Simplify: \(\frac{6}{12}\)
Divide top and bottom by the same number.
6 ÷ 6 = 1 and 12 ÷ 6 = 2, so \(\frac{1}{2}\).
Fractions
Fractions represent parts of a whole and are used in calculations involving division, ratio and proportion. They are closely linked to decimals and percentages, and GCSE Maths requires you to work confidently with all operations involving fractions.
Overview
A fraction shows part of a whole, part of a set, or a division.
In the fraction \( \frac{3}{4} \), the top number shows how many parts you have, and the bottom number shows how many equal parts the whole is split into.
\( \frac{3}{4} \)
Fractions appear across GCSE Maths, including algebra, ratio, probability and geometry, so it is important to be secure with the basics.
What you should understand after this topic
- Understand what numerator and denominator mean
- Simplify fractions
- Find equivalent fractions
- Convert between mixed and improper fractions
- Compare, add, subtract, multiply and divide fractions
- Find fractions of amounts
Key Definitions
Numerator
The top number of a fraction. It shows how many parts you have.
Denominator
The bottom number of a fraction. It shows how many equal parts the whole is split into.
Equivalent Fractions
Fractions that have the same value, such as \( \frac{1}{2} = \frac{2}{4} \).
Simplest Form
A fraction written using the smallest whole numbers possible.
Improper Fraction
A fraction where the numerator is greater than or equal to the denominator.
Mixed Number
A whole number and a fraction written together, such as \( 2\frac{1}{3} \).
Key Rules
Simplifying
Divide the numerator and denominator by the same number.
Equivalent fractions
Multiply or divide the numerator and denominator by the same number.
Add and subtract
Use a common denominator first.
Multiply
Multiply top by top and bottom by bottom.
Divide
Keep, flip, multiply.
Fraction of an amount
Divide by the denominator, then multiply by the numerator.
How to Solve
Understanding fractions
A fraction shows part of a whole. The denominator tells you how many equal parts the whole is split into, and the numerator tells you how many parts are used.
\( \frac{5}{8} \)
5 is the numerator.
8 is the denominator.
This means 5 parts out of 8 equal parts.
Equivalent fractions
Equivalent fractions have the same value even though they look different.
\( \frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} \)
Multiply or divide the numerator and denominator by the same number.
Exam tip: Whatever you do to the top, you must also do to the bottom.
Simplifying fractions
To simplify a fraction, divide the numerator and denominator by their highest common factor.
\( \frac{18}{24} = \frac{3}{4} \)
The highest common factor of 18 and 24 is 6.
Divide both parts by 6.
This uses ideas from factors and multiples.
Mixed numbers and improper fractions
A mixed number has a whole number and a fraction. An improper fraction has a numerator larger than or equal to the denominator.
\( 2\frac{3}{5} = \frac{13}{5} \)
\( \frac{17}{4} = 4\frac{1}{4} \)
To convert \(2\frac{3}{5}\), calculate \(2 \times 5 + 3 = 13\), then keep the denominator 5.
To convert \(\frac{17}{4}\), divide 17 by 4 to get 4 remainder 1.
Comparing and ordering fractions
To compare fractions, use a common denominator or convert them to decimals.
\( \frac{2}{3} \text{ and } \frac{3}{4} \)
\( \frac{2}{3} = \frac{8}{12}, \quad \frac{3}{4} = \frac{9}{12} \)
Since \(\frac{9}{12}\) is greater than \(\frac{8}{12}\), \(\frac{3}{4}\) is greater.
Comparing fractions is closely linked to decimals and equivalent fractions.
Adding and subtracting fractions
To add or subtract fractions, first make the denominators the same.
\( \frac{1}{3} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2} \)
\( \frac{5}{8} - \frac{1}{4} = \frac{5}{8} - \frac{2}{8} = \frac{3}{8} \)
Exam tip: Only add or subtract the numerators once the denominators match.
Multiplying fractions
To multiply fractions, multiply the numerators and multiply the denominators.
\( \frac{2}{5} \times \frac{3}{4} = \frac{6}{20} = \frac{3}{10} \)
You can simplify before multiplying if it makes the calculation easier.
Dividing fractions
To divide by a fraction, multiply by its reciprocal.
\( \frac{3}{7} \div \frac{2}{5} = \frac{3}{7} \times \frac{5}{2} = \frac{15}{14} = 1\frac{1}{14} \)
Exam tip: Keep the first fraction, change division to multiplication, then flip the second fraction.
Finding a fraction of an amount
To find a fraction of an amount, divide by the denominator first, then multiply by the numerator.
\( \frac{3}{5} \text{ of } 40 = 24 \)
Divide by the denominator: \(40 \div 5 = 8\).
Multiply by the numerator: \(8 \times 3 = 24\).
This method is also useful when working with percentages.
Example Questions
Edexcel
Exam-style questions inspired by Edexcel GCSE Mathematics.
Edexcel
Write \( \frac{3}{4} \) as a decimal.
Edexcel
Write \( 1\frac{2}{5} \) as an improper fraction.
Edexcel
Work out \( \frac{2}{3} + \frac{1}{6} \).
Edexcel
Work out \( \frac{5}{8} - \frac{1}{4} \).
Edexcel
Work out \( \frac{3}{5} \times \frac{2}{9} \).
AQA
Exam-style questions based on the AQA GCSE Mathematics specification, focusing on confident fraction skills, clear working, and accurate simplification.
AQA
Simplify \( \frac{18}{24} \).
AQA
Write \( \frac{7}{10} \) as a percentage.
AQA
Work out \( \frac{3}{4} \div \frac{1}{2} \).
AQA
A student says that
\( \frac{1}{3} + \frac{1}{4} = \frac{2}{7} \).
Is the student correct?
Tick one box. Yes ☐ No ☐
Give a reason for your answer.
AQA
Find \( \frac{3}{8} \) of \( 64 \).
OCR
Exam-style questions aligned with OCR GCSE Mathematics, emphasising fraction operations, reasoning, and accurate mathematical communication.
OCR
Arrange these fractions in order of size, starting with the smallest.
\( \frac{1}{2},\; \frac{3}{8},\; \frac{5}{6},\; \frac{2}{3} \)
OCR
Work out \( 2\frac{1}{3} + 1\frac{5}{6} \).
OCR
Work out \( \frac{4}{7} \times \frac{21}{8} \).
OCR
Write \( 0.375 \) as a fraction in its simplest form.
OCR
Explain why \( \frac{2}{5} \) is greater than \( \frac{1}{3} \).
Exam Checklist
Step 1
Check what type of fraction question it is.
Step 2
For addition or subtraction, find a common denominator first.
Step 3
For division, flip the second fraction and multiply.
Step 4
Simplify the final answer if possible.
Most common exam mistakes
Addition
Adding top and bottom numbers directly.
Division
Forgetting to flip the second fraction.
Simplifying
Leaving answers unsimplified.
Mixed numbers
Not converting them before doing operations.
Common Mistakes
These are common mistakes students make when working with fractions in GCSE Maths.
Adding denominators
Incorrect
A student adds both the numerators and denominators, for example \( \frac{1}{2} + \frac{1}{3} = \frac{2}{5} \).
Correct
Only numerators are added. You must first find a common denominator, then add the numerators while keeping the denominator the same.
Not using a common denominator
Incorrect
A student tries to add or subtract fractions directly without matching denominators.
Correct
Fractions must have the same denominator before they can be added or subtracted. Find the lowest common denominator first.
Not simplifying the final answer
Incorrect
A student leaves the fraction in a non-simplified form.
Correct
Always simplify your final answer by dividing the numerator and denominator by their highest common factor.
Confusing multiplication and division
Incorrect
A student uses the wrong method when multiplying or dividing fractions.
Correct
To multiply fractions, multiply numerators and denominators. To divide, multiply by the reciprocal of the second fraction.
Converting mixed numbers incorrectly
Incorrect
A student makes errors when converting between mixed numbers and improper fractions.
Correct
To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place over the same denominator.
Try It Yourself
Practise simplifying and calculating with fractions.
Foundation
Foundation Practice
Simplify and perform basic operations with fractions.
Simplify: \(\frac{8}{20}\)
Divide by the highest common factor.
Divide both by 4 → \(\frac{2}{5}\).
Find: \(\frac{1}{4} + \frac{1}{4}\)
Same denominator → add numerators.
\(1 + 1 = 2\), so \(\frac{2}{4} = \frac{1}{2}\).
Find: \(\frac{3}{5} + \frac{1}{5}\)
Denominators are the same.
Add numerators → \(\frac{4}{5}\).
Find: \(\frac{3}{4} - \frac{1}{4}\)
Subtract numerators.
\(\frac{2}{4} = \frac{1}{2}\).
Find: \(\frac{5}{6} - \frac{2}{6}\)
Same denominator.
\(\frac{3}{6} = \frac{1}{2}\).
Find: \(\frac{2}{3} \times \frac{3}{4}\)
Multiply numerators and denominators.
\(\frac{6}{12} = \frac{1}{2}\).
Find: \(\frac{1}{2} \times \frac{4}{5}\)
Multiply across.
\(\frac{4}{10} = \frac{2}{5}\).
Find: \(\frac{3}{4} \div \frac{1}{2}\)
Flip and multiply.
\(\frac{3}{4} × \frac{2}{1} = \frac{6}{4} = \frac{3}{2}\).
Find: \(\frac{2}{3} \div \frac{1}{3}\)
Flip second fraction.
\(\frac{2}{3} × 3 = 2\).
Higher
Higher Practice
Work with mixed numbers, improper fractions and complex calculations.
Convert \(1 \frac{3}{4}\) to an improper fraction.
Multiply whole number by denominator.
\(1 × 4 + 3 = 7\), so \(\frac{7}{4}\).
Convert \(2 \frac{1}{3}\) to an improper fraction.
Multiply and add.
\(2 × 3 + 1 = 7\), so \(\frac{7}{3}\).
Find: \(\frac{1}{2} + \frac{1}{3}\)
Find common denominator.
LCM is 6 → \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).
Find: \(\frac{3}{4} + \frac{2}{5}\)
Find common denominator (20).
\(\frac{15}{20} + \frac{8}{20} = \frac{23}{20}\).
Find: \(\frac{5}{6} - \frac{1}{4}\)
Use denominator 12.
\(\frac{10}{12} - \frac{3}{12} = \frac{7}{12}\).
Find: \(\frac{7}{8} - \frac{1}{2}\)
Convert \(\frac{1}{2}\) to eighths.
\(\frac{7}{8} - \frac{4}{8} = \frac{3}{8}\).
Find: \(1 \frac{1}{2} \times \frac{2}{3}\)
Convert to improper fraction first.
\(\frac{3}{2} × \frac{2}{3} = 1\).
Find: \(2 \frac{1}{4} \times \frac{4}{5}\)
Convert first.
\(\frac{9}{4} × \frac{4}{5} = \frac{9}{5}\).
A student adds \(\frac{1}{2} + \frac{1}{3}\) and gets \(\frac{2}{5}\). What is wrong?
Denominators must match first.
You must find a common denominator, not add directly.
Find: \(\frac{3}{5} ÷ \frac{6}{7}\)
Flip and multiply.
\(\frac{3}{5} × \frac{7}{6} = \frac{21}{30} = \frac{7}{10}\).
Games
Practise this topic with interactive games.
Games coming soon.
Fractions Video Tutorial
Frequently Asked Questions
When do I need a common denominator?
When adding or subtracting fractions.
How do I multiply fractions?
Multiply the numerators together and the denominators together.
Why should I simplify fractions?
Answers are usually required in their simplest form in exams.