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 Cognito AQA GCSE Maths

Before you start

Recall multiples and common factors; interpret the denominator as the size of an equal part.

Try this first. If you need the answer, revisit the prerequisite before continuing.

Starting check: Which is larger: $\frac12$ or $\frac13$ ? Explain without decimals.

Show answer and reasoning

One half: splitting the same whole into fewer equal pieces makes each piece larger.

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.

How to Add and Subtract Fractions (2026/27 exams)Curated by Kepler, not produced by us. Watch it, then read the notes below.

Open on YouTube

Watch, pause, try

Video by Cognito. This clip explains addition and subtraction. Use the written sections for multiplication, division and mixed numbers.

Viewing focus: Explain why addition needs equal-sized pieces, while multiplication does not.

After watching, attempt these questions before opening the answers. You can also use the written lesson below.

After watching 1: Work out $\frac34+\frac16$ . Explain your denominator.

Show answer and reasoning

Twelfths are equal-sized pieces: $\frac9{12}+\frac2{12}=\frac{11}{12}$ .

After watching 2: Without calculating exactly, explain why $\frac45+\frac13$ must exceed 1.

Show answer and reasoning

The gap from $\frac45$ to 1 is $\frac15$ , and $\frac13>\frac15$ .

KEPLER VISUAL LAB

Explore the idea

Predict what will change, then use the controls to test your explanation.

The shaded amounts stay the same when you repartition them. Select “Make twelfths” to see why the denominator must match.

On a narrow screen, scroll the diagram sideways to read every label.

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: $\frac{2 \times 3 + 1}{3} = \frac{7}{3}$ .

Example 1Simplify

\frac{45}{60}

fully, and write

\frac{23}{6}

as a mixed number.

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.

Show worked solution

Solution

1The HCF of 42 and 56 is 14.
2 $\frac{42 \div 14}{56 \div 14} = \frac{3}{4}$ .

Answer: $\frac{3}{4}$

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}

Example 1Work out

\frac{5}{6} - \frac{3}{8}

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.

Show worked solution

Solution

1Common denominator 20: $\frac{12}{20} + \frac{5}{20}$ .
2 $= \frac{17}{20}$ .

Answer: $\frac{17}{20}$

Example 2Work out

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

1Improper:

\frac{11}{4} + \frac{5}{3}

2Common denominator 12:

\frac{33}{12} + \frac{20}{12} = \frac{53}{12}

\frac{53}{12} = 4\frac{5}{12}

Answer

4\frac{5}{12}

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.

Show worked solution

Solution

1Improper: $\frac{7}{2} - \frac{9}{5}$ .
2Common denominator 10: $\frac{35}{10} - \frac{18}{10} = \frac{17}{10}$ .
3 $= 1\frac{7}{10}$ .

Answer: $1\frac{7}{10}$

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.

Example 1Work out

\frac{4}{9} \times \frac{3}{10}

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}

Example 2Work out

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

1Improper:

\frac{12}{5} \div \frac{4}{3}

2Flip and multiply:

\frac{12}{5} \times \frac{3}{4} = \frac{36}{20}

3Simplify:

\frac{9}{5} = 1\frac{4}{5}

Answer

1\frac{4}{5}

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.

Show hint

Keep, change, flip: $\frac{5}{6} \times \frac{9}{10}$ .

Show worked solution

Solution

1 $\frac{5}{6} \times \frac{9}{10} = \frac{45}{60}$ .
2Simplify: $\frac{3}{4}$ .

Answer: $\frac{3}{4}$

Example 3Work out

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

1Improper:

\frac{3}{2} \times \frac{8}{3}

= \frac{24}{6} = 4

Answer

4

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

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

Your answer

Show worked solution

Solution

1 $\frac{9}{4} \times \frac{4}{3} = \frac{36}{12}$ .
2 $= 3$ .

Answer: $3$

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

Example 1A school has 840 students.

\frac{3}{8}

walk to school. How many walk?

\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

Show worked solution

Solution

1 $168 \div 7 = 24$ .
2 $5 \times 24 = 120$ .

Answer: £120

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{ad + bc}{bd}

Common denominator (use the LCM when you can).

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

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.

Reveal solution

SolutionAfter the coat,

\frac{3}{4}

is left. Shoes cost

\frac{2}{5} \times \frac{3}{4} = \frac{3}{10}

of the original. So she has spent

\frac{1}{4} + \frac{3}{10} = \frac{11}{20}

and has

\frac{9}{20}

left.

\frac{9}{20}

of the total is £54, so

\frac{1}{20}

is £6 and she started with £120.

\blacksquare

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]
Show mark scheme (a)
- B1 $18$
Answer: $18$
(b)Work out $\frac{3}{4} - \frac{2}{3}$ .[2]
Show mark scheme (b)
- M1Correct common denominator, e.g. $\frac{9}{12} - \frac{8}{12}$
- A1 $\frac{1}{12}$
Answer: $\frac{1}{12}$

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]
Show mark scheme
- M1Converts to improper fractions: $\frac{10}{3}$ and $\frac{8}{5}$
- M1Inverts and multiplies: $\frac{10}{3} \times \frac{5}{8}$
- A1 $2\frac{1}{12}$
Answer: $2\frac{1}{12}$

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]
Show mark scheme
- M1Finds the fraction used: $\frac{2}{3} - \frac{3}{8} = \frac{7}{24}$
- M1 $\frac{7}{24}$ of the tank $= 35$ litres, so $35 \div 7 \times 24$
- A1 $120$ litres
Answer: $120$ litres

Unlimited practice

Skill drill

A new question every time, marked instantly, with a full worked solution. Aim for a streak of five, then move up a level.

0 / 0 first-time correct Streak 0Best at this level: 0

Generating a question…

Complete the method

Some of the setup is done. Finish it on paper, then compare your reasoning.

Your turn: Complete the method: $\frac78-\frac13=\frac{\square}{24}-\frac{\square}{24}$ . Give the result in simplest form.

Show a hint

Multiply both parts of each fraction by the same number.

Show answer and reasoning

The numerators are 21 and 8, so the answer is $\frac{13}{24}$ . Neither 13 nor 24 shares a factor greater than 1.

Spot and repair the mistake

Identify the first incorrect step and explain how to fix it.

Diagnose it: A student writes $\frac23\div\frac45=\frac32\times\frac45=\frac65$ . Identify the first incorrect step and repair it.

Show answer and reasoning

They inverted the first fraction. Keep the first fraction and invert the divisor: $\frac23\times\frac54=\frac{10}{12}=\frac56$ . Since the divisor is below 1, the result should exceed $\frac23$ .

Apply it in an exam

Write a full response before revealing the solution. These are Kepler-authored exam-style questions; the suggested marks are a self-check guide.

Question 1 · 3 marks: A jug holds $1\frac12$ litres. You pour out $\frac25$ of its contents. How much remains?

Show answer and reasoning

You keep $\frac35$ of the original: $\frac35\times\frac32=\frac9{10}$ litre.

Identify the remaining fraction as three fifths.
Multiply by the original 1.5 litres.
Give 0.9 litre with a unit.

Explore further

AQA 8300 specification (opens in a new tab)
Check the assessed content, tier requirements and official course documents.
Corbettmaths: adding fractions practice (opens in a new tab)
Extra questions and answer links for adding fractions with different denominators.

Recall and check later

Return with the notes closed. Answer from memory, check the reasoning, then practise anything you missed.

Recall 1: Tomorrow: work out $\frac56\div\frac23$ without the notes.

Show answer and reasoning

$\frac56\times\frac32=\frac54=1\frac14$ .

Recall 2: In a week: explain when a common denominator is needed.

Show answer and reasoning

For adding or subtracting fractions, and often for comparing them. Multiplying and dividing do not require one.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

Simplify $\frac{18}{24}$ .
- $\frac{3}{4}$
- $\frac{9}{12}$
- $\frac{2}{3}$
- $\frac{6}{8}$
Show answer and explanation
Answer: $\frac{3}{4}$
Divide by the HCF, 6.
Work out $\frac{1}{3} + \frac{1}{6}$ .
- $\frac{1}{2}$
- $\frac{2}{9}$
- $\frac{2}{6}$
- $\frac{1}{9}$
Show answer and explanation
Answer: $\frac{1}{2}$
$\frac{2}{6} + \frac{1}{6} = \frac{3}{6}$ .
Find $\frac{3}{5}$ of 45.
- $27$
- $9$
- $75$
- $15$
Show answer and explanation
Answer: $27$
$45 \div 5 \times 3$ .
Work out $\frac{2}{3} \times \frac{9}{10}$ .
- $\frac{3}{5}$
- $\frac{11}{13}$
- $\frac{18}{13}$
- $\frac{20}{27}$
Show answer and explanation
Answer: $\frac{3}{5}$
$\frac{18}{30} = \frac{3}{5}$ .
Work out $\frac{3}{4} \div \frac{1}{8}$ .
- $6$
- $\frac{3}{32}$
- $\frac{1}{6}$
- $\frac{4}{12}$
Show answer and explanation
Answer: $6$
$\frac{3}{4} \times 8 = 6$ .
Write $2\frac{3}{8}$ as an improper fraction.
- $\frac{19}{8}$
- $\frac{13}{8}$
- $\frac{6}{8}$
- $\frac{23}{8}$
Show answer and explanation
Answer: $\frac{19}{8}$
$2 \times 8 + 3 = 19$ .
Work out $1\frac{2}{3} + 2\frac{3}{4}$ .
- $4\frac{5}{12}$
- $3\frac{5}{7}$
- $4\frac{5}{7}$
- $3\frac{17}{12}$
Show answer and explanation
Answer: $4\frac{5}{12}$
$\frac{20}{12} + \frac{33}{12} = \frac{53}{12}$ .
What fraction of 1 kg is 350 g?
- $\frac{7}{20}$
- $\frac{35}{10}$
- $\frac{1}{350}$
- $\frac{7}{2}$
Show answer and explanation
Answer: $\frac{7}{20}$
$\frac{350}{1000} = \frac{7}{20}$ .

Next step

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