GCSE Maths Practice: fractions

Question 11 of 11

This question checks your ability to simplify large fractions efficiently using common factors or prime factorisation. Recognising hidden patterns in numbers helps you work faster and avoid mistakes.

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\( \begin{array}{l}\text{Simplify }\frac{84}{210}\text{ to its lowest terms.}\end{array} \)

Choose one option:

Check divisibility by small primes like 2, 3, 5, or 7 first. Once simplified, verify that the numerator and denominator share no further common factors.

When simplifying large fractions, the aim is to express them in lowest terms — that is, where the numerator and denominator share no common factors other than 1. The most efficient way to do this is to find the highest common factor (HCF) or cancel shared prime factors. Understanding factor relationships allows you to simplify without trial and error, saving time during exams.

Worked Examples

Example 1:
Simplify \( \tfrac{126}{315} \)

  1. Prime factors: 126 = 2 × 3² × 7, 315 = 3² × 5 × 7.
  2. Cancel 3² and 7 (common factors).
  3. Remaining factors: numerator = 2, denominator = 5.
  4. Answer: \( \tfrac{2}{5} \)

Example 2:
Simplify \( \tfrac{96}{144} \)

  1. Both divisible by 48 (the HCF).
  2. Divide: 96 ÷ 48 = 2, 144 ÷ 48 = 3.
  3. Answer: \( \tfrac{2}{3} \)

Example 3:
Simplify \( \tfrac{350}{490} \)

  1. Both divisible by 70.
  2. 350 ÷ 70 = 5, 490 ÷ 70 = 7.
  3. Answer: \( \tfrac{5}{7} \)

Example 4:
Simplify \( \tfrac{168}{252} \)

  1. Find HCF (84).
  2. Divide: 168 ÷ 84 = 2, 252 ÷ 84 = 3.
  3. Answer: \( \tfrac{2}{3} \)

Common Mistakes

  • Dividing by small numbers repeatedly instead of finding the true HCF.
  • Forgetting to simplify fully — always check that no further factors can be cancelled.
  • Incorrectly dividing one part of the fraction only (e.g. top but not bottom).
  • Confusing HCF and LCM — remember, HCF is for simplification.

Exam Tip

GCSE examiners like to test efficiency. Spot obvious factors quickly: if both numbers are even, divide by 2; if both end in 0 or 5, divide by 5; if digit sums are multiples of 3, divide by 3. Continue until the numbers are coprime (no common factors).

Try These Yourself

  • \( \tfrac{180}{420} \)
  • \( \tfrac{135}{225} \)
  • \( \tfrac{294}{490} \)
  • \( \tfrac{288}{432} \)

Apply the same process: look for an HCF, divide both parts, and check whether the fraction is fully simplified. Mastering this will make algebraic fraction simplification much easier later on.