GCSE · Direct & inverse proportion

Proportion with squares, cubes and rootsExam-style questions

Why y ∝ x², y ∝ √x and y ∝ 1/x² all follow the same three-step method — write the equation, find k, use it — and how the power changes the scaling.

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12 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 2 Level 3 Level 4 Level 5

1
From a viewpoint $h$ metres above sea level, the distance $d$ km to the horizon is directly proportional to the square root of $h$ . When $h = 16$ , $d = 4.8$ .

(a) Find a formula for $d$ in terms of $h$ .

(b) Work out the value of $d$ when $h = 100$ .

(c) Work out the value of $h$ when $d = 9$ .
Level 4 · 5 marksTry it
2
$F$ is inversely proportional to the square of $d$ . When $d = 3$ , $F = 20$ .

Work out the value of $F$ when $d = 6$ .
Level 4 · 3 marksTry it
3
The mass of a solid sphere is directly proportional to the cube of its radius. A sphere of radius 2 cm has mass 33.6 g.

(a) Work out the mass of a sphere of radius 5 cm made of the same material.

(b) Write down the number the mass of any of these spheres is multiplied by when its radius is doubled.
Level 4 · 4 marksTry it
4
$y$ is directly proportional to the square root of $x$ . When $x = 25$ , $y = 10$ .

(a) Find a formula for $y$ in terms of $x$ .

(b) Work out the value of $y$ when $x = 64$ .

(c) Work out the value of $x$ when $y = 18$ .
Level 4 · 4 marksTry it
5
$y$ is inversely proportional to the square of $x$ . When $x = 5$ , $y = 8$ .

Work out the value of $y$ when $x = 10$ .
Level 4 · 3 marksTry it

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