GCSE Maths Practice: fractions

Question 5 of 11

This Higher GCSE question involves multiplying fractional scale factors — a key skill for compound changes in geometry, ratio, and proportional reasoning. The total effect of two fractional reductions is found by multiplying the fractions together.

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\( \begin{array}{l}\text{A model car is built at }\frac{2}{3}\text{ of the real size. It is then reduced again to }\frac{3}{5}\text{ of that model.}\\\text{What fraction of the real car’s size is the final model?}\end{array} \)

Choose one option:

When fractional changes happen successively, multiply the fractions — not add them — to find the overall change. Simplify to make the result clear.

In GCSE Maths, multiplying fractions isn’t only a numerical exercise — it models compound effects. When two scale factors, proportions, or fractional changes are applied one after another, the total change is found by multiplying the fractions together.

Real-Life Scenario – Model Building

A designer makes a scale model of a car that is \( \tfrac{2}{3} \) of the real car’s length. Later, the model is digitally reduced again to \( \tfrac{3}{5} \) of its current size for an animation. To find the overall scale factor compared with the real car, multiply the two reductions:

\[ \tfrac{2}{3} \times \tfrac{3}{5} = \tfrac{6}{15} = \tfrac{2}{5}. \]

So the animated version is \( \tfrac{2}{5} \) of the real car’s size. This compound use of fractions is common when working with models, enlargements, or successive percentage changes.

Worked Examples

Example 1 – Area scaling:
A diagram is reduced to \( \tfrac{4}{5} \) of its original side length. The new area is \( \tfrac{4}{5} \times \tfrac{4}{5} = \tfrac{16}{25} \) of the original.

Example 2 – Recipe scaling:
A recipe uses \( \tfrac{3}{4} \) of the original ingredients to make a smaller batch. Then only \( \tfrac{2}{3} \) of that smaller batch is served. Fraction of the original ingredients actually used: \( \tfrac{3}{4} \times \tfrac{2}{3} = \tfrac{6}{12} = \tfrac{1}{2}. \)

Example 3 – Volume scaling:
If each linear dimension of a cube is multiplied by \( \tfrac{3}{5} \), the new volume fraction is \( \tfrac{3}{5} \times \tfrac{3}{5} \times \tfrac{3}{5} = \tfrac{27}{125} \). Compound multiplication shows how volumes shrink faster than lengths.

Common Mistakes

  • Adding scale factors instead of multiplying them.
  • Forgetting to simplify the final result.
  • Misinterpreting what the fraction represents (e.g., thinking \( \tfrac{2}{3} \) means “reduce by two thirds” instead of “keep two thirds”).

Why This Matters

Understanding fractional scaling prepares you for algebraic and geometric reasoning in Higher Maths. It explains why areas and volumes change faster than lengths, and how repeated fractional changes (discounts, dilutions, depreciation) combine through multiplication, not addition.

Quick FAQs

  • Q: Why do we multiply scale factors?
    A: Because each change applies to the result of the previous one, not the original — a compound effect.
  • Q: What if a model is enlarged after being reduced?
    A: Use the actual factors (greater than or less than 1) and multiply them to find the net effect.
  • Q: Does the order of multiplication matter?
    A: No, multiplication of fractions is commutative.

Study Tip

When you see the word “of” in a problem (for example, “\( \tfrac{3}{4} \) of \( \tfrac{2}{5} \)”), replace it with multiplication. This helps translate language into maths quickly.

Try These Yourself

  • A photo is printed at \( \tfrac{4}{5} \) of original size, then reduced again to \( \tfrac{3}{4} \). What is the overall size fraction?
  • A cube’s edges are halved, then halved again. What fraction of the original volume remains?
  • A £60 recipe is made at \( \tfrac{3}{5} \) scale, and only \( \tfrac{2}{3} \) of it is served. What fraction of £60 is used?