# Multiplying and dividing fractions

## Introduction

When multiplying or dividing fractions, you should always look to see if the fractions can be simplified first. This will reduce the need for large multiplication problems!

To multiply fractions, simply multiply the numerators and denominators together.
To divide fractions, multiply the first fraction by the **reciprocal** of the second.

Remember, the **numerator** is the top number in a fraction and the **denominator** is the bottom number.

*Unlike when adding or subtracting fractions, you do not need to convert to a common denominator first.*

## How to multiply fractions

### Example 1 - Multiplying fractions

Work out \(\frac{1}{4} × \frac{4}{5}\).

When multiplying fractions, multiply the numerators and denominators together. Remember to simplify your answer where possible.

\begin{aligned} \frac{1}{4} × \frac{4}{5} & = \frac{4}{20}\\ & = \frac{1}{5}\\ \end{aligned}

### Example 2 - Check if you can simplify before multiplying fraction

Calculate \(\frac{8}{40} × \frac{21}{28}\).

Always simplify the fraction first if possible. This reduces the need for long multiplication calculations.

\begin{aligned} \frac{8}{40} × \frac{21}{28} & = \frac{1}{5} × \frac{3}{4}\\ & = \frac{3}{20}\\ \end{aligned}

### Example 3 - Cross simplifying fractions before multiplying

Work out \(\frac{7}{16} × \frac{8}{35}\).

You can also simplify by cancelling with the other fraction.

\begin{aligned} \frac{7}{16} × \frac{8}{35} & = \frac{1}{2} × \frac{1}{5}\\ & = \frac{1}{10}\\ \end{aligned}

## How to divide fractions

### Example 4 - Dividing fractions

Calculate \(\frac{3}{5} ÷ \frac{2}{7}\).

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

When dividing fractions, flip the second fraction and then multiply as normal.
More correctly, we find the **reciprocal** of the second fraction.

\begin{aligned} \frac{3}{5} ÷ \frac{2}{7} & = \frac{3}{5} × \frac{7}{2}\\ & = \frac{21}{10}\\ & = 2\frac{1}{10}\\ \end{aligned}

### Example 5 - Cancel down first before dividing fractions

Work out \(\frac{7}{24} ÷ \frac{14}{36}\).

After flipping the second fraction, see if you can cancel down the fractions.

\begin{aligned} \frac{7}{24} ÷ \frac{14}{36} & = \frac{7}{24} × \frac{36}{14}\\ & = \frac{1}{2} × \frac{3}{2}\\ & = \frac{3}{4}\\ \end{aligned}

## Multiplying and dividing mixed numbers

### Example 6 - Multiplying mixed numbers

Calculate \(3\frac{1}{3} × 2\frac{2}{5}\).

When multiplying mixed numbers first convert them into **improper** fractions.
Then simplify and multiply as before.

\begin{aligned} 3\frac{1}{3} × 2\frac{2}{5} & = \frac{10}{3} × \frac{12}{5}\\ & = \frac{2}{1} × \frac{4}{1}\\ & = \frac{8}{1}\\ & = 8\\ \end{aligned}

### Example 7 - Dividing mixed numbers

Work out \(2\frac{2}{7} ÷ 4\frac{1}{2}\).

Again, first convert them into **improper** fractions.
Then divide as before.

\begin{aligned} 2\frac{2}{7} ÷ 4\frac{1}{2} & = \frac{16}{7} ÷ \frac{9}{2}\\ & = \frac{16}{7} × \frac{2}{9}\\ & = \frac{32}{63}\\ \end{aligned}

## Test yourself!

Now try these worksheets to practise your skills.