Proportion

Description

Proportion describes how two quantities relate as they both change.

Direct proportion: as one quantity increases, the other increases at the same rate. Written $y \propto x$, so $y = kx$ for some constant $k$. $$\text{If } y \propto x \text{ and } y = 15 \text{ when } x = 5, \text{ then } k = 3, \text{ so } y = 3x$$

Inverse proportion: as one quantity increases, the other decreases. Written $y \propto \frac{1}{x}$, so $y = \frac{k}{x}$.

A direct proportion graph passes through the origin; an inverse proportion graph is a reciprocal curve.

Unitary method: find the value for 1 unit first, then scale. $$\text{6 items cost £8.40} \implies 1 \text{ item costs £1.40} \implies 9 \text{ items cost £12.60}$$

Common error: assuming a linear relationship is proportional — it is only proportional if the line passes through the origin.

Links

Ratios Direct & inverse proportion Equation of a straight line