Fractions — GCSE Maths

Why do fractions matter at GCSE?

• Fractions appear throughout GCSE Maths — in algebra, ratio, probability and more.
• You must be able to: simplify fractions, add and subtract fractions, and multiply and divide fractions.
• Working with fractions without a calculator is a key skill.
• A fraction has a numerator (top) and a denominator (bottom).

Key Words

• Numerator — The top number in a fraction
• Denominator — The bottom number in a fraction
• Equivalent fraction — A fraction with the same value but different numbers (e.g. ½ = 2/4)
• Reciprocal — The flipped version of a fraction (e.g. reciprocal of ¾ is 4/3)

Adding and subtracting fractions

• To add or subtract fractions, they must have the same denominator.
• Example: 1/4 + 2/4 = 3/4 (denominators already match).
• Example: 1/3 + 1/4. Find common denominator: 12. → 4/12 + 3/12 = 7/12.
• To find a common denominator, find the lowest common multiple (LCM) of both denominators.
• Always simplify your answer if possible.

Multiplying fractions

• To multiply fractions: multiply the numerators, then multiply the denominators.
• Example: 2/3 × 3/5 = (2×3)/(3×5) = 6/15 = 2/5 (simplified).
• You can cross-cancel before multiplying to keep numbers smaller.
• Mixed numbers: convert to improper fractions first.
• 2½ × 1⅓ → 5/2 × 4/3 = 20/6 = 10/3 = 3⅓

Dividing fractions

• To divide fractions: Keep, Change, Flip (KCF).
• Keep the first fraction, Change ÷ to ×, Flip the second fraction.
• Example: 3/4 ÷ 2/5 → 3/4 × 5/2 = 15/8 = 1⅞.
• Always convert mixed numbers to improper fractions first.
• Simplify your final answer.

Worked Examples

Practice Questions

• 1/2 + 2/3 = ?
  Answer: 7/6
  Common denominator is 6. 3/6 + 4/6 = 7/6.
• 3/4 × 8/9 = ?
  Answer: 2/3
  Cross-cancel: 1/1 × 2/3 = 2/3.
• 5/6 − 1/4 = ?
  Answer: 7/12
  LCM = 12. 10/12 − 3/12 = 7/12.
• 2/3 ÷ 4/5 = ?
  Answer: 5/6
  KCF: 2/3 × 5/4 = 10/12 = 5/6.
• Simplify 18/24.
  Answer: 3/4
  HCF of 18 and 24 is 6. 18÷6 = 3, 24÷6 = 4. Answer: 3/4.