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MathidiaMathematics Academy
Ratio, proportion and rates
Lesson 38 of 75
Standard practice

Direct proportion

Objective. Model direct proportion with equations and graphs.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Direct proportion describes two quantities that increase or decrease together at a constant rate, so one is always a fixed multiple of the other. This relationship can be written as y = kx, where k is the constant of proportionality, and its graph is a straight line through the origin. If 4 kg of apples costs £6, then k = 6 ÷ 4 = 1.5, so the cost of 10 kg is 1.5 × 10 = £15. Always find k first by dividing one known pair of values before using it to find others.

Worked example

If 5 litres of paint cost £22.50, use direct proportion to find the cost of 8 litres.

£22.50
Money illustration for £22.5
  1. 1Find the constant of proportionality: k = 22.50 ÷ 5 = 4.5
  2. 2This means each litre costs £4.50
  3. 3Multiply by 8 litres: 4.5 × 8
  4. 4The cost of 8 litres is £36

Answer: £36

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Common mistake. Pupils sometimes use proportion wrongly on non-proportional situations, such as assuming a taxi with a fixed starting fee is directly proportional to distance. Always check the relationship passes through zero before treating it as direct proportion.

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