Direct proportion to the cube

Last updated on May 2, 2026

Scaffolded direct proportion to the cube worksheet

Direct proportion to the cube is when one quantity increases or decreases as the cube of another. This relationship often appears in physics, such as when calculating the volume of a sphere, which is directly proportional to the cube of its radius. Jump to the questions

Practise now

Question

y is directly proportional to p³

y = when p =

Find the equation for y and p

Step 1 - describe the relationship using the correct symbol

y ∝ 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

= k

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 - calculate the value of p³

p³ =

Step 5 - calculate the value of p

p =

Topic guide

What this worksheet practises

This worksheet focuses on direct proportion to the cube of a number. In standard direct proportion, if one variable doubles, the other doubles. When proportion is linked to a cube, a small increase in one variable causes a massive, exponential increase in the other.

Key method

To solve these problems, you must set up an algebraic equation involving a constant of proportionality, 'k'.

Write the proportionality statement: y ∝ x³.
Convert this into an equation by adding 'k': y = kx³.
Substitute the pair of known values given in the question into the equation to find 'k'.
Rewrite the full equation with your calculated value of 'k'.
Use this specific equation to find any other missing values.

Worked example

y is directly proportional to the cube of x. When x = 2, y = 24. Find the value of y when x = 5.

Step 1: Write the equation.

y = kx³

Step 2: Substitute the known values to find k.

24 = k × (2)³

24 = k × 8

k = 24 ÷ 8 = 3.

Step 3: Write the full equation.

y = 3x³

Step 4: Use the equation to answer the question.

When x = 5, y = 3 × (5)³

y = 3 × 125 = 375.

Common mistakes to avoid

The most common error is forgetting to cube the 'x' value before multiplying by 'k'. For example, calculating 3 × 5 first (getting 15) and then cubing it. Because of the order of operations (BIDMAS/BODMAS), you must calculate the index (the cube) before you multiply by the constant.

Things to remember

With cubic proportion, expect your answers to grow very quickly. If x doubles (from 2 to 4), y doesn't just double; it is multiplied by 2³ (which is 8). Recognising this rapid growth pattern helps you spot calculation errors early.