Learn4MA1Papers 1H, 2H · Higher tierEquations, formulae and identitiesDirect and inverse proportion

Equations, formulae and identities · Subtopic 6 of 6

Direct and Inverse Proportion

Proportion links two quantities by one constant $k$ . Direct: $y=kx^n$ , so $y$ grows with $x$ . Inverse: $y=\dfrac k{x^n}$ , so $y$ shrinks as $x$ grows. Write the equation, find $k$ from the given pair, then use it.

Cambridge 4024 Paper 1Cambridge 4024 Paper 2Edexcel 4MA1 Papers 1H, 2HEdexcel 4MA1 Papers 1F, 2FAQA 8300 Papers 1–3≈ 30 minRevision 5 · updated 2026-10-27

Section 1 of 14

Learning objectives

Write direct and inverse proportion as equations with a constant $k$ .
Include squares, cubes and square roots.
Find $k$ and use it to find unknown values.
Recognise proportion graphs and the effect of scaling $x$ .

Pause and think

A pizza's area is proportional to the square of its diameter. Is a $30$ cm pizza "twice as much" as a $15$ cm one?

Section 2 of 14

Core idea

Start hereDouble the side, quadruple the paint

Double the side of a square and its area doesn't double, it quadruples. Area is proportional to the side squared. Knowing the type of proportion lets you scale anything correctly.

The big idea

$y\propto x\Rightarrow y=kx$ ; $y\propto\frac1x\Rightarrow y=\frac kx$ ; similarly for $x^2$ , $\sqrt x$ , $\frac1{x^2}$ . Find $k$ from one pair, then use.

1. Graphs

Direct (y = 2x) and inverse (y = 12/x) proportion

Cost of apples is direct: twice as many, twice the cost. Time for a job is inverse: twice as many workers, half the time.

2. Method

Worked example $y\propto x^2$

$y=12$ when $x=2$ . Find $y$ when $x=5$ .

Goal: $y$ .
Strategy: Write, find $k$ , use.

$y=kx^2$
Write with $k$ .
$12=4k\Rightarrow k=3$
Given pair.
$y=3(25)=75$
Use.

Answer

75

✓ Check

x

×2.5 so

y

×6.25:

12\times6.25=75

✓.

Worked example $t\propto\frac1{\sqrt d}$

$t=6$ when $d=4$ . Find $d$ when $t=3$ .

Goal: $d$ .
Strategy: Same method.

$t=\frac k{\sqrt d}\Rightarrow k=6\times2=12$
Find $k$ .
$3=\frac{12}{\sqrt d}\Rightarrow\sqrt d=4\Rightarrow d=16$
Solve.

Answer

16

✓ Check

t

halved needs

\sqrt d

doubled:

d

×4 ✓.

Common misconceptionInverse ≠ negative Inversely proportional means

y=\frac kx

, not

y=-kx

Pause and think

If $y\propto x^3$ , what happens to $y$ when $x$ doubles?

Summary

Write with $k$ , find $k$ , use.

Constant of proportionality: $k$ .

HistoryBoyle's law. In

1662

Robert Boyle found that, at constant temperature, the pressure of a gas is inversely proportional to its volume: squeeze the gas into half the space and the pressure doubles. It was one of the first physical laws written as a proportion.

Fun factWhy giants can't exist. Strength depends on the cross-section of bones (

\propto\text{height}^2

) but weight depends on volume (

\propto\text{height}^3

). A giant ten times taller would be

1000

times heavier but only

100

times stronger. Galileo worked this out in

1638

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