# Adding and Subtracting Fractions

## Adding and Subtracting Fractions Revision

**Adding and Subtracting Fractions**

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:

- Use the lowest common multiple (LCM) of the two denominators.
- Use the product of the two denominators.

**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}

**Example 1: Adding Fractions**

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}

**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.

**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}

**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.

## Adding and Subtracting Fractions Example Questions

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

**[2 marks]**

Give your answer in its simplest form.

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}

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

**[2 marks]**

Give your answer in its simplest form.

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}

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

**[2 marks]**

Give your answer in its simplest form.

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}

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

**[3 marks]**

Give your answer in its simplest form.

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}

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

**[3 marks]**

Give your answer in its simplest form.

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}