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Subtracting integers

Equivalent fractionsCancelling fractions

Improper fractions and mixed numbersThis topic is relevant for:

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.

**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.

In order to subtract fractions:

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

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

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

DOWNLOAD FREEWork out

\[\frac{3}{5}-\frac{1}{5}\]

**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.

2**Subtract the numerators (top numbers).**

\[\frac{3}{5}-\frac{1}{5}=\frac{3-1}{5}=\frac{2}{5}\]

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

The final answer is:

\[\frac{2}{5}\]

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

Work out

\[\frac{5}{8}-\frac{3}{8}\]

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

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

**Subtract the numerators (top numbers).**

\[\frac{5}{8}-\frac{3}{8}=\frac{5-3}{8}=\frac{2}{8}\]

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

The answer is:

\[\frac{2}{8}\]

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:

\[\frac{2}{8}=\frac{2\times1}{2\times4}=\frac{1}{4}\]

The final answer is:

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

Work out

\[\frac{3}{4}-\frac{2}{5}\]

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

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:

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

**Subtract the numerators (top numbers).**

\[\frac{15}{20}-\frac{8}{20}=\frac{15-8}{20}=\frac{7}{20}\]

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

The final answer is:

\[\frac{7}{20}\]

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:

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

Work out

\[\frac{5}{6}-\frac{1}{2}\]

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

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:

\[\frac{5}{6}-\frac{1}{2}=\frac{5}{6}-\frac{3}{6}\]

**Subtract the numerators (top numbers).**

\[\frac{5}{6}-\frac{3}{6}=\frac{5-3}{6}=\frac{2}{6}\]

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

The answer is:

\[\frac{2}{6}\]

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:

\[\frac{2}{6}=\frac{2\times1}{2\times3}=\frac{1}{3}\]

The final answer is:

\[\frac{1}{3}\]

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

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

Work out

\[2\frac{1}{2}-1\frac{1}{3}\]

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

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:

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

**Subtract the numerators (top numbers).**

\[\frac{15}{6}-\frac{8}{6}=\frac{15-8}{6}=\frac{7}{6}\]

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

The answer is:

\[\frac{7}{6}\]

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:

\[\frac{7}{6}=\frac{6+1}{6}=\frac{6}{6}+\frac{1}{6}=1+\frac{1}{6}=1\frac{1}{6}\]

The final answer is:

\[1\frac{1}{6}\]

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}

Work out

\[3\frac{1}{2}-1\frac{3}{5}\]

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

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:

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

**Subtract the numerators (top numbers).**

\[\frac{35}{10}-\frac{16}{10}=\frac{35-16}{10}=\frac{19}{10}\]

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

The answer is:

\[\frac{19}{10}\]

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.

\[\frac{19}{10}=\frac{10+9}{10}=\frac{10}{10}+\frac{9}{10}=1+\frac{9}{10}=1\frac{9}{10}\]

The final answer is:

\[1\frac{9}{10}\]

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}

**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.

Subtracting 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. 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}

Give your answer as a fraction in its simplest form

\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}

Give your answer as a fraction in its simplest form

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}

Give your answer as a fraction in its simplest form

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}

1. Work out

\frac{4}{5}-\frac{1}{3}**(2 marks)**

Show answer

\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}

*For the correct final answer*

**(1)**

2. Work out

2\frac{5}{8}-1\frac{1}{3}**(2 marks)**

Show answer

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}

*For the correct final answer*

**(1)**

3. Which of these two fractions is closer to 1?

You must show all your working.

**(3 marks)**

Show answer

\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)**

You have now learned how to:

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

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