GCSE Maths Practice: fractions

Question 9 of 11

This Higher GCSE question involves multiplying three fractions. Simplify strategically by cancelling common factors first — it reduces working time and keeps numbers manageable.

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\( \begin{array}{l}\text{Evaluate: }\frac{7}{8}\times\frac{5}{12}\times\frac{7}{5}.\\\text{Simplify your answer.}\end{array} \)

Choose one option:

Before multiplying three fractions, check for cancellations. Simplify across diagonals, then multiply remaining numerators and denominators to get the result in lowest terms.

At Higher GCSE, multi-step fraction multiplication tests your ability to identify cancellation patterns and simplify strategically. With three or more fractions, arithmetic alone is inefficient — cancelling common factors early makes the process faster and more accurate.

Method Summary

  1. List all numerators and denominators. Write them clearly to spot factors that appear top and bottom.
  2. Cross-cancel before multiplying. If a number appears once in a numerator and once in a denominator, divide both by their greatest common factor (GCF).
  3. Multiply remaining numerators and denominators. Do this only after simplification.
  4. Reduce the final result. Simplify again if possible, checking for common factors.

Worked Example 1

\( \tfrac{7}{8}\times\tfrac{5}{12}\times\tfrac{7}{5} \)

  1. Cancel the 5s (top and bottom).
  2. Multiply remaining: \( \tfrac{7}{8}\times\tfrac{7}{12}=\tfrac{49}{96}. \)
  3. Check simplification: 49 and 96 share no factors → \( \tfrac{49}{96} \).
  4. If numerators or denominators differ, adjust cancellation: for example, if the final 7 was 1, the answer simplifies to \( \tfrac{35}{96} \).

Worked Example 2 – Factor-rich expression

\( \tfrac{9}{10}\times\tfrac{15}{8}\times\tfrac{4}{27} \)

  1. Cancel 9 and 27 → divide both by 9 → \( 1 \text{ and } 3 \).
  2. Cancel 4 and 8 → divide both by 4 → \( 1 \text{ and } 2 \).
  3. Now \( \tfrac{1}{10}\times\tfrac{15}{2}\times\tfrac{1}{3}=\tfrac{15}{60}=\tfrac{1}{4}. \)

Worked Example 3 – Introducing negatives

\( -\tfrac{2}{3}\times\tfrac{5}{6}\times-\tfrac{3}{10} \)

  1. Two negatives make a positive.
  2. Cancel the 3s and 5s: result \( \tfrac{1}{6}\times\tfrac{1}{2}=\tfrac{1}{12}. \)

Common Mistakes

  • Skipping cross-cancellation: leads to large, error-prone numbers.
  • Forgetting to multiply all denominators: e.g., missing one term in triple products.
  • Not simplifying early: higher risk of mistakes and wasted time.
  • Incorrect sign handling: always check negatives if any are present.

Why This Matters

Three-fraction products appear in GCSE contexts such as compound scaling, ratio composition, and probability chains. Learning to cancel efficiently prepares you for algebraic fraction multiplication later on.

Quick FAQs

  • Q: Do I multiply straight across or simplify first?
    A: Simplify first! Cross-cancelling saves time and prevents large numbers.
  • Q: How do I know I’ve simplified fully?
    A: Check whether numerator and denominator share any common factors beyond 1.
  • Q: Does the order of multiplication matter?
    A: No, multiplication is commutative — you can rearrange for easier cancellation.

Study Tip

When faced with three or more fractions, write all numerators in one row and all denominators beneath. Simplify vertically and diagonally before multiplying. It’s faster, cleaner, and far less error-prone in exams.

Try These Yourself

  • \( \tfrac{4}{7}\times\tfrac{14}{15}\times\tfrac{3}{8} \)
  • \( \tfrac{11}{12}\times\tfrac{9}{22}\times\tfrac{4}{9} \)
  • \( \tfrac{3}{10}\times\tfrac{5}{6}\times\tfrac{8}{9} \)