GCSEKS3Level 1-3AQAEdexcelOCRWJEC

## Adding and Subtracting Fractions Revision

In order to add and subtract fractions, you need to find a common denominator – some value that can become the denominator of both fractions. There are two main methods for choosing a common denominator:

## Take Note

Before adding and subtracting fractions, it’s important to understand that if you multiply both top and bottom of a fraction by the same value, the fraction’s value doesn’t change. For example,

$\dfrac{2}{3}=\dfrac{4}{6},\,\,\,\,\text{ and }\,\,\,\,\dfrac{2}{3}=\dfrac{10}{15}$

Evaluate $\dfrac{3}{5} + \dfrac{1}{4}$

[2 marks]

To find a common denominator here, we will take the product of the two denominators: $5\times 4=20$.

To make sure we aren’t changing the value of the fraction, we also multiply the top by $4$.

$\dfrac{3}{5}=\dfrac{3\times \textcolor{red}{4}}{5\times \textcolor{red}{4}}=\dfrac{12}{20}$

For the second fraction’s denominator to be $20$, we’ll have to multiply it by $5$. So, we will also have to multiply the top by $5$.

$\dfrac{1}{4}=\dfrac{1\times \textcolor{red}{5}}{4\times \textcolor{red}{5}}=\dfrac{5}{20}$

Now, to add two fractions with the same denominator, simply add the numerators together. Doing so, we get,

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

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## Example 2: Subtracting Fractions

Evaluate $\dfrac{4}{7} - \dfrac{2}{5}$

[2 marks]

Our choice of common denominator in this case will be $\textcolor{red}{7}\times \textcolor{red}{5}=35$

To make the denominator of the first fraction $35$, we’ll have to multiply its top and bottom by $5$. To make the denominator of the second fraction be $35$, we’ll have to multiply its top and bottom by $7$. This looks like,

\begin{aligned}\dfrac{4}{7}-\dfrac{2}{5} &= \dfrac{5\times 4}{35}-\dfrac{7\times 2}{35} \\ \\ &=\dfrac{20}{35} - \dfrac{14}{35} =\dfrac{6}{35}\end{aligned}

Remember, in the final step you subtract the numerators, and the denominator is unchanged.

Level 1-3GCSEKS3AQAEdexcelOCRWJEC

## Example 3: Adding Fractions and Whole Numbers

Evaluate $8+\dfrac{5}{6}$

[2 marks]

To add a whole number to a fraction, we turn the whole number into a fraction by dividing by $1$. Thus,

$\dfrac{8}{1}+\dfrac{5}{6}$

This time, since $1\times 6=6$, the common denominator will be $6$, meaning we’ll only have to change the first fraction – we will multiply its top and bottom by $6$. Doing so, we get

\begin{aligned}\dfrac{8}{1}+\dfrac{5}{6} &= \dfrac{8\times 6}{1\times 6}+\dfrac{5}{6} \\ \\ &=\dfrac{48}{6} + \dfrac{5}{6} = \dfrac{53}{6} \end{aligned}

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## Example 4: Adding Mixed Fractions

Evaluate $2\dfrac{5}{9}+\dfrac{1}{3}$

[3 marks]

To add a mixed number to a fraction, first convert the mixed number to an improper fraction.

$2\dfrac{5}{9}=\dfrac{(2\times 9)+5}{9}=\dfrac{23}{9}$

Now the calculation looks like

$\dfrac{23}{9}+\dfrac{1}{3}$

Now, for the common denominator we could use $27$ (since it’s the product of $9$ and $3$). However, the LCM of $9$ and $3$ is just $9$, and if we use $9$ as our common denominator we will only have to change the second fraction. Doing this, we get

\begin{aligned}\dfrac{23}{9}+\dfrac{1}{3} &= \dfrac{23}{9}+\dfrac{1\times 3}{3\times 3} \\ \\ &=\dfrac{23}{9} + \dfrac{3}{9} = \dfrac{26}{9} \end{aligned}

This fraction is already in its simplest form, so we’re done. However, if we had chosen $27$ as our common denominator, we would’ve had to simplify the fraction at the end.

Level 1-3GCSEKS3AQAEdexcelOCRWJEC

## Adding and Subtracting Fractions Example Questions

Question 1: Evaluate $\dfrac{1}{8}+\dfrac{5}{12}$

[2 marks]

Level 1-3GCSEKS3 AQAEdexcelOCRWJEC

To add fractions, they must first share a common denominator.

This can be achieved by first multiplying the top and bottom of the first fraction by $3$, and then multiplying the top and bottom of the second fraction by $2$. Thus,

\begin{aligned}\dfrac{1}{8}+\dfrac{5}{12} &= \dfrac{1\times 3}{24}+\dfrac{5\times 2}{24} \\ \\ &=\dfrac{3}{24} + \dfrac{10}{24} =\dfrac{13}{24} \end{aligned}

Gold Standard Education

Question 2: Evaluate $\dfrac{9}{10} + 5$

[2 marks]

Level 1-3GCSEKS3 AQAEdexcelOCRWJEC

Writing $5$ as $\dfrac{5}{1}$, the calculation becomes,

$\dfrac{9}{10}+\dfrac{5}{1}$

To add fractions, they must first share a common denominator. This can be achieved by first multiplying the top and bottom of the second fraction by $10$. Thus,

\begin{aligned}\dfrac{9}{10}+\dfrac{5}{1} &= \dfrac{9}{10}+\dfrac{5\times 10}{1\times10} \\ \\ &=\dfrac{9}{10} + \dfrac{50}{10} =\dfrac{59}{10} \end{aligned}

Gold Standard Education

Question 3: Evaluate $\dfrac{4}{5} + \dfrac{5}{3}$

[2 marks]

Level 1-3GCSEKS3 AQAEdexcelOCRWJEC

To add fractions, they must first share a common denominator.

This can be achieved by first multiplying the top and bottom of the first fraction by $3$, and then multiplying the top and bottom of the second fraction by $5$. Thus,

\begin{aligned}\dfrac{4}{5}+\dfrac{5}{3} &= \dfrac{4\times 3}{15}+\dfrac{5\times 5}{15} \\ \\ &=\dfrac{12}{15} + \dfrac{25}{15} = \dfrac{37}{15} \end{aligned}

Gold Standard Education

Question 4: Evaluate $4\dfrac{1}{2} + \dfrac{4}{3}$

[3 marks]

Level 1-3GCSEKS3 AQAEdexcelOCRWJEC

Firstly we have to convert the mixed fraction to an improper fraction,

$4\dfrac{1}{2} + \dfrac{4}{3} = \dfrac{9}{2} + \dfrac{4}{3}$

To add fractions, they must first share a common denominator. This can be achieved by first multiplying the top and bottom of the first fraction by $3$, and then multiplying the top and bottom of the second fraction by $2$. Thus,

\begin{aligned}\dfrac{9}{2} + \dfrac{4}{3} &= \dfrac{9\times 3}{2\times3}+\dfrac{4\times 2}{3\times2} \\ \\ &=\dfrac{27}{6} + \dfrac{8}{6} = \dfrac{35}{6} \end{aligned}

Gold Standard Education

Question 5: Evaluate $5\dfrac{2}{3} + 2\dfrac{3}{4}$

[3 marks]

Level 1-3GCSEKS3 AQAEdexcelOCRWJEC

Firstly we have to convert the mixed fractions to an improper fraction,

$5\dfrac{2}{3} + 2\dfrac{3}{4} = \dfrac{17}{3} + \dfrac{11}{4}$

To add fractions, they must first share a common denominator. This can be achieved by first multiplying the top and bottom of the first fraction by $4$, and then multiplying the top and bottom of the second fraction by $3$. Thus,

\begin{aligned}\dfrac{17}{3} + \dfrac{11}{4} &= \dfrac{17\times 4}{3\times4}+\dfrac{11\times 3}{4\times3} \\ \\ &=\dfrac{68}{12} + \dfrac{33}{12} = \dfrac{101}{12 }\end{aligned}

Gold Standard Education

## Adding and Subtracting Fractions Worksheet and Example Questions

### Adding and subtracting fractions - Exam Questions - MME

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