Equivalent Fractions

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.

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 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.

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

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 is 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 is 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 is 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

• 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.

$\frac{20}{60}$

20 and 60 have a common factor of 10.

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

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

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

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 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}

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)

\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)

\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)

(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

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