# 3.1 Using equivalent fractions

Equivalent fractions are fractions that are the same as each other, but are expressed in different ways. The BBC Skillswise website has an explanation of equivalent fractions. [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)]

To make an equivalent fraction, you multiply or divide the numerator (top) and denominator (bottom) by the same number. The size of the fraction is not altered. For example:

In the fraction , the numerator is 4 and the denominator is 6.

4 × 2 = 8

6 × 2 = 12

So =

In the fraction , the numerator is 10 and the denominator is 15.

10 ÷ 5 = 2

15 ÷ 5 = 3

So =

## Example: Looking at equivalent fractions

Arrange the following fractions in order of size, starting with the smallest:

- , ,

### Method

You need to look at the bottom number in each fraction (the denominator) and find the lowest common multiple. In this case, the bottom numbers are 6, 3 and 12, so the lowest common multiple is 12:

- 6 × 2 = 12
- 3 × 4 = 12
- 12 × 1 = 12

Whatever you do to the bottom of the fraction you must also do to the top of the fraction, so that it holds the equivalent value. The third fraction, , already has 12 as its denominator, so we don’t need to make any further calculations for this fraction. But what about and ?

- 2 × means calculating (2 × 3 = 6) and (2 × 6 = 12), so the equivalent fraction is
- 4 × means calculating (4 × 1 = 4) and (4 × 3 = 12), so the equivalent fraction is

Now you can now see the size order of the fractions clearly:

- , ,

So the answer is:

- , ,

Use the examples above to help you with the following activity. Remember to check your answers once you have completed the questions.

## Activity 19: Fractions in order of size

- Put these fractions in order of size, with the smallest first:

- , , , ,

### Answer

Remember that when the numerator of a fraction is 1, the larger the denominator, the smaller the fraction.

From smallest to largest, the order is:

- , , , ,

- What should you replace the question marks with to make these fractions equivalent?

- =
- =
- =
- =

### Answer

- =
- =
- =
- =

- Put these fractions in order of size, with the smallest first:

- , ,

### Answer

You need to change to equivalent fractions to compare like-for-like. To do this, you need to look at the bottom numbers of the fractions (3, 5 and 10) and find the lowest common multiple. The lowest common multiple of 3, 5 and 10 is 30:

3 × 10 = 30

5 × 6 = 30

10 × 3 = 30

Whatever you do to the bottom of each fraction, you must also do to the top:

With , you need to multiply the top and bottom numbers by 10 to make .

With , you need to multiply the top and bottom number by 6 to equal .

With , you need to multiply the top and bottom number by 3 to equal.

The order of the fractions from smallest to largest is therefore:

()

()

()