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Lesson 4

The Meaning of Percent

The Metric System

Thanks to our system of positional numeration, it is a simple matter to multiply or divide by a power of 10. It never calls for the written methods we use with other numbers (Lesson 9). And it certainly does not call for a calculator. For the remainder of this course, multiplying and dividing by powers of 10 will be a basic skill.

1.

How do we multiply a whole number by a power of 10?

36 × 10

Add on as many 0's as appear in the power.

Examples.

36 × 10

=

360

Add on one 0.

36 × 100

=

3600

Add on two 0's.

36 × 1000

=

36,000

Add on three 0's.

Doing that is an example of skill in arithmetic, which is to be able to do a problem as quickly and as easily as possible. Traditional multiplication would eventually give the right answer (we hope). But it is not skillful, and it does not take advantage of positional numeration. The student should let go of that written method immediately.

2.

How do we multiply a decimal by a power of 10?

7.32 × 10

Move the decimal point right as many places
as there are 0's in the power. If there are not enough digits, add on 0's.

Examples.

7. 32 × 10

= 73. 2

Move the decimal point one place right.

7. 32 × 100

= 732

Move the point two digits right: 732. However, since all the digits fall to the left of the decimal point, the answer is a whole number, 732, which we write without a decimal point.

7. 32 × 1000

= 7,320

Move the point three digits right. To do this, we must add on a 0.
Again, the answer is a whole number.

Problem. If 5 pounds of sugar cost $2.79, how much will 50 pounds cost?

Answer. Since 50 pounds are ten times 5 pounds, they will cost ten times more. Move the decimal point one place right: $27.90. Since money has two decimal digits, we added on a 0. ( Lesson 3, Question 8)

3.

How do we divide a decimal by a power of 10?

63.4 ÷ 10

"63.4 divided by 10."

Move the decimal point left as many places as there are 0's in the power. If there are not enough digits, add on 0's.

Examples.

63.4 ÷ 10

=

6.34

Move the point one place left.

63.4 ÷ 100

=

.634

Move the point two digits left.

63.4 ÷ 1000

=

.0634

Move the point three digits left. To
do this, add on a 0.

These example illustrate that, whenever we multiply or divide by a power of 10, the digits do not change We simply move the decimal point or add on 0's.

Finally, we must see how to divide a whole number by a power of 10. Now in Lesson 2 we saw that when a whole number ends in 0's, we simply take off 0's. ( Lesson 2, Question 10)

265,000 ÷ 100 = 2,650

But when a whole number does not end in 0's -- as 265 -- then there are no 0's to chop off We will see that we must place a decimal point to separate digits on the right.

4.

How do we divide a whole number by a power of 10?

265 ÷ 10

Starting from the right of the whole number, separate as many decimal digits as there are 0's in the power. If there are not enough digits,
add on 0's.

Examples.

265 ÷ 10

= 26. 5

Starting from the right of 265, separate one decimal digit.

265 ÷ 100

= 2. 65

Separate two decimal digits.

265 ÷ 1000

= . 265

Separate three decimal digits.

When we divide a whole number by a power of 10, the answer will have as many decimal digits are there are 0's.

8 ÷ 100 = . 08

Two 0's. Two decimal digits.

Again, as in Lesson 2, consider this array:

As we move up the list -- as we push the digits one place right -- the number has been 2 divided by 10 because each place to the right is worth 10 times less. (As we move from 26.58 to .658, we go from 2 tens to 2 ones.) It appears, though, as if the decimal point has shifted one place left, or, with the whole number 26580, that a 0 has been taken off.

As we move down the list -- as we push the digits to the left through the decimal point -- each number has been multiplied by 10.

And so we can easily multiply or divide by a power of 10 because of our system of positional numeration. Each place belongs to the next power of 10.

At this point, please "turn" the page and do some Problems.

or

Continue on to the Section 2: The meaning of percent

Introduction | Home | Table of Contents

Copyright © 2020 Lawrence Spector

Questions or comments?

E-mail: themathpage@yandex.com

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Multiply and divide by powers of 10 -- A complete course in arithmetic

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Lesson 4 Thanks to our system of positional numeration, it is a simple matter to multiply or divide by a power of 10. It never calls for the written methods we use with other numbers (Lesson 9). And it certainly does not call for a calculator.

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FactMonster.com

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A math game requiring logic, strategy and a little luck!

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funbrain.com

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SafeShare.tv reduces advertisements and other unwanted distractions from Vimeo and YouTube videos, so they can be viewed safely. Perfect for Schools!

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Brain Pop

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Safeshare.tv - The safest way to share YouTube and Vimeo videos

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SafeShare.tv reduces advertisements and other unwanted distractions from Vimeo and YouTube videos, so they can be viewed safely. Perfect for Schools!

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Brain Pop

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Safeshare.tv - The safest way to share YouTube and Vimeo videos

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SafeShare.tv reduces advertisements and other unwanted distractions from Vimeo and YouTube videos, so they can be viewed safely. Perfect for Schools!

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Website: https://www.brainpop.com/ Twitter: https://twitter.com/brainpop Facebook: https://www.facebook.com/brainpopofficial/ Numbers are sort of like your bedroom closet: the less clutter, the better. So how do you express a number that gets really big, really fast-say, the growth of a rabbit population, or the distance between two planets-without it taking up your whole notebook?

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Compute 23 times 44. And maybe the hardest part of this problem, or maybe the first hard part, is to recognize that that dot even means multiplication. This could have also been written as 23 times 44, or they could have written it as 23 in parentheses times 44, so you just put the two parentheses next to each other. That also implies multiplication. So now that we know we're multiplying, let's actually do the problem. So we're going to multiply 23-- I'll write it bigger. We're going to multiply 23 by 44. I'll write the traditional multiplication sign there, just so that we know we're multiplying. When you write it vertically like this, you very seldom put a dot there. So let's do some multiplication. Let's start off multiplying this 4 in the ones place times 23. So you have 3 times 4 is 12. We can write 2 in the ones place, but then we want to carry the 1, or we want to regroup that 1 in the tens place. So it's 12, so you put the 1 over here. And now you have 4 times 2 is 8 plus 1 is 9. So you can think about it as 4, this 4 right here, times 23 is 92. That's what we just solved for. Now, we want to figure out what this 4 times 23 is. Now what we do here is, when you just do it mechanically, when you just learn the process, you stick a 0 here. But the whole reason why you're putting a 0 here is because you're now dealing with a 4 in the tens place. If you had another-- I don't know, a 3 or a 4 or whatever digit, and you're dealing with the hundreds place, you'd put more zeroes here, because we're going to find out 4 times 23 is 92. We just figured that out. If we just multiplied this 4 times 23 again, we would get 92 again. But this 4 is actually a 40, so it actually should be 920, and that's why we're putting that 0. Now you're going to see it in a second. So we have-- so let me put this in a different color. So this 4 now we're multiplying. 4 times 3 is 12. Let's put the 2 right here. It should be in the tens place because this is really a 40 times the 3. Just think about it, or you could just think of the process. It's the next space that's free. 4 times 3 is 12. Carry the 1. This blue 1 is from last time. You ignore it now. You don't want to make that mess it up. That's when we multiplied this 4. So now we have 4 times 2 is 8 plus 1 is 9. So what we figured out so far is 4 times 23 is 92, and this green 4 times 23 is 920, and that's because this green 4 actually represents 40. It's in the tens place. So when you multiply 44 times 23, it's going to be 4 times 23, which is 92, plus 40 times 23, which is 920. I just want to make sure we understand what we're doing here. And so we can take their sum now. Let's add them up. 2 plus 0 is 2. 9 plus 2 is 11. Carry the 1. 1 plus 9 is 10. Put a comma here, just so it's easy to read, every third digit. So 23 times 44 is 1,012.

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Multiplying 2-digit by 2-digit: 23x44 (video) | Khan Academy

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Compute 23 times 44. And maybe the hardest part of this problem, or maybe the first hard part, is to recognize that that dot even means multiplication. This could have also been written as 23 times 44, or they could have written it as 23 in parentheses times 44, so you just put the two parentheses next to each other.

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