GCSE Maths Number FDP Fractions

Subtracting fractions

# Subtracting Fractions

Here we will learn about subtracting fractions including how to subtract fractions with the same denominator and with unlike denominators (different denominators), and how to subtract mixed numbers.
There are also subtracting 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 is subtracting fractions?

Subtracting fractions is when we take away one fraction from another or we find the difference between fractions.

To subtract fractions they need to have the same denominator (the same bottom numbers).
Then we can subtract the fractions by subtracting the numerators (the top numbers) and keeping the denominator the same.

The method for subtracting fractions can be modified to add fractions.

E.g.

## How to subtract fractions

In order to subtract fractions:

1. Look at the denominators (bottom numbers) to see if they have a common denominator.
2. Subtract the numerators (top numbers).
3. Write your answer as a fraction, making sure it is in its simplest form.

## Subtracting fractions examples

### Example 1: subtracting fractions with a common denominator

Work out

$\frac{3}{5}-\frac{1}{5}$

1. Look at the denominators (bottom numbers) to see if they have a common denominator.

5 is the denominator for both fractions, so 5 is the common denominator.

2Subtract the numerators (top numbers).

$\frac{3}{5}-\frac{1}{5}=\frac{3-1}{5}=\frac{2}{5}$

3Write your answer as a fraction, making sure it is in its simplest form.

$\frac{2}{5}$

2 and 5 do not have a common factor so the fraction cannot be simplified. This fraction is in its simplest form.

### Example 2: subtracting fractions with a common denominator

Work out

$\frac{5}{8}-\frac{3}{8}$

8 is the denominator for both fractions, so 8 is the common denominator.

$\frac{5}{8}-\frac{3}{8}=\frac{5-3}{8}=\frac{2}{8}$

$\frac{2}{8}$

This fraction is not in its simplest form as it can be cancelled.  Both 2 and 8 are multiples of 2. So 2 is a common factor and can be cancelled:

$\frac{2}{8}=\frac{2\times1}{2\times4}=\frac{1}{4}$

$\frac{1}{4}$

### Example 3: subtracting fractions with different denominators

Work out

$\frac{3}{4}-\frac{2}{5}$

4 is the denominator of the first fraction and 5 is the denominator of the second fraction. These fractions do not have a common denominator.

To be able to subtract the fractions they need to have a common denominator.

4 and 5 have a lowest common multiple of 20, so we change both fractions so that they have a common denominator of 20:

$\frac{3}{4}\times\frac{5}{5}=\frac{3\times5}{4\times5}=\frac{15}{20}$
$\frac{2}{5}\times\frac{4}{4}=\frac{2\times4}{5\times4}=\frac{8}{20}$

We have converted the fractions so that they have a common denominator and can now be subtracted:

$\frac{3}{4}-\frac{2}{5}=\frac{15}{20}-\frac{8}{20}$

$\frac{15}{20}-\frac{8}{20}=\frac{15-8}{20}=\frac{7}{20}$

$\frac{7}{20}$

7 and 20 do not have a common factor so the fraction cannot be simplified. This fraction is in its simplest form.

Alternatively, this question could have been solved by using decimals:

$\frac{3}{4}-\frac{2}{5}=0.75-0.4=0.35=\frac{35}{100}=\frac{7}{20}$

### Example 4: subtracting fractions with different denominators

Work out

$\frac{5}{6}-\frac{1}{2}$

6 is the denominator of the first fraction and 2 is the denominator of the second fraction.  These fractions do not have a common denominator.

To be able to subtract the fractions they need to have a common denominator.

6 and 2 have a lowest common multiple of 6, so we make sure that both fractions have a common denominator of 6 by changing the second fraction:

$\frac{1}{2}\times\frac{3}{3}=\frac{1\times3}{2\times3}=\frac{3}{6}$

So we have converted the fractions so that they have a common denominator and can now be subtracted:

$\frac{5}{6}-\frac{1}{2}=\frac{5}{6}-\frac{3}{6}$

$\frac{5}{6}-\frac{3}{6}=\frac{5-3}{6}=\frac{2}{6}$

$\frac{2}{6}$

This fraction is not in its simplest form as it can be simplified because both 2 and 6 are multiples of 2.
So 2 is a common factor and can be cancelled:

$\frac{2}{6}=\frac{2\times1}{2\times3}=\frac{1}{3}$

$\frac{1}{3}$

Alternatively, you could have chosen 12 as a common denominator and cancelled by the common factor of 4 to simplify:

$\frac{5}{6}-\frac{1}{2}=\frac{10}{12}-\frac{6}{12}=\frac{10-6}{12}=\frac{4}{12}=\frac{4\times1}{4\times3}=\frac{1}{3}$

### Example 5: subtracting mixed numbers

Work out

$2\frac{1}{2}-1\frac{1}{3}$

As they are mixed numbers it is a good idea to write them as improper fractions (where the numerator is larger than the denominator):

$2\frac{1}{2}=\frac{4}{2}+\frac{1}{2}=\frac{4+1}{2}=\frac{5}{2}$
$1\frac{1}{3}=\frac{3}{3}+\frac{1}{3}=\frac{3+1}{3}=\frac{4}{3}$

So the question is:

$2\frac{1}{2}-1\frac{1}{3}=\frac{5}{2}-\frac{4}{3}$

2 is the denominator of the first fraction and 3 is the denominator of the second fraction. These fractions do not have a common denominator.

To be able to subtract the fractions they need to have a common denominator.

2 and 3 have a lowest common multiple of 6, so we change both fractions so that they have a common denominator of 6:

$\frac{5}{2}=\frac{5}{2}\times\frac{3}{3}=\frac{5\times3}{2\times3}=\frac{15}{6}$
$\frac{4}{3}=\frac{4}{3}\times\frac{2}{2}=\frac{4\times2}{3\times2}=\frac{8}{6}$

We have converted the fractions so that they have a common denominator and can now be subtracted:

$\frac{5}{2}-\frac{4}{3}=\frac{15}{6}-\frac{8}{6}$

$\frac{15}{6}-\frac{8}{6}=\frac{15-8}{6}=\frac{7}{6}$

$\frac{7}{6}$

7 and 6 do not have a common factor to cancel, but this fraction is an improper fraction
It is usually expected that improper fractions are written as mixed numbers:

$\frac{7}{6}=\frac{6+1}{6}=\frac{6}{6}+\frac{1}{6}=1+\frac{1}{6}=1\frac{1}{6}$

$1\frac{1}{6}$

Alternatively you could subtract the whole numbers and then subtract the fractions.

\begin{align} 2\frac{1}{2}-1\frac{1}{3}=&(2+\frac{1}{2})-(1+\frac{1}{3})\\ =&(2-1)+(\frac{1}{2}-\frac{1}{3})\\ =&(2-1)+(\frac{3}{6}-\frac{2}{6})\\ =&1+\frac{1}{6}\\ =&1\frac{1}{6} \end{align}

### Example 6: subtracting mixed numbers

Work out

$3\frac{1}{2}-1\frac{3}{5}$

As they are mixed numbers it is a good idea to write them as improper fractions (where the numerator is larger than the denominator):

$3\frac{1}{2}=\frac{6}{2}+\frac{1}{2}=\frac{6+1}{2}=\frac{7}{2}$

$1\frac{3}{5}=\frac{5}{5}+\frac{3}{5}=\frac{5+3}{5}=\frac{8}{5}$

So the question is:

$3\frac{1}{2}-1\frac{3}{5}=\frac{7}{2}-\frac{8}{5}$

2 is the denominator of the first fraction and 5 is the denominator of the second fraction. These fractions do not have a common denominator.

To be able to subtract the fractions they need to have a common denominator.

2 and 5 have a lowest common multiple of 10, so we change both fractions so that they have a common denominator of 10:

$\frac{7}{2}=\frac{7}{2}\times\frac{5}{5}=\frac{7\times5}{2\times5}=\frac{35}{10}$
$\frac{8}{5}=\frac{8}{5}\times\frac{2}{2}=\frac{8\times2}{5\times2}=\frac{16}{10}$

We have converted the fractions so that they have a common denominator and can now be subtracted:

$\frac{7}{2}-\frac{8}{5}=\frac{35}{10}-\frac{16}{10}$

$\frac{35}{10}-\frac{16}{10}=\frac{35-16}{10}=\frac{19}{10}$

$\frac{19}{10}$

19 and 10 do not have a common factor to cancel, but this is an improper fraction.
It is usually expected that improper fractions are written as mixed numbers.

$\frac{19}{10}=\frac{10+9}{10}=\frac{10}{10}+\frac{9}{10}=1+\frac{9}{10}=1\frac{9}{10}$

$1\frac{9}{10}$

Alternatively, you could subtract the whole numbers and then subtract the fractions:

\begin{align} 3\frac{1}{2}-1\frac{3}{5}=&(3+\frac{1}{2})-(1+\frac{3}{5})\\ =&(3-1)+(\frac{1}{2}-\frac{3}{5})\\ =&(3-1)+(\frac{5}{10}-\frac{6}{10})\\ =&(2)+(-\frac{1}{10})\\ =&1\frac{9}{10} \end{align}

### Common misconceptions

• Subtracting numerators
When subtracting fractions with common denominators, only the numerators are subtracted.
E.g.
$\frac{5}{7}-\frac{2}{7}=\frac{5-2}{7}=\frac{3}{7}$

• Mixing up addition and subtraction

Sometimes we can get so involved in making sure the fractions have a common denominator that we forget it is a subtraction.

E.g

$\frac{3}{5}-\frac{1}{4}=\frac{12}{20}-\frac{5}{20}=\frac{12+5}{20}=\frac{17}{20}$

This would be wrong as the question has turned into an addition question.

• Remember to use the correct order of operations

When there is addition and subtraction in the same maths sentence, perform the operations in order from left to right as you read them.

E.g.

$\frac{6}{8}-\frac{1}{8}+\frac{2}{8}=\frac{6-1+2}{8}=\frac{7}{8}$

The correct order is 6 – 1 and then + 2.

### Practice Subtracting fractions questions

1. Work out \frac{2}{5}-\frac{1}{5}

\frac{3}{5}

\frac{1}{5}

\frac{1}{0}

\frac{2}{5}
\frac{2}{5}-\frac{1}{5}=\frac{2-1}{5}=\frac{1}{5}

2. Work out \frac{3}{8}-\frac{1}{8}

\frac{2}{5}

\frac{2}{0}

\frac{2}{8}

\frac{1}{4}
\frac{3}{8}-\frac{1}{8}=\frac{3-1}{8}=\frac{2}{8}=\frac{1}{4}

3. Work out \frac{4}{5}-\frac{1}{2}

\frac{3}{10}

\frac{3}{3}

\frac{9}{10}

\frac{1}{5}
\frac{4}{5}-\frac{1}{2}=\frac{8}{10}-\frac{5}{10}=\frac{8-5}{10}=\frac{3}{10}

4. Work out \frac{5}{8}-\frac{1}{4}

\frac{1}{8}

\frac{4}{4}

\frac{3}{8}

\frac{3}{12}
\frac{5}{8}-\frac{1}{4}=\frac{5}{8}-\frac{2}{8}=\frac{5-2}{8}=\frac{3}{8}

5. Work out 2\frac{3}{4}-1\frac{1}{3}

4\frac{1}{12}

2\frac{5}{12}

1\frac{5}{12}

1\frac{7}{12}
2\frac{3}{4}-1\frac{1}{3} =\frac{11}{4}-\frac{4}{3} =\frac{33}{12}-\frac{16}{12} =\frac{33-16}{12} =\frac{17}{12} =1\frac{5}{12}

6. Work out 2\frac{1}{5}-1\frac{2}{3}

1\frac{1}{2}

3\frac{13}{15}

1\frac{8}{15}

\frac{8}{15}
2\frac{1}{5}-1\frac{2}{3} =\frac{11}{5}-\frac{5}{3} =\frac{33}{15}-\frac{25}{15} =\frac{33-25}{15} =\frac{8}{15}

### Subtracting fractions GCSE questions

1. Work out

\frac{4}{5}-\frac{1}{3}

(2 marks)

\frac{4}{5}-\frac{1}{3}=\frac{12}{15}-\frac{5}{15}

For using a correct common denominator with at least one correct fraction

(1)

=\frac{7}{12}

(1)

2. Work out

2\frac{5}{8}-1\frac{1}{3}

(2 marks)

2\frac{5}{8}-1\frac{1}{3}=\frac{21}{8}-\frac{4}{3}=\frac{63}{24}-\frac{32}{24}

For using a correct common denominator with at least one correct fraction

(1)

\frac{63}{24}-\frac{32}{24}=\frac{31}{24}=1\frac{7}{24}

(1)

3. Which of these two fractions is closer to 1?
You must show all your working.

\frac{7}{4}     \frac{4}{7}

(3 marks)

\frac{7}{4}=\frac{49}{28}
and
\frac{4}{7}=\frac{16}{28}

For writing fractions with a common denominator

(1)

\frac{49}{28}-\frac{28}{28}=\frac{21}{28}
and
\frac{28}{28}-\frac{16}{28}=\frac{12}{28}

For finding the difference of the fractions to 1

(1)

\frac{12}{28} is the smallest difference,

So \frac{4}{7} is closer to 1.

For the correct answer with workings

(1)

## Learning checklist

You have now learned how to:

• Subtract fractions with the same denominator
• Subtract fractions with different denominators
• Subtract mixed numbers

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