GCSE Maths Number FDP Fractions

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.

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

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

Work out:

$\frac{1}{5}+\frac{1}{5}$

1. Ensure the fractions have a common denominator.

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

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

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

$\frac{2}{5}$

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

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

Work out:

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

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

$\frac{1}{8}+\frac{5}{8}=\frac{1+5}{8}=\frac{6}{8}$

$\frac{6}{8}$

This fraction is not in its simplest form as it can be simplified.
Both 6 and 8 are multiples of 2, so 2 is a common factor and can be cancelled.

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

$\frac{3}{4}$

### Example 3: adding fractions with different denominators

Work out:

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

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 add the fractions they need to have a common denominator.

4 and 5 have a Lowest Common Multiple of 20, so we need to change both fractions so that they have a common denominator of 20.
(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.

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

$\frac{5}{20}+\frac{8}{20}=\frac{5+8}{20}=\frac{13}{20}$

$\frac{13}{20}$

13 and 20 do not have a common factor. The fraction cannot be simplified. This fraction is in its simplest form.

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

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

### Example 4: adding fractions with different denominators

Work out:

$\frac{5}{8}+\frac{1}{2}$

8 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 add the fractions they need to have a common denominator.

8 and 2 have a Lowest Common Multiple of 8, so we make sure that both fractions have a common denominator of 8.
(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.

$\frac{5}{8}+\frac{1}{2}=\frac{5}{8}+\frac{4}{8}$

$\frac{5}{8}+\frac{4}{8}=\frac{5+4}{8}=\frac{9}{8}$

$\frac{9}{8}$

9 and 8 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.

$\frac{9}{8}=\frac{8+1}{8}=\frac{8}{8}+\frac{1}{8}=1+\frac{1}{8}=1\frac{1}{8}$

$1\frac{1}{8}$

Alternatively, you could have chosen 16 as a common denominator and cancelled by the common factor of 2 to simplify.

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

### Example 5: adding mixed numbers

Work out:

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

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

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

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 add 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 added.

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

$\frac{15}{6}+\frac{8}{6}=\frac{15+8}{6}=\frac{23}{6}$

$\frac{23}{6}$

23 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.

$\frac{23}{6}=\frac{18+5}{6}=\frac{18}{6}+\frac{5}{6}=3+\frac{5}{6}=3\frac{5}{6}$

$3\frac{5}{6}$

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

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

### Example 6: adding mixed numbers

Work out:

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

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

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

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 add 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.

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

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

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

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

$\frac{55}{20}+\frac{28}{20}=\frac{55+28}{20}=\frac{83}{20}$

$\frac{83}{20}$

83 and 20 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.

$\frac{83}{20}=\frac{80+3}{20}=\frac{80}{20}+\frac{3}{20}=4+\frac{3}{20}=4\frac{3}{20}$

$4\frac{3}{20}$

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

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

### Common misconceptions

• 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.
$\frac{1}{2}+\frac{1}{2}=\frac{1+1}{2}=\frac{2}{2}=1$

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)

\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}

(1)

2. Calculate:

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

(3 marks)

\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}

(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)

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}

(1)

## Learning checklist

You have now learned how to:

• Add fractions with the same denominator
• Add fractions with different denominators