Direct proportion

Last updated on May 2, 2026

Scaffolded direct proportion interactive worksheet

Direct proportion is a concept where two quantities increase or decrease at the same rate. It’s useful in situations like scaling recipes, converting measurements, or calculating travel times. Understanding direct proportion helps you solve real-world problems where one quantity directly affects another. Jump to the questions

Practise now

Question

y is directly proportional to m

y = when m =

Find the equation for y and m

Step 1 - describe the relationship using the correct symbol

y ∝ m

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

y = m

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 m

Find y when m =

Step 1 - restate the equation

Step 2 - substitute the value you know

y = ×

Step 3 - determine the final value of y

y =

Find m when y =

Step 1 - restate the equation

Step 2 - substitute the value you know

= × m

Step 3 - rearrange the equation to isolate m

÷ = m

Step 4 - determine the value of m

m =

Topic guide

What this worksheet practises

This worksheet provides practice on simple direct proportion. Direct proportion means that two quantities increase or decrease at the exact same rate. For example, if you buy twice as many apples, it will cost exactly twice as much. Their ratio remains constant.

Key method

The most reliable method for solving direct proportion problems is the "unitary method" (finding the value of a single unit).

Identify the two quantities given in the complete pair (e.g., the cost for a certain number of items).
Divide the total amount by the number of items to find the value of exactly one unit.
Multiply this single unit value by the new number of items you need to find.
Alternatively, you can use the algebraic method: write y = kx, substitute the first pair of values to find the constant 'k', and then use the equation to find the missing value.

Worked example

If 5 identical books cost £30, how much will 8 books cost?

Step 1: Find the value of a single unit (1 book).

30 ÷ 5 = 6.

One book costs £6.

Step 2: Multiply this unit value by the required amount (8 books).

6 × 8 = 48.

The answer is £48.

Common mistakes to avoid

The most common error is performing the division backwards when finding the single unit. In the example above, calculating 5 ÷ 30 instead of 30 ÷ 5. Always think about what the resulting number means: does it represent "books per pound" or "pounds per book"? You almost always want to find the cost or weight of a single object.

How to check your answer

Use simple estimation and logic to verify your answer. If 5 books cost £30, then 10 books would cost double that (£60). Since you are trying to find the cost of 8 books, the final answer must fall somewhere between £30 and £60. Our answer of £48 fits perfectly into this range.