Fractions are one of the most fundamental topics in GCSE Maths and appear across every exam board — AQA, Edexcel, and OCR. Whether you are sitting Foundation or Higher tier, you need to be confident with adding, subtracting, multiplying, and dividing fractions, as well as converting between improper fractions and mixed numbers. This page walks you through the key rules, gives you clear worked examples at both tiers, highlights the mistakes examiners see most often, and provides practice questions so you can test yourself. If you want a broader picture of what to revise, take a look at our complete GCSE Maths topics list.
What Is a Fraction?
A fraction represents a part of a whole. It is written as one number over another, separated by a line. The top number is the numerator (how many parts you have) and the bottom number is the denominator (how many equal parts the whole is divided into).
There are three types you need to know:
- Proper fraction — the numerator is smaller than the denominator, e.g. 3/4.
- Improper fraction — the numerator is equal to or larger than the denominator, e.g. 7/3.
- Mixed number — a whole number combined with a proper fraction, e.g. 2 1/3.
Key Formulas
Equivalent Fractions and Simplifying
Two fractions are equivalent if they represent the same value. You create equivalent fractions by multiplying or dividing both numerator and denominator by the same number. To simplify (or cancel down) a fraction, divide top and bottom by their highest common factor (HCF). For example, 8/12 simplifies to 2/3 because the HCF of 8 and 12 is 4.
Step-by-Step Method
Adding and Subtracting Fractions
- Look at the denominators. If they are the same, simply add or subtract the numerators and keep the denominator.
- If the denominators are different, find the lowest common denominator (LCD). The LCD is the lowest common multiple (LCM) of both denominators.
- Convert each fraction so that both have the LCD as their denominator.
- Add or subtract the numerators. Keep the denominator the same.
- Simplify the result if possible.
Multiplying Fractions
- Multiply the numerators together to get the new numerator.
- Multiply the denominators together to get the new denominator.
- Simplify the answer. You can also cross-cancel before multiplying to make the arithmetic easier.
Dividing Fractions
- Keep the first fraction as it is.
- Change the division sign to multiplication.
- Flip the second fraction (take its reciprocal).
- Multiply using the method above.
- Simplify the result.
Working with Mixed Numbers
If a question involves mixed numbers, always convert them to improper fractions first, carry out the operation, then convert back to a mixed number if required.
Worked Example 1 — Foundation Level
Question: Work out 2/5 + 3/4. Give your answer as a fraction in its simplest form.
Working:
Step 1 — Find the LCD of 5 and 4. The LCM of 5 and 4 is 20.
Step 2 — Convert each fraction:
- 2/5 = (2 × 4) / (5 × 4) = 8/20
- 3/4 = (3 × 5) / (4 × 5) = 15/20
Step 3 — Add the numerators: 8/20 + 15/20 = 23/20
Step 4 — Convert to a mixed number: 23/20 = 1 3/20
The fraction 3/20 cannot be simplified further.
Answer: 1 3/20
Worked Example 2 — Higher Level
Question: Work out 2 3/5 ÷ 1 1/4. Give your answer as a mixed number in its simplest form.
Working:
Step 1 — Convert to improper fractions:
- 2 3/5 = (2 × 5 + 3) / 5 = 13/5
- 1 1/4 = (1 × 4 + 1) / 4 = 5/4
Step 2 — Change division to multiplication and flip the second fraction: 13/5 ÷ 5/4 = 13/5 × 4/5
Step 3 — Multiply: (13 × 4) / (5 × 5) = 52/25
Step 4 — Convert to a mixed number: 52 ÷ 25 = 2 remainder 2, so 52/25 = 2 2/25
Answer: 2 2/25
Common Mistakes
- Forgetting to find a common denominator when adding or subtracting. You cannot simply add numerators and denominators — 1/3 + 1/4 is NOT 2/7. Always find the LCD first.
- Not converting mixed numbers to improper fractions before multiplying or dividing. Working directly with mixed numbers leads to errors. Convert first, then operate.
- Forgetting to simplify the final answer. Examiners expect answers in their simplest form unless stated otherwise. Always check if the numerator and denominator share a common factor.
- Flipping the wrong fraction when dividing. Remember: Keep, Change, Flip — keep the first fraction, change ÷ to ×, flip the second fraction only.
- Losing negative signs. When one or both fractions are negative, track the sign carefully through each step.
Exam Tips
- Show every step. Even if you can do the calculation mentally, writing each step earns method marks if your final answer is wrong.
- Simplify at the end or cross-cancel early. Both approaches are valid, but cross-cancelling before you multiply keeps numbers small and reduces arithmetic errors.
- Check whether the question asks for a fraction, mixed number, or decimal. Giving the answer in the wrong form can cost you the final accuracy mark.
- Use fractions on your calculator wisely. On calculator papers, the fraction button can verify your answer, but on non-calculator papers you must show full working. See our formulas guide for more on what to memorise.
Practice Questions
Q1 (Foundation): Work out 5/6 − 1/3. Give your answer in its simplest form.
Q2 (Foundation): Calculate 3/7 × 2/5.
Q3 (Higher): Work out 3 1/2 ÷ 2 1/3. Give your answer as a mixed number in its simplest form.
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Summary
- A fraction has a numerator (top) and a denominator (bottom).
- To add or subtract fractions, you must use a common denominator.
- To multiply fractions, multiply numerators together and denominators together.
- To divide fractions, keep the first, flip the second, and multiply.
- Always convert mixed numbers to improper fractions before calculating.
- Simplify your final answer by dividing by the HCF.
- Show clear working in exams to secure method marks even if you slip up on the arithmetic.