Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Direct & Inverse Proportion

In one sentence. Quantities that scale together, quantities that trade off, and the algebraic y = kx² machinery of Higher Tier proportion.

Ratio & Proportion — one of 3 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Robert Boyle. Learn Direct & Inverse Proportion, teach it, then defend it against Robert Boyle.

Where this sits in the specification

Specification 4MA1, 1.7 Proportion · 2.5 Algebraic proportion (Higher). Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 1.7 — Proportion
  • 2.5 — Algebraic proportion (Higher)

1 · Understand it

What it is

Direct proportion: double one, double the other (cost of pens). Inverse proportion: double one, HALVE the other (workers vs time). At Higher Tier proportion becomes algebra: 'y is proportional to x²' means y = kx², and one known pair of values pins down k.

Why it works

Direct proportion is a constant RATE hiding in the relationship (price per pen), and inverse proportion is a constant PRODUCT (total worker-days: a job that takes 5 workers 12 days is 60 worker-days however you split it). The constant k is that hidden invariant made explicit.

When a question is asking for it

Recipes, best-buy and unit-cost problems (direct); 'how long with more workers/taps/machines' (inverse); and any statement with the symbol ∝ or the words 'proportional to' — write y = k(…) immediately, find k from the given pair, then answer.

How to hold it in your head

Ask: 'if I double this, what happens to that?' Same direction → direct (divide to find the unit rate). Opposite direction → inverse (multiply to find the invariant total). The k-method is just this instinct with a letter attached.

2 · The mistakes that cost marks

  • Treating everything as direct: 'if 5 workers take 12 days, 4 take 9.6' — fewer workers must take LONGER. Sense-check direction before calculating.
  • Forgetting to find k. 'y ∝ x², y = 20 when x = 2' does NOT mean y = 20x²; the pair gives k = 5, so y = 5x².
  • Proportional to x² read as 'proportional to 2x'. Squaring is not doubling: tripling x multiplies y by NINE.
  • In inverse proportion, dividing when you should multiply: the invariant is the PRODUCT (workers × days), not the quotient.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

3 identical taps fill a tank in 40 minutes. How long would 8 of these taps take?

  1. More taps → less time: inverse proportion, so find the invariant PRODUCT.
  2. Invariant: 3 × 40 = 120 tap-minutes.
  3. With 8 taps: 120 ÷ 8 = 15 minutes.
  4. (Sense check: more taps, fewer minutes ✓)

Answer: 15 minutes

d is inversely proportional to the square of t, and d = 10 when t = 3. (a) Find a formula for d in terms of t. (b) Find d when t = 5.

  1. Write the law with k: d = k/t² ('inversely' puts the t² underneath).
  2. Pin k with the given pair: 10 = k/9 → k = 90.
  3. (a) The formula — usually the first marks: d = 90/t².
  4. (b) d = 90/5² = 90/25 = 3.6.
  5. (Sense check: t grew, so d had to shrink ✓)

Answer: (a) d = 90/t² (b) 3.6

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — 12 identical pens cost £4.44. Work out the cost of 7 of these pens. (answer in £)
  2. Grade 7 — 5 workers take 12 days to complete a job. Working at the same rate, how long would 4 workers take? (answer in days)
  3. Grade 8 — y is directly proportional to x². When x = 2, y = 20. Work out the value of y when x = 5.
  4. Grade 9 — y is inversely proportional to √x. When x = 9, y = 8. Work out the value of y when x = 36.
  5. 9+ — p is directly proportional to q³. When q = 3, p = 54. Work out the value of q when p = 128.

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Direct & Inverse Proportion, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Direct & Inverse Proportion works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Direct & Inverse Proportion against Robert Boyle — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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