Adding and multiplying fractions

Foundation and Higher

The idea

To add or subtract fractions, give them the same denominator first. To multiply, multiply tops and bottoms. To divide, multiply by the reciprocal.

Add: $\tfrac{2}{3} + \tfrac{1}{4} = \tfrac{8}{12} + \tfrac{3}{12} = \tfrac{11}{12}$ .
Multiply: $\tfrac{2}{3} \times \tfrac{3}{5} = \tfrac{6}{15} = \tfrac{2}{5}$ .
Divide: $\tfrac{2}{3} \div \tfrac{4}{5} = \tfrac{2}{3} \times \tfrac{5}{4} = \tfrac{5}{6}$ . Turn mixed numbers into improper fractions first.

Watch out Never add the denominators: $\tfrac{1}{2} + \tfrac{1}{3}$ is $\tfrac{5}{6}$ , not $\tfrac{2}{5}$ .

See it worked

Worked exampleNon-calculator4 marks

Answer each part without a calculator.

(a) Work out $\frac{3}{8}$ of $64$ .

(b) A circle has an area of $25\pi\ \text{cm}^2$ . A sector is $\frac{2}{5}$ of the circle. Work out the area of the sector. Give your answer in terms of $\pi$ .

(a) $\frac{1}{8}$ of $64$ is $8$ , so $\frac{3}{8}$ is $3 \times 8 = 24$ .
(b) $\frac{2}{5} \times 25\pi = \frac{50\pi}{5} = 10\pi$ .

Answer (a) $24$ ; (b) $10\pi\ \text{cm}^2$

Your turn

Question 1Non-calculator6 marks

Answer each part without a calculator. Give each answer as a fraction in its simplest form.

(a) Work out $\frac{2}{5} + \frac{1}{3}$

(b) Work out $\frac{3}{4} \times \frac{2}{9}$

Question 2Non-calculator5 marks

A charity has 72 badges. It sells one quarter in the morning, then two thirds of the remaining badges in the afternoon.

(a) How many badges remain unsold?

(b) What fraction of the original badges was sold altogether?

Question 3Non-calculator5 marks

A water tank is $\frac{3}{5}$ full. After $42$ litres are used, the tank is $\frac{1}{4}$ full.

(a) Work out the capacity of the tank, in litres.

(b) Next, $\frac{2}{3}$ of the water left in the tank is used. What fraction of the tank is now full?

In past papers

233 official questions in 155 papers since 2017. The latest:

All GCSE Maths past papers

GCSE Maths Pro

£119 once

At least 12 months of access

By paying you agree to our Terms and get access straight away, which ends your 14-day cancellation right.

Start free

No VAT is charged.