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Adding integers

Equivalent fractionsCancelling fractions

Mixed numbers and improper fractionsThis topic is relevant for:

Here we will learn about adding fractions including how to add fractions with the same denominators and with unlike denominators (different denominators) and how to add mixed numbers.

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

**Adding fractions** is when we add two or more fractions together or when we calculate the total of several numbers written as fractions.

To do this the fractions must have a** common denominator** (bottom number). Then we can add the fractions by adding the numerators (top numbers).

The method for adding fractions can be modified to subtract fractions.

E.g.

In order to add fractions:

- Ensure the fractions have a common denominator.
- Add the numerators (top numbers).
- Write your answer as a fraction, making sure it is in its simplest form.

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

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

DOWNLOAD FREEWork out:

\[\frac{1}{5}+\frac{1}{5}\]

- Ensure the fractions have a common denominator.

2Add the numerators (top numbers).

\[\frac{1}{5}+\frac{1}{5}=\frac{1+1}{5}=\frac{2}{5}\]

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

The final answer is

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

Work out:

\[\frac{1}{8}+\frac{5}{8}\]

**Ensure the fractions have a common denominator.**

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

\[\frac{1}{8}+\frac{5}{8}=\frac{1+5}{8}=\frac{6}{8}\]

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

The answer is

\[\frac{6}{8}\]

This fraction is not in its simplest form as it can be simplified.

Both

\[\frac{6}{8}=\frac{2\times3}{2\times4}=\frac{3}{4}\]

The final answer is

\[\frac{3}{4}\]

Work out:

\[\frac{1}{4}+\frac{2}{5}\]

**Ensure the fractions have a common denominator.**

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

(Lowest Common Multiple is sometimes called LCM or Least Common Multiple.)

\[\frac{1}{4}=\frac{1}{4}\times\frac{5}{5}=\frac{1\times5}{4\times5}=\frac{5}{20}\]

\[\frac{2}{5}=\frac{2}{5}\times\frac{4}{4}=\frac{2\times4}{5\times4}=\frac{8}{20}\]

We have converted fractions so that they have a common denominator and can now be added.

\[\frac{1}{4}+\frac{2}{5}=\frac{5}{20}+\frac{8}{20}\]

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

\[\frac{5}{20}+\frac{8}{20}=\frac{5+8}{20}=\frac{13}{20}\]

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

The final answer is

\[\frac{13}{20}\]

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

\[\frac{1}{4}+\frac{2}{5}=0.25+0.4=0.65=\frac{65}{100}=\frac{13}{20}\]

Work out:

\[\frac{5}{8}+\frac{1}{2}\]

**Ensure the fractions have a common denominator.**

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

(Lowest Common Multiple is sometimes called LCM or Least Common Multiple.)

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

We have converted the fractions so that they have a common denominator and can now be added.

\[\frac{5}{8}+\frac{1}{2}=\frac{5}{8}+\frac{4}{8}\]

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

\[\frac{5}{8}+\frac{4}{8}=\frac{5+4}{8}=\frac{9}{8}\]

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

The answer is

\[\frac{9}{8}\]

** improper fraction**.

It is usually expected that improper fractions are written as mixed numbers.

\[\frac{9}{8}=\frac{8+1}{8}=\frac{8}{8}+\frac{1}{8}=1+\frac{1}{8}=1\frac{1}{8}\]

The final answer is

\[1\frac{1}{8}\]

Alternatively, you could have chosen

\[\frac{5}{8}+\frac{1}{2}=\frac{10}{16}+\frac{8}{16}=\frac{10+8}{16}=\frac{18}{16}=\frac{2\times9}{2\times8}=\frac{9}{8}=1\frac{1}{8} \]

Work out:

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

**Ensure the fractions have a common denominator.**

As the values 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

\[2\frac{1}{2}+1\frac{1}{3}=\frac{5}{2}+\frac{4}{3}\]

To be able to** add the fractions** they need to have a **common denominato**r.

\[\frac{5}{2}=\frac{5}{2}\times\frac{3}{3}=\frac{5\times3}{2\times3}=\frac{15}{6}\]

\[\frac{4}{3}=\frac{4}{3}\times\frac{2}{2}=\frac{4\times2}{3\times2}=\frac{8}{6}\]

We have converted the fractions so that they have a common denominator and can now be added.

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

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

\[\frac{15}{6}+\frac{8}{6}=\frac{15+8}{6}=\frac{23}{6}\]

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

The answer is

\[\frac{23}{6}\]

**improper fraction**.

It is usually expected that improper fractions are written as mixed numbers.

\[\frac{23}{6}=\frac{18+5}{6}=\frac{18}{6}+\frac{5}{6}=3+\frac{5}{6}=3\frac{5}{6}\]

The final answer is

\[3\frac{5}{6}\]

Alternatively, you could add the whole numbers together and then add the fractions together.

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

Work out:

\[2\frac{3}{4}+1\frac{2}{5}\]

**Ensure the fractions have a common denominator.**

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

\[2\frac{3}{4}=\frac{8}{4}+\frac{3}{4}=\frac{8+3}{4}=\frac{11}{4}\]

\[1\frac{2}{5}=\frac{5}{5}+\frac{2}{5}=\frac{5+2}{5}=\frac{7}{5}\]

So the question is

\[2\frac{3}{4}+1\frac{2}{5}=\frac{11}{4}+\frac{7}{5}\]

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

\[\frac{11}{4}=\frac{11}{4}\times\frac{5}{5}=\frac{11\times5}{4\times5}=\frac{55}{20}\]

\[\frac{7}{5}=\frac{7}{5}\times\frac{4}{4}=\frac{7\times4}{5\times4}=\frac{28}{20}\]

We have converted the fractions so that they have a common denominator and can now be added.

\[\frac{11}{4}+\frac{7}{5}=\frac{55}{20}+\frac{28}{20}\]

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

\[\frac{55}{20}+\frac{28}{20}=\frac{55+28}{20}=\frac{83}{20}\]

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

The answer is

\[\frac{83}{20}\]

**improper fraction**.

It is usually expected that improper fractions are written as mixed numbers.

\[\frac{83}{20}=\frac{80+3}{20}=\frac{80}{20}+\frac{3}{20}=4+\frac{3}{20}=4\frac{3}{20}\]

The final answer is

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

Alternatively, you could add the whole numbers together and then add the fractions together.

\[2\frac{3}{4}+1\frac{2}{5}\\
=2+\frac{3}{4}+1+\frac{2}{5}\\
=2+1+\frac{3}{4}+\frac{2}{5}\\
=2+1+\frac{15}{20}+\frac{8}{20}\\
=3+\frac{15+8}{20}\\
=3+\frac{23}{20}\\
=3+1+\frac{3}{20}\\
=4\frac{3}{20}\]

**To be able to add fractions they must have common denominators**

When fractions are added together they must have the same denominator. If not, they must be converted to an equivalent fraction with a common denominator.

**Adding the correct parts of the fraction together**

When adding fractions with a common denominator we only add the numerators.

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

Adding 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{3}{9}+\frac{4}{9}

\frac{7}{9}

\frac{7}{18}

\frac{8}{8}

\frac{11}{18}

\frac{3}{9}+\frac{4}{9}=\frac{3+4}{9}=\frac{7}{9}

2. Work out – give your answer as a fraction in its simplest form:

\frac{5}{12}+\frac{3}{12}

\frac{8}{12}

\frac{8}{24}

\frac{1}{3}

\frac{2}{3}

\frac{5}{12}-\frac{3}{12}=\frac{5+3}{12}=\frac{8}{12}=\frac{2}{3}

3. Work out:

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

\frac{4}{9}

\frac{17}{20}

\frac{4}{20}

\frac{19}{20}

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

4. Work out:

\frac{3}{5}+\frac{1}{3}

\frac{4}{8}

\frac{14}{15}

\frac{4}{15}

\frac{11}{15}

\frac{3}{5}+\frac{1}{3}=\frac{9}{15}+\frac{5}{15}=\frac{9+5}{15}=\frac{14}{15}

5. Work out: 2\frac{1}{2}+1\frac{2}{5}

1\frac{1}{2}

3\frac{13}{15}

1\frac{8}{15}

3\frac{9}{10}

2\frac{1}{2}+1\frac{2}{5}
=\frac{5}{2}+\frac{7}{5}
=\frac{25}{10}+\frac{14}{10}
=\frac{25+14}{10}
=\frac{39}{10}
=3\frac{9}{10}

6. Work out:

1\frac{1}{6}+1\frac{1}{5}

2\frac{11}{30}

2\frac{2}{11}

2\frac{7}{30}

1\frac{17}{30}

1\frac{1}{6}+1\frac{1}{5}
=\frac{7}{6}+\frac{6}{5}
=\frac{35}{30}+\frac{36}{30}
=\frac{35+36}{30}
=\frac{71}{30}
=2\frac{11}{30}

1. Work out:

\frac{5}{12}+\frac{1}{6}

**(2 marks)**

Show answer

\frac{1}{6}=\frac{2}{12}

*For using a correct common denominator of 12 (or 36)*

**(1)**

\frac{5}{12}+\frac{2}{12}=\frac{7}{12}

*For the correct final answer*

**(1)**

2. Calculate:

\frac{4}{5}+\frac{5}{7}

Give your answer as a mixed number in its simplest terms.

**(3 marks)**

Show answer

\frac{4}{5}+\frac{5}{7}=\frac{28}{35}+\frac{25}{35}

*For writing fractions with a common denominator of 35*

**(1)**

\frac{28+25}{35}

*For adding the fractions*

**(1)**

\frac{53}{35}=1\frac{18}{35}

*For writing the final answer as a mixed number*

**(1)**

3. Work out:

2\frac{1}{8}+1\frac{1}{3}

**(2 marks)**

Show answer

2\frac{1}{8}+1\frac{1}{3}=\frac{17}{8}+\frac{4}{3}=\frac{51}{24}+\frac{32}{24}

OR

2\frac{1}{8}+1\frac{1}{3}=2\frac{3}{24}+1\frac{8}{24}

*For a method to add fractions with a correct common denominator*

**(1)**

\frac{51}{24}+\frac{32}{24}=\frac{83}{24}

OR

2\frac{3}{24}+1\frac{8}{24}=3\frac{11}{24}

*For the correct final answer*

**(1)**

You have now learned how to:

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

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