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

**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 so that they can be written in order of size.

Here are some examples of the concept of equivalent fractions:

E.g.

\[\frac{1}{2}=\frac{2}{4}\]

E.g.

\[\frac{2}{3}=\frac{6}{9}\]

In order to simplify fractions:

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

In order to calculate equivalent fractions:

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

Get your free equivalent fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free equivalent fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEWrite the following fraction in its simplest form:

\[\frac{12}{14}\]

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

The numerator of the fraction is

The denominator of the fraction is

They are both multiples of

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

\[\frac{12}{14}=\frac{6\times2}{7\times2}\]

The HCF of

The new numerator is

3**Write the fraction in its simplest terms**.

\[\frac{12}{14}=\frac{6}{7}\]

Write the following fraction in its simplest form:

\[\frac{15}{20}\]

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

The numerator for the fraction is

The denominator of the fraction is

They are both multiples of

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

\[\frac{15}{20}=\frac{3\times5}{4\times5}\]

The HCF of

The new numerator is

**Write the fraction in its simplest terms**.

\[\frac{15}{20}=\frac{3}{4}\]

Write the following fraction in its simplest form:

\[\frac{24}{30}\]

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

The numerator for the fraction is

The denominator of the fraction is

They are both multiples of

The Highest Common Factor of

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

\[\frac{24}{30}=\frac{4\times6}{5\times6}\]

The HCF of

The new numerator is

**Write the fraction in its simplest terms**.

\[\frac{24}{30}=\frac{5}{6}\]

Find the missing value of these equivalent fractions:

\[\frac{3}{5}=\frac{?}{20}\]

**Look at the denominators in both fractions and work out the multiplier**.

The denominator of the first fraction is

\[20\div5=4\]

The multiplier is

**Multiply the numerator of one of the fractions by the multiplier**.

\[3\times4=12\]

\[\frac{3}{5}=\frac{3\times4}{5\times4}\]

The missing value is

**Complete the second fraction**.

\[\frac{3}{5}=\frac{12}{20}\]

Find the missing value of these equivalent fractions:

\[\frac{2}{7}=\frac{?}{21}\]

**Look at the denominators in both fractions and work out the multiplier**.

The denominator of the first fraction is

\[21\div7=3\]

The multiplier is

**Multiply the numerator of one of the fractions by the multiplier**.

\[2\times3=6\]

\[\frac{2}{7}=\frac{2\times3}{7\times3}\]

The missing value is

**Complete the second fraction**

\[\frac{2}{7}=\frac{6}{21}\]

Find the missing value of these equivalent fractions:

\[\frac{5}{8}=\frac{?}{72}\]

**Look at the denominators in both fractions and work out the multiplier**.

The denominator of the first fraction is

\[72\div8=9\]

The multiplier is

**Multiply the numerator of one of the fractions by the multiplier**.

\[5\times9=45\]

\[\frac{5}{8}=\frac{5\times9}{8\times9}\]

The missing value is

**Complete the second fraction**.

\[\frac{5}{8}=\frac{45}{72}\]

**Fractions in the simplest form**

You can simplify a fraction by using any of the common factors of the numerator and the denominator of a fraction. However 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

\[\frac{20}{60}=\frac{2\times10}{6\times10}=\frac{2}{6}\]

However the new fraction has not been written in its simplest form as the new numerator

\[\frac{2}{6}=\frac{1\times2}{3\times2}=\frac{1}{3}\]

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

**Equivalent fractions and 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 and 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.

\[\frac{5}{10}=\frac{1\times5}{2\times5}=\frac{1}{2}\]

Equivalent fractions is part of our series of lessons to support revision on fractions. You may find it helpful to start with the main fractions lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

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 of 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 of 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 of 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}

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}

You have now learned how to:

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

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