GCSE Maths Practice: fractions

Question 2 of 11

This question tests your ability to evaluate a complex fractional expression step by step. Simplify each part carefully, then express the final result as a single simplified fraction.

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\( \begin{array}{l} \text{Solve the fraction to its simplest form: }\\\dfrac{\dfrac{12}{(9-5)}}{\dfrac{25}{5}}.\end{array} \)

Choose one option:

Use BIDMAS to simplify step by step. Always handle the numerator and denominator separately before combining them into a single fraction.

How to think about complex (stacked) fractions. A stacked or compound fraction such as \( \dfrac{\dfrac{12}{(9-5)}}{\dfrac{25}{5}} \) is just a division of two simpler fractions. The most reliable approach is to tidy each line first, then perform one clean division at the end. This keeps your working structured and prevents order-of-operations slips.

Two efficient methods

  1. Line-by-line evaluation (recommended first pass).
    • Deal with any brackets in the numerator and denominator.
    • Reduce the top to a single number or single fraction.
    • Reduce the bottom to a single number or single fraction.
    • Finally do “top ÷ bottom”. If you are dividing by a fraction, multiply by its reciprocal.
  2. Flatten to one multiplication. Once each line is a single fraction, use the identity \( \dfrac{a/b}{c/d} = \dfrac{a}{b}\times\dfrac{d}{c} \). This avoids carrying a big division sign and often exposes cancellations early.

Why structure matters (Higher GCSE mindset)

Complex fraction questions at Higher level are rarely about raw arithmetic; they test your method control. The examiner wants to see that you can: (i) respect BIDMAS (brackets first), (ii) simplify strategically, and (iii) express the final result in lowest terms without losing accuracy.

Where simplification helps most

  • Inside the brackets: Always clear these immediately so each line becomes a single clean fraction.
  • Before the final multiplication: When you convert “divide by a fraction” into “multiply by its reciprocal”, look for cross-cancellation (a factor in a numerator that also appears in the other denominator). Cancelling first keeps numbers small and reduces errors.
  • At the end: If your result is \( \tfrac{A}{B} \), check for any common factors of \(A\) and \(B\) and divide them out.

Worked examples (different outcomes)

Example A: \( \dfrac{\dfrac{32}{(20-8)}}{\dfrac{18}{9}} \). Brackets: \(20-8=12\). Top: \(32\div12=\tfrac{8}{3}\). Bottom: \(18\div9=2\). Divide: \( \tfrac{8}{3} \div 2 = \tfrac{8}{3}\times\tfrac{1}{2}=\tfrac{4}{3} \).

Example B: \( \dfrac{\dfrac{45}{(18-3)}}{\dfrac{28}{14}} \). Brackets: \(18-3=15\). Top: \(45\div15=3\). Bottom: \(28\div14=2\). Result: \(3\div2=\tfrac{3}{2} \).

Example C: \( \dfrac{\dfrac{50}{(11-6)}}{\dfrac{27}{3^3}} \). Brackets: \(11-6=5\). Top: \(50\div5=10\). Bottom: \(3^3=27\Rightarrow 27\div27=1\). Result: \(10\div1=10\).

Common mistakes (and fixes)

  • Forgetting that dividing by a fraction means multiplying by its reciprocal. Write a small arrow: \( \div \tfrac{p}{q} \rightarrow \times \tfrac{q}{p} \).
  • Doing cancellations too early across the big fraction bar. First reduce the top and bottom separately; only cancel once both are single fractions.
  • Leaving answers unsimplified. Examiners expect lowest terms unless a specific form is requested.

Quality checks before you move on

  • Estimate size: If the top is smaller than the bottom, expect a fraction less than 1. Here, top \(=3\), bottom \(=5\), so \( \tfrac{3}{5} \) makes sense.
  • Units & context: In applied problems, make sure numerator and denominator refer to compatible quantities before dividing.

Try these (practice, no answers shown)

  • \( \dfrac{\dfrac{36}{(22-10)}}{\dfrac{40}{8}} \)
  • \( \dfrac{\dfrac{63}{(21-12)}}{\dfrac{16}{4}} \)
  • \( \dfrac{\dfrac{48}{(19-7)}}{\dfrac{30}{6}} \)
  • \( \dfrac{\dfrac{75}{(30-18)}}{\dfrac{54}{9}} \)