Direct & Inverse Variation (Years 10–11) — Maths

Model direct and inverse proportion with y = kx and y = k/x, find k, then solve.

Some quantities rise together: buy twice as much fuel and it costs twice as much — that is direct variation. Others trade off: with twice as many workers a job takes half the time — that is inverse variation. Algebra captures both exactly using a single number k, the constant of proportionality.

Why this matters: proportion is the maths behind physics laws (Ohm's law, gas laws), currency conversion, scaling recipes and pricing. Cambridge Extended sets a variation question most series — often "y is proportional to x², find y when x = …". Finding k first is the key skill.

1. Direct variation: y = kx

"y is directly proportional to x" is written y ∝ x, which becomes y = kx.
The graph is a straight line through the origin; double x and y doubles.
Variations include y = kx² (proportional to the square) and y = k√x (proportional to the square root).

2. Inverse variation: y = k/x

"y is inversely proportional to x" is written y ∝ 1/x, giving y = k/x.
The graph is a curve that never touches the axes; double x and y halves.
Variations include y = k/x² (inverse square, as in gravity and light intensity).

Statement	Equation	Effect of doubling x
y ∝ x	y = kx	y doubles
y ∝ x²	y = kx²	y becomes 4×
y ∝ 1/x	y = k/x	y halves
y ∝ 1/x²	y = k/x²	y becomes ¼

Direct variation is a straight line through the origin; inverse variation is a falling curve approaching the axes.

3. The method — always find k first

Whatever the type, use the given pair of values to find k, then rewrite the full equation and answer the question.

The reliable three steps: write the relationship, use a known pair to find k, then solve.

Direct worked: y is proportional to x², and y = 18 when x = 3. So y = kx². Substitute: 18 = k(3²) = 9k, so k = 2. Equation: y = 2x². When x = 5, y = 2(25) = 50.

Inverse worked: y is inversely proportional to x, and y = 12 when x = 4. So y = k/x. Substitute: 12 = k/4, so k = 48. Equation: y = 48/x. When x = 6, y = 48/6 = 8.

Contrast with the unitary method

The unitary method (find one, then scale) works for simple direct proportion, but only variation with k handles squares, roots and inverse relationships cleanly — so it is the general tool.

Common exam mistake: for y ∝ x², substituting x = 3 as 3k instead of 9k. Square the x before multiplying by k. And for inverse problems, don't forget y = k/x means k = xy.

Exam technique: the command word Find an equation connecting y and x means write y = kx (or the right form), compute k, and quote the finished equation with the number in place — not just "y = kx".