GCSE Maths revision · Ratio and proportion

Direct and inverse proportion

Foundation and Higher

The idea

In direct proportion, both quantities change by the same factor: double one and the other doubles. In inverse proportion, one goes up as the other goes down by the same factor: twice as many workers take half the time.

Decide which kind it is: does more of one mean more or less of the other?
Direct: scale both the same way. Inverse: multiply up one, divide the other.
Check: in direct proportion $\frac{y}{x}$ stays the same; in inverse proportion $xy$ stays the same.

Watch out More workers means less time: do not scale an inverse relationship directly.

See it worked

Worked exampleNon-calculator4 marks

Answer each part without a calculator.

(a) 6 cinema tickets cost £45. Work out the cost of 10 tickets.

(b) A recipe for 4 people uses $300$ g of flour. How much flour is needed for 10 people?

(a) One ticket costs $45 \div 6 = £7.50$ .
10 tickets cost $10 \times 7.50 = £75$ .
(b) For 2 people: $150$ g. For 10 people: $5 \times 150 = 750$ g.

Answer (a) £75; (b) $750$ g

Your turn

Question 1Non-calculator3 marks

A recipe for 6 portions uses 420 g of oats. How many grams of oats are needed for 9 portions of the same size?

Question 2Non-calculator3 marks

Positive quantities x and y satisfy y = 72/x. Find the change in y when x increases from 6 to 9. Give the amount by which y decreases.

Question 3Non-calculator4 marks

A job is planned to take 12 workers 15 days. Each worker works at the same constant rate. After 6 days, the deadline changes: the whole job must be finished by the end of day 12. How many additional workers are needed for the remaining days? You must show your working.

In past papers

158 official questions in 124 papers since 2017. The latest:

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