GCSE Maths Number Fractions

Equivalent Fractions

# Equivalent Fractions

Here we will learn about equivalent fractions including how to simplify fractions and how to work out equivalent fractions.

There are also equivalent fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are equivalent fractions?

Equivalent fractions are fractions that have the same value. We use equivalent fractions to write a fraction in its simplest terms. To do this we look at the numerator (the top number) and the denominator (the bottom number) and find a common factor to cancel. The numerator and denominator are always whole numbers.

Equivalent fractions can be used to compare fractions when they have different denominators. Knowledge of equivalent fractions can also be used to write a set of values in order of size.

Here are some examples of the concept of equivalent fractions:

This is one half:

$\frac{1}{2}=\frac{2}{4}$

This is two thirds:

$\frac{2}{3}=\frac{6}{9}$

## How to write equivalent fractions

In order to simplify fractions:

1. Look at the numerator and the denominator and find the Highest Common Factor.
2. Divide both the numerator and the denominator by the HCF.
3. Write the fraction in its simplest terms.

In order to calculate equivalent fractions:

1. Look at the denominators in both fractions and work out the multiplier.
2. Multiply the numerator of one of the fractions by the multiplier.
3. Complete the second fraction.

## Equivalent fractions examples

### Example 1: simplify fractions

Write the following fraction in its simplest form:

$\frac{12}{14}$

1. Look at the numerator and the denominator and find the Highest Common Factor.

The numerator of the fraction is 12.

The denominator of the fraction is 14.

They are both multiples of 2. The Highest Common Factor of 12 and 14 is 2.

2Divide both the numerator and the denominator by the HCF.

$\frac{12}{14}=\frac{6\times2}{7\times2}$

The HCF of 2 will cancel.

The new numerator is 6 and the new denominator is 7. 6 and 7 have no common factors other than 1. So the fraction will be in its simplest form.

3Write the fraction in its simplest terms.

$\frac{12}{14}=\frac{6}{7}$

### Example 2: simplify fractions

Write the following fraction in its simplest form:

$\frac{15}{20}$

The numerator for the fraction is 15.

The denominator of the fraction is 20.

They are both multiples of 5. The Highest Common Factor of 15 and 20 is 5.

$\frac{15}{20}=\frac{3\times5}{4\times5}$

The HCF of 5 will cancel.

The new numerator is 3 and the new denominator is 4. 3 and 4 have no common factors other than 1. So the fraction will be in its simplest form.

$\frac{15}{20}=\frac{3}{4}$

### Example 3: simplify fractions

Write the following fraction in its simplest form:

$\frac{24}{30}$

The numerator for the fraction is 24.

The denominator of the fraction is 30.

They are both multiples of 2, 3 and 6. The Highest Common Factor of 24 and 30 is 6.

$\frac{24}{30}=\frac{4\times6}{5\times6}$

The HCF of 6 will cancel.

The new numerator is 4 and the new denominator is 5. 4 and 5 have no common factors other than 1. So the fraction will be in its simplest form.

$\frac{24}{30}=\frac{5}{6}$

### Example 4: calculating equivalent fractions

Find the missing value of these equivalent fractions:

$\frac{3}{5}=\frac{?}{20}$

The denominator of the first fraction is 4 and the denominator of the second fraction is 20.

$20\div5=4$

The multiplier is 4.

$3\times4=12$
$\frac{3}{5}=\frac{3\times4}{5\times4}$

The missing value is 12.

$\frac{3}{5}=\frac{12}{20}$

### Example 5: calculating equivalent fractions

Find the missing value of these equivalent fractions:

$\frac{2}{7}=\frac{?}{21}$

The denominator of the first fraction is 7 and the denominator of the second fraction is 21.

$21\div7=3$

The multiplier is 3.

$2\times3=6$
$\frac{2}{7}=\frac{2\times3}{7\times3}$

The missing value is 6.

$\frac{2}{7}=\frac{6}{21}$

### Example 6: calculating equivalent fractions

Find the missing value of these equivalent fractions:

$\frac{5}{8}=\frac{?}{72}$

The denominator of the first fraction is 8 and the denominator of the second fraction is 72.

$72\div8=9$

The multiplier is 9.

$5\times9=45$
$\frac{5}{8}=\frac{5\times9}{8\times9}$

The missing value is 45.

$\frac{5}{8}=\frac{45}{72}$

### Common misconceptions

• Write a fraction in its simplest form

You can simplify a fraction by using any of the common factors of the numerator and the denominator of a fraction. But you may need to cancel more than once to make sure you have written the fraction in its simplest form. Using the Highest Common Factor means that you will only have to cancel once

E.g.

Write the following fraction in its simplest form.

$\frac{20}{60}$

2

0 and 60 have a common factor of 10.  We can cancel the common factor of 10.

$\frac{20}{60}=\frac{2\times10}{6\times10}=\frac{2}{6}$

But the first fraction has not been written in its simplest form as the new numerator 2 and the new denominator 6 also share a common factor of 2.

$\frac{2}{6}=\frac{1\times2}{3\times2}=\frac{1}{3}$

The fraction in its simplest form is \frac{1}{3}

• Equivalent fractions can only be made by using common factors

You cannot make equivalent fractions by using addition.

E.g.

This is incorrect cancelling.

$\frac{5}{10}=\frac{2+3}{7+3}=\frac{2}{7}$ ✘

This is correct cancelling.

$\frac{5}{10}=\frac{1\times5}{2\times5}=\frac{1}{2}$ ✔

• Fractions that appear to be different can have equivalence

E.g.

Because \frac{5}{10} and \frac{1}{2} use different numbers (they have different numerators and denominators) they appear to be different fractions, however they are actually equivalent to each other.

$\frac{5}{10}=\frac{1\times5}{2\times5}=\frac{1}{2}$

### Practice equivalent fractions questions

1. Write the following fraction in its simplest form: \frac{5}{20}

\frac{1}{4}

\frac{1}{3}

\frac{2}{5}

\frac{1}{15}
\frac{5}{20}=\frac{1\times5}{4\times5}=\frac{1}{4}

2. Write the following fraction in its simplest form: \frac{8}{24}

\frac{4}{12}

\frac{2}{6}

\frac{1}{3}

\frac{5}{12}

\frac{8}{24}=\frac{1\times8}{3\times8}=\frac{1}{3}

3. Write the following fraction in its simplest form: \frac{36}{48}

\frac{6}{8}

\frac{18}{24}

\frac{7}{12}

\frac{3}{4}

\frac{36}{48}=\frac{3\times12}{4\times12}=\frac{3}{4}

4. Find the missing value is these equivalent fractions: \frac{3}{4}=\frac{?}{24}

\frac{23}{24}

\frac{18}{24}

\frac{19}{24}

\frac{21}{24}

\frac{3}{4}=\frac{3\times6}{4\times6}=\frac{18}{24}

5. Find the missing value is these equivalent fractions: \frac{6}{7}=\frac{?}{28}

\frac{18}{28}

\frac{24}{28}

\frac{12}{28}

\frac{24}{28}

\frac{6}{7}=\frac{6\times4}{7\times4}=\frac{24}{28}

6. Find the missing value is these equivalent fractions: \frac{5}{9}=\frac{?}{63}

\frac{35}{63}

\frac{25}{63}

\frac{30}{63}

\frac{42}{63}

\frac{5}{9}=\frac{5\times7}{9\times7}=\frac{35}{63}

### Equivalent fractions GCSE questions

1. Here is a list of four fractions.

\frac{15}{20}     \frac{5}{25}       \frac{3}{15}       \frac{2}{10}

One of these fractions is not equivalent to \frac{1}{5}

Write down this fraction.

(1 mark)

Show answer

\frac{15}{20}

(1)

2. Here is a list of four fractions.

\frac{18}{24}       \frac{15}{20}       \frac{28}{40}       \frac{36}{48}

One of these fractions is not equivalent to \frac{3}{4}

Write down this fraction.

(1 mark)

Show answer

\frac{28}{40}

(1)

3. (a) Show that \frac{3}{5} is bigger than \frac{5}{9} .

(b) Find a fraction which is bigger than \frac{3}{5} but smaller than \frac{3}{4} .

(4 marks)

Show answer

(a)

\frac{3\times9}{5\times9}=\frac{27}{45}       \frac{5\times5}{9\times5}=\frac{25}{45}

for converting one fraction to an equivalent fraction

(1)

for both fractions with a common denominator

(1)

\frac{3}{5}=\frac{27}{45}       \frac{5}{9}=\frac{25}{45}

(b)

\frac{3\times4}{5\times4}=\frac{12}{20}       \frac{3\times5}{4\times5}=\frac{15}{20}

for converting one fraction to an equivalent fraction

(1)

for a correct answer between 0.6 and 0.75

(1)

\frac{13}{20} or \frac{14}{20} or \frac{7}{10}

## Learning checklist

You have now learned how to:

• Simplify fractions
• Write an equivalent fraction for a given fraction

## The next lessons are

• Ordering fractions
• Adding fractions
• Subtracting fractions
• Improper fractions
• Using the bar model
• Mixed numbers
• Decimals

## Still stuck?

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