Proportion

Revise Proportion for Mathematics A 4MA1 (O Level) — revision notes, 52 practice questions and instant AI marking.

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Pearson Edexcel International GCSE Mathematics A (4MA1)

Proportion

The big idea: Proportion problems all follow one routine: write the relationship with a constant k, use the given pair of values to find k, then use the formula. The only thing that changes is the power of x.

Summary — What This Chapter Covers

Higher tier only.
Direct proportion: y ∝ x, x², x³ or √x, written y = kx, kx², kx³ or k√x.
Inverse proportion: y ∝ 1/x, 1/x², 1/x³ or 1/√x, written y = k/x and so on.
Find k from one pair of values, then write the full formula before answering anything else.
Recognise the graph of each relationship.

1. Direct Proportion

Higher tier

y = kxⁿ

"y is directly proportional to x²" means y ∝ x², so y = kx² for some constant k. The 4MA1 specification limits the relationships to x, x², x³ and √x (and their reciprocals for inverse proportion).

Routine 1. Write y = k × (power of x) → 2. Substitute the given pair to find k → 3. Write the formula with k → 4. Use it

Worked Example

y is directly proportional to x². When x = 3, y = 45. (a) Find a formula for y in terms of x. (b) Find y when x = 4. (c) Find x when y = 125, given x > 0.

Practice Question 1

The mass, m grams, of a metal sphere is directly proportional to the cube of its radius, r cm. When r = 2, m = 40. Find m when r = 3.

Practice Question 2

y is directly proportional to √x. When x = 9, y = 12. Find y when x = 30. Give your answer correct to 3 significant figures.

2. Inverse Proportion

Higher tier

y = k/xⁿ

"y is inversely proportional to x" means y ∝ 1/x, so y = k/x: as x increases, y decreases, and the product xy stays constant. The same routine applies.

Worked Example

F is inversely proportional to the square of d. When d = 3, F = 20. (a) Find F when d = 5. (b) Find d when F = 5, given d > 0.

Practice Question 3

y is inversely proportional to x³. When x = 3, y = 2. Find y when x = 6.

Practice Question 4

y is directly proportional to x². x is increased by 20%. Find the percentage increase in y.

3. Proportion and Graphs

Higher tier

Relating the algebra to the shape

Each relationship has a recognisable graph (for positive x). Matching a graph to a statement is a common short question.

Statement	Formula	Graph for x > 0
y ∝ x	y = kx	straight line through the origin
y ∝ x²	y = kx²	curve through the origin, getting steeper (half a parabola)
y ∝ x³	y = kx³	through the origin, steeper still
y ∝ √x	y = k√x	through the origin, getting less steep
y ∝ 1/x	y = k/x	curve that never meets either axis (reciprocal graph)
y ∝ 1/x²	y = k/x²	similar, but falls away faster

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Proportion

Proportion

y = kxn

y = k/xn

Relating the algebra to the shape

Past papers that test Proportion

More Mathematics A topics

y = kxⁿ

y = k/xⁿ