Inverse proportion to the square

Last updated on May 2, 2026

Inverse proportion to the square worksheet

Inverse proportion to the square occurs when one quantity decreases in proportion to the square of another. This is commonly seen in physics, such as in gravitational force or light intensity, where distance affects the strength of the force or brightness. Understanding this concept helps in solving problems where small changes in one value lead to much larger changes in another. Jump to the questions

Practise now

Question

y is inversely proportional to p²

y is when p is

Find the equation for y and p

Step 1 - describe the relationship using the correct symbol

y ∝ 1 / p²

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

y = / p²

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 p²

y = / p²

Find y when p =

Step 1 - restate the equation

y = / p²

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 = / p²

Step 2 - substitute the value you know

= ÷ p²

Step 3 - rearrange the equation to isolate p²

÷ = p²

Step 4 - determine the value of p²

p² =

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 square of another variable (e.g., y ∝ 1/x²). In physics, this is known as the "inverse-square law" and governs things like gravity and light intensity. As you move away from a light source (x increases), the brightness (y) drops off very sharply.

Key method

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

Write out the relationship algebraically: y = k / x².
Substitute the pair of known values (the 'x' and 'y' given in the question) into your equation.
Square the x value first, and then solve the equation to find 'k'. Usually, this means multiplying the 'y' value by the squared '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 square of x. When x = 3, y = 4. Find the value of y when x = 6.

Step 1: Write the base equation.

y = k / x²

Step 2: Substitute the known values (x=3, y=4) to find 'k'.

4 = k / 3²

4 = k / 9

k = 4 × 9 = 36.

Step 3: Write the full formula.

y = 36 / x²

Step 4: Answer the question (Find y when x = 6).

y = 36 / 6²

y = 36 / 36

y = 1.

Common mistakes to avoid

The most fatal error is ignoring the word "square" and using y = k/x. Always read the phrasing carefully. The second most common error is forgetting to actually square the number when calculating 'k'. In the example above, students often write 4 = k / 3, finding k=12, which ruins the rest of the calculation.

How to check your answer

Because it is an inverse-square proportion, doubling the 'x' value will cause the 'y' value to divide by 4. In our example, 'x' went from 3 to 6 (it doubled). Our starting 'y' was 4, and it dropped to 1 (divided by 4). This proves our calculation is absolutely correct.