GCSE Maths Practice: fractions

Question 10 of 11

This question tests your Higher GCSE understanding of dividing fractions through a realistic comparison problem. You’ll determine how many times one rate exceeds another by converting a division into multiplication using the reciprocal.

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\( \begin{array}{l}\text{Recipe A uses }\frac{3}{4}\text{ cup of sugar per batch. Recipe B uses }\frac{2}{3}\text{ cup per batch.}\\\text{How many times more sugar does Recipe A use compared to Recipe B?}\end{array} \)

Choose one option:

When comparing rates or ratios, divide one by the other. Turn division into multiplication using the reciprocal of the second fraction, then simplify.

At Higher GCSE level, fraction division often appears in real-world rate or proportion questions. It’s not just about flipping and multiplying — it’s about comparing two fractional quantities accurately. Here, dividing fractions helps determine how many times one rate fits into another.

Example Context – Recipe Comparison

Suppose Recipe A uses \( \tfrac{3}{4} \) cup of sugar per cake, while Recipe B uses \( \tfrac{2}{3} \) cup. To compare how much more sugar Recipe A uses, we divide the two fractions:

\[ \tfrac{3}{4} \div \tfrac{2}{3} = \tfrac{3}{4} \times \tfrac{3}{2} = \tfrac{9}{8} = 1\tfrac{1}{8}. \]

So Recipe A needs 1.125 times as much sugar per batch as Recipe B. This approach works for comparing rates, efficiency, or proportional scaling — all key topics in Higher GCSE.

Worked Examples

Example 1 – Work rate:
Machine A completes \( \tfrac{3}{5} \) of a task per hour. Machine B completes \( \tfrac{1}{2} \) per hour. How many times faster is Machine A?

  1. Divide rates: \( \tfrac{3}{5} \div \tfrac{1}{2} \).
  2. Multiply by reciprocal: \( \tfrac{3}{5} \times \tfrac{2}{1} = \tfrac{6}{5} \).
  3. Answer: Machine A is \( \tfrac{6}{5} \) times faster.

Example 2 – Comparing densities:
Liquid X has density \( \tfrac{7}{8} \) g/cm³ and Liquid Y has \( \tfrac{3}{4} \) g/cm³. Ratio X:Y = \( \tfrac{7}{8} \div \tfrac{3}{4} = \tfrac{7}{8} \times \tfrac{4}{3} = \tfrac{7}{6} \).

So Liquid X is about 1.17 times denser than Y.

Example 3 – Speed comparison:
Car A travels \( \tfrac{9}{10} \) km per minute; Car B travels \( \tfrac{3}{4} \) km per minute.
\( \tfrac{9}{10} \div \tfrac{3}{4} = \tfrac{9}{10} \times \tfrac{4}{3} = \tfrac{6}{5} \).
Car A’s speed is \( 1.2 \) times Car B’s.

Common Mistakes

  • Reversing the order of division (\(a\div b\neq b\div a\)).
  • Forgetting to take the reciprocal of the second fraction.
  • Failing to simplify before multiplying, leading to large numbers.

Real-Life Links

Fraction division underpins concepts such as scaling recipes, comparing energy efficiency (output/input), exchange rates, and calculating relative probabilities. Understanding the inverse relationship helps you think multiplicatively rather than procedurally.

Quick FAQs

  • Q: Why do we multiply by the reciprocal instead of dividing directly?
    A: Because dividing by a fraction is equivalent to multiplying by its inverse — it simplifies the calculation.
  • Q: Can we divide mixed numbers directly?
    A: Convert them to improper fractions first, then follow the same reciprocal rule.
  • Q: What if both fractions are less than 1?
    A: The result can be greater than 1 — you’re finding how many smaller parts fit into another small part.

Study Tip

Before diving into arithmetic, pause and estimate: if both fractions are less than 1, expect a result greater than 1. Estimation helps confirm whether your answer makes sense — a crucial Higher GCSE habit.

Try These Yourself

  • Compare \( \tfrac{5}{6} \) with \( \tfrac{3}{4} \). How many times larger is the first?
  • Two printers complete \( \tfrac{7}{9} \) and \( \tfrac{1}{2} \) of a job per hour. Find the ratio of their speeds.
  • Find \( \tfrac{11}{12} \div \tfrac{3}{8} \) and write your answer as a mixed number.