3.05 Dividing with decimals

Sometimes, before you can really understand how to perform some number problems, it helps to understand why you need to do them.

We use the same approach we use for dividing with whole numbers, it's just that we work on places to the right of the decimal point.

Now we’ll look at using short division as a way to evaluate a division with decimals. In this example we need to rename some of our digits, just like we would when dividing whole numbers.

Consider the quotient 54.05\div5. We can break this up by partitioning the decimal into 5 tens, 4 units, 0 tenths and 5 hundredths. The 5 tens and 5 hundredths parts are easily divisible by 5, but how shall we try to divide the 4 units by 5?

We can use the fact that 4 units is the same as 40 tenths, which is now divisible by 5. Now the number 54.05 is partitioned into 5 tens, 0 units, 40 tenths and 5 hundredths. When we renamed 4 units to 40 tenths, we made use of a zero placeholder.

The process we went through in partitioning and renaming is what gets used behind the scenes when we perform short division, as shown in the video below.

Examples

Example 1

We want to find 92.1\div3.

a

Choose the most reasonable estimate for 92.1\div3

A

31

B

310

C

3.1

D

0.31

Worked Solution

Create a strategy

Round the dividend to the nearest multiple of 3 then perform division.

Apply the idea

92.1 rounded to the nearest multiple of 3 is 93.

\displaystyle 93\div3

\displaystyle =

\displaystyle 31

Evaluate

The most reasonable estimate is 31, option A.

Watch question walkthrough

b

Complete the short division to find 92.1\div3.

Worked Solution

Create a strategy

Perform short division with the same method as whole numbers.

Apply the idea

9 divided by 3 is 3

3 does not go into 2 so we put 0above the units place.

Copy the decimal point.

Write the 2 above the 1.

Divide 21 by 3 to get 7.

Watch question walkthrough

But what about when we want to divide by a decimal number?

We've already seen how to divide decimal numbers by whole numbers, so it would be great if we could just keep using this strategy. Using our knowledge of place value, we can.

We know that 10\div5=2, but what is 100\div50? Well, obviously that is just 2 as well. We can see that even though both numbers were ten times larger, we ended up with exactly the same answer. We can use the same strategy but with numbers ten times smaller. This means 1\div0.5=2 as well.

Remember: To divide a decimal by a decimal, we can first multiply both numbers by a suitable power of 10 to make them whole numbers. This can make the division easier.

Examples