Edexcel · GCSE Maths · 1MA1 · Foundation and Higher
M24 · Direct and inverse proportion
Revision notes, worked examples and methods for direct and inverse proportion.
Notes and quizzes ready · 50 questions · Video coming soon.
Foundation shows shared notes and questions. Higher includes the shared content and labelled Higher extensions. Changing tier starts the notes and a new quiz set.
Revise the key ideas
Direct and inverse relationships
Direct proportion means y changes by the same factor as x: y = kx for constant k. The graph is a straight line through the origin. A straight line with a non-zero intercept is not direct proportion. Direct and inverse proportion
Worked example: Five notebooks cost £12.50 at a constant unit price. One costs £2.50, so eight cost £20. Here cost = 2.5 × number of notebooks.
For direct proportion, yx is constant. Doubling x doubles y; tripling x triples y. Test ratios rather than differences.
Inverse proportion means y = kx, so xy is constant. For positive quantities, doubling x halves y. Its graph is a reciprocal curve, not a straight line.
Worked example: Four equally productive workers take 9 hours for a fixed job. Six workers take (4 × 9) ÷ 6 = 6 hours, assuming work is shared perfectly and each worker's rate is unchanged.
For a fixed distance, time is inversely proportional to speed. This needs a constant journey length; a general time-versus-speed situation may not be inverse proportion.
Using equations
To use a proportional equation, find the constant from known values, then substitute the new input. Keep the complete relation, not just the value of k.
Worked example: y = 24x. When x = 3, y = 8; when x = 8, y = 3. x = 0 is not allowed.
Read the context carefully: a fixed starting fee, changing productivity or a changing total can invalidate a proportional model.
Higher — powers and constructing models
If y is directly proportional to x2, write y = kx2. If y is inversely proportional to x2, write y = kx2. State which power the question specifies.
Worked example: y ∝ x2, and y = 18 when x = 3. Then 18 = 9k, so k = 2 and y = 2x2. At x = 5, y = 50.
Worked example: y ∝ 1x2, and y = 12 when x = 2. Then k = 48, so y = 48x2. At x = 4, y = 3.
If y ∝ √x and y = 15 at x = 9, then k = 5 and y = 5√x. To find x when y = 20, √x = 4, so x = 16.
Plotting y against x2 gives a straight line through the origin for y = kx2. Its gradient is k. Plotting against 1x does the same for inverse proportion.
Test yourself
50 questions · Sets of 10 from the selected tier. For fractions, use / when typing; for powers, use superscripts or ^. Follow each question's answer format. These quick checks support revision; practise full written solutions and proofs too.
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