Add, subtract, multiply and divide proper fractions and mixed numbers by hand, and leave every answer in lowest terms. It is on Foundation and Higher, aimed at grade 3. Free questions with answers are on this page. Worked examples show each line before you try the questions.
A fraction is a division written in a stack: 125 means 5÷12. Once you treat every fraction that way, the four operations stop being four separate tricks. Addition and subtraction need the pieces to be the same size, so you rewrite both fractions with a common denominator. Multiplication does not: you multiply the tops, multiply the bottoms, and cancel anything that appears on both. Division is multiplication by the reciprocal — flip the fraction you are dividing by, then multiply.
Mixed numbers such as 352 are a whole number sitting next to a proper fraction. For multiplying and dividing you must glue them into one improper fraction first (352=517). For adding and subtracting you can either convert both to improper fractions, or add the whole parts and the fraction parts separately and then tidy up. Both routes are fine; pick one and stick to it on the page.
Non-calculator papers love this topic because every slip is visible. Write the rewritten fractions on their own line, cancel before you multiply, and finish as a mixed number if the question started with mixed numbers. 'Of' on this page is the same as multiply: 52 of 43 is 52×43.
Objectives
The denominators must match before you add the numerators.
Multiply the tops and multiply the bottoms, then simplify.
Dividing by a fraction is multiplying by its reciprocal.
Key points
Build an equivalent fraction by multiplying (or dividing) the numerator and the denominator by the same number.
Add or subtract by rewriting both fractions with the lowest common denominator, then combining the numerators only.
Multiply by multiplying numerators and multiplying denominators — cancel any common factor first.
Divide by keeping the first fraction, flipping the second, and changing ÷ to ×.
Worked examples
Example 1
Work out 1/2 + 1/3. Give the fraction in its lowest terms.
2 marks
Common denominator 6: 1/2 = 3/6 and 1/3 = 2/6.
3/6 + 2/6 = 5/6.
The result is 5/6.
Answer: 5/6
Common slips
Slip: Common denominator 8: 1/2 = 3/6 and 1/3 = 2/6. That changes a number from the question, so it answers a different question.
Slip: 6/6 + 2/6 = 5/6. That changes a number in the working, so the step no longer follows from the one before.
Slip: The result is 9/6. That does not match the working above it.
Example 2
Work out 2/3 × 3/5. Give the fraction in its lowest terms.
2 marks
Tops and bottoms: 2 × 3 = 6 and 3 × 5 = 15.
That gives 6/15, which simplifies to 2/5.
The result is 2/5.
Answer: 2/5
Common slips
Slip: Tops and bottoms: 4 × 3 = 6 and 3 × 5 = 15. That changes a number from the question, so it answers a different question.
Slip: That gives 9/15, which simplifies to 2/5. That changes a number in the working, so the step no longer follows from the one before.
Slip: The result is 6/5. That does not match the working above it.
Example 3
Work out 1/2 ÷ 1/4. Give the fraction in its lowest terms.
2 marks
Dividing by 1/4 is the same as multiplying by its reciprocal, 4/1.
1/2 × 4/1 = 4/2, which simplifies to 2.
The result is 2.
Answer: 2
Common slips
Slip: Dividing by 3/4 is the same as multiplying by its reciprocal, 4/1. That changes a number from the question, so it answers a different question.
Slip: 4/2 × 4/1 = 4/2, which simplifies to 2. That changes a number in the working, so the step no longer follows from the one before.
Slip: The result is 6. That does not match the working above it.
Questions and answers
GCSE Maths fractions questions and answers. Try each one, then open the answer. Easy questions come first.
Question 1 Easy2 marks
Work out 2/5 + 1/5.
Show answer
Answer: 3/5
The denominators already match.
Add the numerators over 5.
M1 — The denominators already match.
A1 — Answer 3/5
Question 2 Easy2 marks
Work out 1/2 + 1/3.
Show answer
Answer: 5/6
The common denominator is 6.
Convert both fractions, then add.
M1 — The common denominator is 6.
A1 — Answer 5/6
Question 3 Medium2 marks
Work out 3/4 − 1/4.
Show answer
Answer: 1/2
Use a common denominator.
Subtract the numerators.
M1 — Use a common denominator.
A1 — Answer 1/2
Question 4 Medium2 marks
Work out 1 1/2 + 2 1/4.
Show answer
Answer: 15/4
Change the mixed numbers into improper fractions.
Add, using a common denominator.
M1 — Change the mixed numbers into improper fractions.
A1 — Answer 15/4
Question 5 Hard2 marks
Work out 2 1/3 − 1 1/2.
Show answer
Answer: 5/6
Change both mixed numbers into improper fractions.
Subtract using a common denominator.
M1 — Change both mixed numbers into improper fractions.
A1 — Answer 5/6
Question 6 Hard2 marks
A rope is 36 m. 2/9 of it is cut off. How many metres are left?
Show answer
Answer: 28
7/9 of the rope is left.
36 × 7 ÷ 9.
M1 — 7/9 of the rope is left.
A1 — Answer 28
Question 7 Exam-style2 marks
Work out 3 1/2 − 1 3/4.
Show answer
Answer: 7/4
Change both mixed numbers into improper fractions.
Subtract using a common denominator.
M1 — Change both mixed numbers into improper fractions.
A1 — Answer 7/4
Common mistakes
Most lost marks on this topic are one slip.
Adding the denominators.
Keep a common denominator and add the numerators.
Dividing by a fraction by dividing the tops only.
Multiply by the reciprocal.
Forgetting to simplify.
Divide the top and bottom by their highest common factor.
Spot the slip
What is 1/5 + 2/5?
3/5
3/10
2/5
Show the slip
3/5 The denominators already match, so add the tops.
Exam tips
Be ready to do this without a calculator. Paper 1 is the non-calculator paper.
If the question says “show that”, the result is given. The marks are for the steps that reach it.
The method is the same on AQA-style and Edexcel-style papers.
The method mark is usually on the rewritten fractions with a common denominator, or on the flipped fraction in a division. Write that line even if you can do it in your head.
If the question says 'show that', you must land on the printed fraction with no skipped cancelling.
Printable worksheet
The same questions are in a PDF you can print. Answers start on a later page.