N2, N8, N12NumberFoundation & HigherGrades 1–5

Fractions

Fractions appear on every GCSE paper, and on the non-calculator paper you must handle them by hand: simplifying, adding with different denominators, multiplying, dividing and working with mixed numbers. Each operation has a short, dependable method — the trick is knowing which one to use.

45 min Video by Maths Genie AQA GCSE Maths

The big picture

A fraction is a division: $\frac{3}{4}$ means $3 \div 4$ . Equivalent fractions such as $\frac{3}{4} = \frac{9}{12}$ are the same number written differently, and that single idea powers adding (make the denominators match) and simplifying (divide top and bottom by the same number).

Multiplying and dividing are actually simpler than adding. You never need a common denominator to multiply — and dividing is just multiplying by the reciprocal.

Adding and Subtracting FractionsCurated by Kepler, not produced by us. Watch it, then read the notes below.

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What you'll learn

Simplify fractions and find equivalent fractions
Add and subtract fractions and mixed numbers
Multiply and divide fractions and mixed numbers
Find a fraction of an amount and express one quantity as a fraction of another

Equivalent fractions and simplifying

Multiplying or dividing the numerator and denominator by the same number gives an equivalent fraction.

To simplify fully, divide both by their highest common factor: $\frac{24}{36} = \frac{2}{3}$ (HCF 12). A mixed number such as $2 \frac{1}{3}$ converts to an improper fraction: $\frac{2 \times 3 + 1}{3} = \frac{7}{3}$ .

1HCF of 45 and 60 is 15:

\frac{45 \div 15}{60 \div 15} = \frac{3}{4}

23 \div 6 = 3

remainder 5, so

\frac{23}{6} = 3 \frac{5}{6}

Answer

\frac{3}{4}

and

3 \frac{5}{6}

Your turnSame method, new numbers — try it before moving on.

Simplify $\frac{42}{56}$ fully.

Your answer

Type a fraction like 5/8.

Adding and subtracting

Write both fractions with a common denominator — ideally the lowest common multiple of the denominators — then add or subtract the numerators. The denominator stays the same.

With mixed numbers, convert to improper fractions first. (You can also deal with whole numbers and fractions separately when adding.)

Cutting both bars into twelfths makes the pieces the same size, so they can be added:

\frac{2}{3} + \frac{1}{4} = \frac{11}{12}

1LCM of 6 and 8 is 24.

\frac{20}{24} - \frac{9}{24} = \frac{11}{24}

Answer

\frac{11}{24}

Your turnSame method, new numbers — try it before moving on.

Work out $\frac{3}{5} + \frac{1}{4}$ .

Your answer

Type a fraction like 5/8.

Your turnSame method, new numbers — try it before moving on.

Work out $3 \frac{1}{2} - 1 \frac{4}{5}$ . Give your answer as a mixed number.

Your answer

Type a mixed number like 1 3/10.

Tip — Never add denominators: $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$ . Check the size — the answer must be bigger than $\frac{1}{2}$ .

Multiplying and dividing

To multiply, multiply the numerators and multiply the denominators. Cancelling common factors first keeps the numbers small.

To divide, keep the first fraction, change $\div$ to $\times$ , and flip the second fraction (use its reciprocal). Always convert mixed numbers to improper fractions before multiplying or dividing.

1Cancel: 4 and 10 share 2; 3 and 9 share 3.

\frac{2}{3} \times \frac{1}{5}

= \frac{2}{15}

Answer

\frac{2}{15}

Your turnSame method, new numbers — try it before moving on.

Work out $\frac{5}{6} \div \frac{10}{9}$ .

Your answer

Type a fraction like 5/8.

Your turnSame method, new numbers — try it before moving on.

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

Your answer

Why "flip and multiply" works

Dividing by $\frac{1}{2}$ asks "how many halves fit in?" — and there are two halves in every whole, so $3 \div \frac{1}{2} = 6$ . That is exactly $3 \times 2$ : multiplying by the reciprocal.

Fractions of amounts

To find $\frac{3}{5}$ of an amount, divide by 5 (to find one fifth), then multiply by 3.

To write one quantity as a fraction of another, put the part over the whole — in the same units — and simplify: 40 minutes as a fraction of 2 hours is $\frac{40}{120} = \frac{1}{3}$ .

\frac{1}{8}

of 840 is

840 \div 8 = 105

\frac{3}{8}

3 \times 105

Answer

315

students

Your turnSame method, new numbers — try it before moving on.

Find $\frac{5}{7}$ of £168.

Your answer

Think like an examiner

"Give your answer in its simplest form" or "as a mixed number" is worth a mark on its own — do the final step.
Show the common-denominator fractions in your working ( $\frac{20}{24} - \frac{9}{24}$ ); this is the method mark.
In "show that" questions with fractions, every step must be written — including converting mixed numbers.

Methods

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

Common denominator (use the LCM when you can).

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

Multiply straight across.

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

Keep, change, flip.

Watch out for these

\frac{2}{3} + \frac{1}{4} = \frac{3}{7}

Use a common denominator:

\frac{8}{12} + \frac{3}{12} = \frac{11}{12}

2 \frac{1}{2} \times 3 \frac{1}{3} = 6 \frac{1}{6}

Convert first:

\frac{5}{2} \times \frac{10}{3} = \frac{25}{3} = 8 \frac{1}{3}

Flipping the first fraction when dividing.

Keep the first, flip the second.

Writing 40 minutes as a fraction of 2 hours as

\frac{40}{2} = 20

Use the same units:

\frac{40}{120} = \frac{1}{3}

Stretch yourself

Ella spends $\frac{1}{4}$ of her money on a coat and $\frac{2}{5}$ of what is left on shoes. She has £54 left. How much did she start with?

Hint — After the coat, $\frac{3}{4}$ remains. The shoes use $\frac{2}{5}$ of that remainder, not of the original.

Questions students ask

Do I always need the lowest common denominator?

No — any common denominator works, such as multiplying the two denominators. The LCM just keeps the numbers smaller and saves simplifying later.

Should I give answers as improper fractions or mixed numbers?

Either is usually accepted unless the question specifies. Always simplify fully.

Can I use a calculator?

On calculator papers, yes — the fraction button handles all of this. But Paper 1 is non-calculator, so the written methods must be secure.

Key takeaways

Simplify by dividing top and bottom by their HCF.
Add and subtract with a common denominator.
Multiply across; divide by multiplying by the reciprocal.
Convert mixed numbers to improper fractions before multiplying or dividing.
Fraction of an amount: divide by the denominator, multiply by the numerator.

Exam-style questions

Practise it the way it is examined

3 original questions · 9 marks, written to match the style and mark allocation of the real papers. Work on paper, then open the mark scheme and tick the marks you earned — method marks count even if the final answer slips.

Question 1Grade 23 marks

(a)Work out $\frac{2}{7}$ of 63.[1]
(b)Work out $\frac{3}{4} - \frac{2}{3}$ .[2]

Question 2Grade 43 marks

Work out $3 \frac{1}{3} \div 1 \frac{3}{5}$ . Give your answer as a mixed number in its simplest form.[3]

Question 3Grade 53 marks

A tank is $\frac{2}{3}$ full of water. After 35 litres are used, the tank is $\frac{3}{8}$ full.

Work out how many litres the tank holds when full.[3]

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Real past-paper questions on Fractions, marked mark-by-mark. How you do feeds straight into your weak-topic list, so your revision keeps targeting what actually needs work.

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Fractions

Equivalent fractions and simplifying

Adding and subtracting

Multiplying and dividing

Fractions of amounts

Practise it the way it is examined

Skill drill

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Related topics

Fractions

Equivalent fractions and simplifying

Adding and subtracting

Multiplying and dividing

Fractions of amounts

Practise it the way it is examined

Skill drill

Practise this topic

Related topics