Multiplying and Dividing Decimals by Powers of 10

Our number system is set up so that multiplying and dividing by 10 (or by 100, or 1000 etc.) is particularly easy. The numbers just have to be moved the correct number of places to the left or to the right.

The decimal point in a number never moves. We can think of this as like the framework that the number is held up on. Although we don’t write it, in whole numbers, like 37, there is always a decimal point on the far right hand side of the number.

In general, multiplying by 10 moves the number to the left and dividing the number by 10 moves the number to the right. Because 100 is 10×10, this moves the number twice (i.e. two places), because 1000 is 10x10x10 this moves the number three places, etc.

So 2.4 x 10 = 24

What about 2.4 ÷ 10?

What about 2.4 x 100?

What about 2.4 ÷ 100?

Let’s try some other numbers with the teacher.

If we have to multiply by numbers like 1/10 or 1/100, the situation is reversed, so multiplying by 1/10 is the same as dividing by ten. And dividing by 1/10 is the same as multiplying by 10, etc.

So 2.4 x 1/10 = 0.24
What about 2.4 ÷ 1/10?

What about 2.4 x 1/100?

What about 2.4 ÷ 1/100?

Let’s try some other numbers with the teacher.

And let’s try this exercise in our notebooks (from exercise 5D on pages 74 and 75 of your textbooks):

Below are the answers:

Multiplying Numbers by Decimals

Multiplying by decimals is made easier if we remember that 0.1 = 1/10 and 0.01 = 1/100 (and 0.001 = 1/1000, etc.)

So 0.7 = 7 x 1/10 and 0.08 = 8 x 1/100

So if we want to calculate 8 x 0.7, we can rewrite it as 8 x 7 x 1/10, so it is 56 x 1/10, which we know how to do.

How about 21 x 0.6?

Or 17 x 0.02?

Or 9 x 0.008?

Let’s try these questions from exercise 5E on page 76 of our textbook:

The answers to the above questions are below:

Multiplying Decimals by Decimals

Multiplying decimals by decimals is effectively the same. So if we want to calculate 0.7 x 0.08, we can rewrite it as 7 x 0.1 x 8 x 0.01.

When we multiply the order is not important, so this can be rewritten as 7 x 8 x 0.1 x 0.01

So it is 56 x 0.001 = 0.056.

Examples

Let’s try some of these with our teacher:

Exercise

Now we can do exercise 10D on pages 149 and 150 of our textbook:

Below are the answers:

Dividing Decimals by Decimals

Surprisingly, dividing decimals by decimals is even easier than multiplications, because we can multiply our numbers by anything we want, and as long as we multiply them both by the same thing it will not change our calculations.

This is because a/b = 3a/3b = 14a/14b = ca/cb.

So, for instance, if we need to calculate 0.3 ÷ 0.015, we can multiply both of our numbers by 1000 to give 300 ÷ 15 = 20.

Example: Let’s try some of these with the teacher:

Exercise

Now we can do the exercise 10E on pages 150 to 151 of our textbook:

Below are the answers: