Dividing Fractions

What you will learn

• Explain what dividing by a fraction means.
• Find the reciprocal of a fraction.
• Divide fractions, whole numbers and mixed numbers.

Before you start: you should be able to multiply fractions. See Multiplying Fractions.

The idea

$6 \div 2$ asks “how many 2s fit into 6?” Dividing by a fraction asks the same thing. $3 \div \frac{1}{4}$ asks “how many quarters fit into 3?” Each whole has 4 quarters, so 3 wholes have 12.

Dividing by a fraction less than one gives a bigger answer, because the pieces are small and many fit.

Fractions in fractions

$\frac{3}{4} \div \frac{1}{8}$ asks how many eighths fit into three quarters. Three quarters is six eighths, so the answer is $6$.

The reciprocal rule

The reciprocal of a fraction is the fraction turned upside down. The reciprocal of $\frac{1}{8}$ is $\frac{8}{1} = 8$. Dividing by $\frac{1}{8}$ is the same as multiplying by $8$.

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

Keep the first fraction, change the division to a multiplication, and flip the second fraction.

Worked examples

Example 1: Use the rule

Work out $\frac{4}{5} \div \frac{2}{3}$.

1. Flip the second fraction: $\frac{3}{2}$.
2. Multiply: $\frac{4}{5} \times \frac{3}{2} = \frac{12}{10}$.
3. Simplify: $\frac{12}{10} = \frac{6}{5} = 1\frac{1}{5}$.

Answer: $1\frac{1}{5}$

Example 2: A whole number

Work out $6 \div \frac{3}{4}$.

1. Write $6$ as $\frac{6}{1}$ and flip $\frac{3}{4}$ to $\frac{4}{3}$.
2. $\frac{6}{1} \times \frac{4}{3} = \frac{24}{3} = 8$.
3. Check: $8$ pieces of $\frac{3}{4}$ make $6$.

Answer: $8$

Example 3: A mixed number

Work out $1\frac{1}{2} \div \frac{3}{4}$.

1. Write $1\frac{1}{2}$ as $\frac{3}{2}$.
2. $\frac{3}{2} \times \frac{4}{3} = \frac{12}{6} = 2$.

Answer: $2$

Watch out: only flip the second fraction, the one you divide by.

Why it matters

Cutting lengths of wood, pouring portions of a drink and working out how many times one amount fits into another all need division by a fraction. It is also the key to solving equations such as $\frac{x}{3} = 4$ and to understanding rates.