Direct and inverse proportion
Proportion problems test whether candidates can spot the relationship between two quantities and write a function for it.
Direct proportion
y is directly proportional to x means: as x increases, y increases at a constant rate. Algebraically:
y = kx for some constant k > 0.
Equivalently, y / x is constant, or the graph of y against x is a straight line through the origin with gradient k.
Notation: y ∝ x.
Inverse proportion
y is inversely proportional to x means: as x increases, y decreases (and their product stays constant). Algebraically:
y = k / x for some constant k > 0.
Equivalently, xy is constant, or the graph of y against x is a hyperbola.
Notation: y ∝ 1/x.
Higher: powers in proportion
WJEC Higher includes y ∝ x², y ∝ √x, y ∝ 1/x², etc. Method is identical:
- Write y = k × (function of x).
- Substitute one given pair to find k.
- Use the formula to answer the rest.
Worked: direct proportion (Foundation)
The cost of fuel is directly proportional to the number of litres bought. 12 litres cost £18.
- y = kx where y = cost, x = litres.
- 18 = 12k → k = 1.5 (£/litre).
- y = 1.5x.
- Cost of 25 litres: £37.50.
Worked: inverse proportion (Higher)
The pressure of a gas is inversely proportional to its volume. P = 80 when V = 5.
- P = k / V.
- 80 = k / 5 → k = 400.
- P = 400 / V. When V = 25, P = 16.
WJEC exam tip
Always begin a proportion question with the phrase "y = kx" (or k/x, kx², etc.) on the page — that single line is the M1 method mark even if the rest goes wrong. Then substitute, find k, and write the FULL EQUATION before answering further parts.
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