is by using fraction notation.

Fraction notation can be used to show a part to whole ratio relationships.

For example,

The bar model below shows the ratio of blue : red as 3 : 2 (3 to 2). There are 3 blue blocks, 2 red blocks and 5 blocks in total.

This part to whole relationship allows us to make statements like...

\cfrac{3}{5} of the blocks are blue

\cfrac{2}{5} of the blocks are red

\cfrac{5}{5} of the blocks are blue or red

The ratio of blue : red as 3 : 2 can also be shown as a part to part fraction...

For example,

The fractions show the ratio relationship BETWEEN the blue and red blocks. This allows us to make statements like...

The number of blue blocks is \cfrac{3}{2} larger than red

The number of red blocks is \cfrac{2}{3} the amount of blue

Step-by-step guide: Ratio to fraction

Ratios can also be written with percents.

For example,

The ratio of pencils to crayons is 4 : 6.

The ratio has 10 parts, so the fractions are

\cfrac{4}{10} : \cfrac{6}{10}.

The numerator represents the numbers of the ratio, which show how many pencils or crayons there are. The denominator represents the total number of pencils and crayons.

You may be able to recognize what the fractions are as percents or you may need to use long division to help convert your fractions.

\cfrac{4}{10}=40 \%, so 40 \% are pencils.

\cfrac{6}{10}=60 \%, so 60 \% are crayons.

Step-by-step guide: Ratio to percent

Solving problems with ratios is common in the real world. One place that this shows up is in calculating exchange rates. An exchange rate is the rate at which the money of one country can be exchanged for the money of another country.

Using a currency's exchange rate you can convert between US dollars and foreign currencies.

For example,

To convert from US dollars (USD) to Japanese yen (JPY), you must multiply by the exchange rate.

For example,

So \$ 15 \; USD would be ¥2,134.35 \; JPY because,

\$ 15 \; USD \times 142.29=¥ 2,134 .35 \; JPY.

Step-by-step guide: How to calculate exchange rates

All the skills above are examples of ratio problem solving. When solving problems with ratios, it is important to ask:

What is the ratio involved?

What order are the quantities in the ratio?

What is the total amount / what is the part of the total amount known?

What are you trying to calculate?

In the classroom, ratio problem solving often comes in the form of real world scenarios or word problems.

For example,

\cfrac{8}{10} students are right handed. What is the ratio of left handed students to right handed students? (2 : 8)

Step-by-step guide: Ratio problem solving

Common Core State Standards

How does this relate to 6 th grade math and 7 th grade math?

Grade 6 - Ratios and Proportions (6.RP.A.1)
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2 : 1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."

Grade 6 - Ratios and Proportions (6.RP.A.2)
Understand the concept of a unit rate \cfrac{a}{b} associated with a ratio a : b with b ≠ 0, and use rate language in the context of a ratio relationship.

For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is \cfrac{3}{4} cup of flour for each cup of sugar." "We paid \$ 75 for 15 hamburgers, which is a rate of \$ 5 per hamburger."

Grade 6 - Ratios and Proportions (6.RP.A.3)
Use ratio and rate reasoning to solve real-world and mathematical problems, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Grade 6 - Ratios and Proportions (6.RP.A.3b)
Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

Grade 7 - Ratios and Proportions (7.RP.A.1)
Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

For example, if a person walks \cfrac{1}{2} mile in each \cfrac{1}{4} hour, compute the unit rate as the complex fraction \cfrac{\cfrac{1}{2}}{\cfrac{1}{4}} miles per hour, equivalently 2 miles per hour.

How to work with a ratio

There are a lot of ways to work with a ratio. For more specific step-by-step guides, check out the ratio pages linked in the "What are ratios?" section above or read through the examples below.

Example 1: how to write a ratio

Write the ratio of apples to pears.

Identify the different quantities being compared and their order.

There are 5 pears and 2 apples.

The order of the ratio is apples to pears.

2 Write the ratio using a colon.

Apples : Pears

\hspace{0.7cm} 5 : 2

3 Check if the ratio can be simplified.

5 and 2 only have a common factor of 1, so this ratio is already in its lowest terms (simplest form).

Example 2: unit rate calculation - decimal

A car travels 303 miles in 6 hours. If the car travels the same number of miles each hour, what is the miles per hour rate?

Example 3: simplifying ratios

Write the ratio 48 : 156 in lowest terms.

GCF(48,156)=12

Example 4: solving a problem involving ratio to percents

The ratio of adults to children in a park is 11 : 14.

One-fourth of the adults are women. What percent of the people in the park are men?

11 : 14 becomes \cfrac{11}{25} : \cfrac{14}{25}.

\cfrac{1}{4} of 44 \%=11 \%.

Example 5: converting from KRW / USD

₩5,000 \; KRW is equal to \$ 3.85 \; USD. What is the exchange rate from \$ \; (USD) to ₩ \; (KRW)?

Example 6: ratio problem solving - mixed numbers

Fruit Salad Recipe:

2 \cfrac{1}{2} cups of blueberries

2 \cfrac{1}{5} cups of orange slices

1 \cfrac{1}{4} cups of strawberries

2 cups of apple slices

Write the ratio of the total cups of berries for every 1 cup of strawberries in the salad.

\cfrac{15}{4} : \cfrac{5}{4}

3 : 1

Teaching tips for ratio

There are many ways to engage students in ratios. One way is to introduce the golden ratio (based on Fibonacci's sequence) and challenge students to look for it in the real world. Keep a chart or wall in the classroom for students to add any examples of this ratio that they find.

Incorporate as many examples of ratios in the classroom as you can - even across subjects. For example, have students write ratios about the "Word of the day" - from an English or Science class. Such as "Write the ratio of nouns to adjectives" or "Write the ratio of words with the letter 'e' to total words."

As students work with ratios in different ways, keep track of successful solving strategies on a bulletin board or on chart paper. This allows students to see and utilize another students' strategy, make connections between strategies and feel ownership in any ideas they help create.

Be mindful of how to progress with ratio topics. Typically whole number ratios are introduced first, then ratios with rational numbers. Ratios that involve compare only fractional (or decimal values), such as \cfrac{2}{3} : \cfrac{5}{6} or 0.45 : 0.34 are the most difficult for students. As always, be mindful of your state's curriculum when making decisions on when to introduce certain ratio topics.

Easy mistakes to make

Writing the ratio in the wrong order
A common error is to write the parts of the ratio in the wrong order.
For example,
The number of dogs to cats is given as the ratio 12 : 13 but the solution is incorrectly written as 13 : 12.

Confusing ratios and fractions
You can write a ratio with fraction notation. A part to whole fraction will have the same fractional language as a fraction. However, a part to part fraction will not.
For example,
The ratio of boys to girls is 2 : 3. Two ways to express this ratio are \cfrac{2}{3} or \cfrac{2}{5}.
However, you must be careful how you read these fractions. You can say " \cfrac{2}{5} of the kids are boys" but you cannot say " \cfrac{2}{3} of the kids are boys." Instead, say "The number of boys is \cfrac{2}{3} the number of girls."

Not fully simplified
A common error is to not find the greatest common factor when simplifying a ratio.
For example,
Simplify the ratio 12 : 18.
Dividing both numbers by only 2 leaves a ratio of 6 : 9, which is not fully simplified. This can be simplified further by dividing by 3 to get the ratio 2 : 3, which is the correct answer. By dividing both numbers by the greatest common factor, 6, would get the ratio 2 : 3 in one step.

Practice ratio questions

Ratio FAQs

What does ratio mean in math?

While the term ratio is used in a variety of ways in the real world, the definition of ratio in math is the comparison of two or more values that have a constant relationship. Some examples of ratios are " 2 dogs to 5 cats" or " 24 miles per hour."

Are 'ratio' and 'rate' synonyms?

A rate is a special type of ratio that compares different units. They are not synonyms, since not all ratios are rates. However, all rates are ratios, so they can be called by either name.

How are ratios used in upper level mathematics?

Ratio understanding is expanded to include more complex comparisons that involve exponents, variables and/or polynomials. This extends to include ratio relationships in proportions and linear equations. As students progress in their learning, they will become comfortable graphing, creating tables and equations that represent ratio relationships.

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs.

[FREE] Common Core Practice Tests (3rd to 8th Grade)

Prepare for math tests in your state with these 3rd Grade to 8th Grade practice assessments for Common Core and state equivalents.

Get your 6 multiple choice practice tests with detailed answers to support test prep, created by US math teachers for US math teachers!

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Ratio - Math Guide (Steps, Examples & Questions)

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"Please excuse my dear Aunt Sally."

Now, I know what you're thinking: "What does that phrase actually mean?" Quite a bit, actually, because that saying provides the key to remembering an important math concept: the order of operations.

The order of operations is one of the more critical mathematical concepts you'll learn because it dictates how we calculate problems. It gives us a template so that everyone solves math problems the same way.

Let's start off with a simple question. What is an operation?

An operation is a mathematical action. Addition, subtraction, multiplication, division, and calculating the root are all examples of a mathematical operation. Let's take a look at this problem:

It looks easy, right? Well, it wouldn't be so easy if we didn't understand the order in which the math operation occurs. If we didn't have rules to determine what calculations we should make first, we'd come up with different answers.

Should you start by subtracting 4 minus 6 and then multiplying by 7?

No. The order of operations tells us how to solve a math problem. And this brings us back to Aunt Sally.

What is the Order of Operations?

Operations have a specific order, and this is what "Please Excuse My Dear Aunt Sally" helps us to understand. It's an acronym that tells us in which order we should solve a mathematical problem.

"Please" stands for " Parentheses," so we solve everything inside of the parentheses first.

Then, "Excuse," which is for " Exponents." We solve that after we solve everything in parentheses.

Multiplication, which is the "My," and this happens from left to right.

And then division, which is the "Dear," which also happens left to right.

And then we have addition and subtraction, which also happens from left to right, and this is "Aunt" and "Sally."

Order of Operations Examples

Okay, so now that we know the order of operations, let's apply it to our problem that we have here and solve.

We don't have parentheses and we don't have exponents, but we do have multiplication, so we do that before we do any addition or subtraction. Let's go ahead and multiply \(7\times 4\). That gives us 28.

And now we're subtracting 6, which gives us 22.

Now, let's look at another problem.

Without the operations, you could calculate this problem as \(7+7=14\times 3=42\).

And this would be wrong!

Remember, you multiply before you add. Therefore, the equation should look like this:

So when we do problems like this, we can use parentheses to group together our numbers that are going to take place first. So in this case, it's \(7\times 3\), and when we do that we get 21 and we have 7 left over. When we add those together, we get 28, and that's our answer!

Let's look at some more complex problems.

The order of operations dictates how to solve this problem. Remember, you multiply exponents first. Here's the wrong way to solve the problem:

Why is that wrong? Because you violated the order of operations. You do not multiply first! You perform an operation on the exponent first. This is how it should be done:

See? Solving the equation in the right order provides the correct answer.

Let's try out one more problem. This one is a little bit more challenging, but it perfectly illustrates the order of operations.

\(5\times 10-(8\times 6\)\(-15)+4\times 20\div 4\)

Remember the order. What do we do first? The numbers inside the parentheses. So \(8\times 6=48\), then we subtract 15 and that gives us 33. Here's how the problem looks now:

\(5\times 10-33+4\times 20\div 4\)

So our next step is multiplication and division, so let's perform all our multiplication and division problems and then see what we have left.

Now we finish with addition and subtraction, so here's what we have:

And our answer is 37!

There is an exception. If an equation only has one expression, you don't have to follow the order of operations.

Here are some examples of single expressions.

\(10+10\): Well, there are no other operations, so you just know to go ahead and add them together and you get 20. Same thing with subtraction, multiplication, and division. All those are single expressions.

Alright guys, that's our video on the order of operations. I hope that this was helpful!

See you guys next time!

Fact Sheet

Practice Questions

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What are the Order of Operations? (Video & Practice)

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"Please excuse my dear Aunt Sally." Now, I know what you're thinking: "What does that phrase actually mean?" Quite a bit, actually, because that saying provides the key to remembering an important math concept: the order of operations.

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GED Order of Operations Practice | Get Sum Math

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GED Order of Operations Practice. This free GED PEMDAS practice will prepare you for the GED Math Test!