Inverse proportion to the cube

Last updated on May 25, 2026

Inverse proportion to the cube worksheet

Inverse proportion to the cube occurs when one quantity decreases in proportion to the cube of another quantity. This relationship is commonly seen in physics, such as in gravitational or electrostatic forces, where the strength of the force diminishes rapidly with increasing distance. Understanding this concept is essential for solving problems where changes become dramatically smaller as one value grows larger. Jump to the questions

Practise now

Question

y is inversely proportional to p³

y is when p is

Find the equation for y and p

Step 1 - describe the relationship using the correct symbol

y ∝ 1 / p³

Step 2 - replace the symbol with the constant of proportionality, k

y = / p³

Step 3 - substitute the values from the question into the equation

= ÷

Step 4 - rearrange the equation to isolate k

× = k

Step 5 - determine the value of k

k =

Step 6 - write the final equation linking y and p³

y = / p³

Find y when p =

Step 1 - restate the equation

y = / p³

Step 2 - substitute the value you know

y = ÷

Step 3 - determine the final value of y

y =

Find p when y =

Step 1 - restate the equation

y = / p³

Step 2 - substitute the value you know

= ÷ p³

Step 3 - rearrange the equation to isolate p³

÷ = p³

Step 4 - determine the value of p³

p³ =

Step 5 - calculate p

p =

Topic guide

What this worksheet practises

This worksheet focuses on solving inverse proportion problems where one variable is inversely proportional to the cube of another variable (e.g., y ∝ 1/x³). Inverse proportion means as one value gets larger, the other gets smaller. Because it involves a cube, this change happens extremely aggressively.

Key method

Every proportion question requires you to construct a formula and find a constant 'k'.

Write out the relationship algebraically: y = k / x³.
Substitute the pair of known values (the 'x' and 'y' given in the question) into your equation.
Cube the x value first, and then solve the equation to find 'k'. Usually, this means multiplying the 'y' value by the cubed 'x' value.
Rewrite your full, final formula with the actual number for 'k' placed into it.
Use this completed formula to answer the final part of the question.

Worked example

y is inversely proportional to the cube of x. When x = 2, y = 5. Find the value of y when x = 4.

Step 1: Write the base equation.

y = k / x³

Step 2: Substitute the known values (x=2, y=5) to find 'k'.

5 = k / 2³

5 = k / 8

k = 5 × 8 = 40.

Step 3: Write the full formula.

y = 40 / x³

Step 4: Answer the question (Find y when x = 4).

y = 40 / 4³

y = 40 / 64

Simplify the fraction: y = 5/8 (or 0.625).

Common mistakes to avoid

The most fatal error is ignoring the word "cube" and using y = k/x. Always read the phrasing carefully. The second most common error is forgetting to actually cube the number when calculating 'k'. In the example above, students often write 5 = k / 2, finding k=10, which ruins the rest of the calculation.

How to check your answer

Because it is inverse proportion, as x gets bigger, y must get smaller. Our starting x was 2, and the new x was 4 (it got bigger). Our starting y was 5, and our new y was 0.625 (it got smaller). The direction of change is correct.