Inverse Variation

Two quantities vary inversely when their product is fixed: double one and the other halves. The constant of variation k is that product, found from any single matching pair. Worked example: if y = 8 when x = 6 then k = 48, so at x = 16 the value is y = 48/16 = 3. Plotted, the relation is a hyperbola that approaches both axes without ever touching them.

The pattern is everywhere in the physical world. Boyle's law of 1662 — pressure times volume is constant at fixed temperature — was the first quantitative gas law and is pure inverse variation; so are gear ratios (a wheel with twice the teeth turns at half the speed), lever arms about a fulcrum, the time a job takes against the number of workers on it, and the wavelength of light against its frequency. Two traps. First, "inversely proportional" means k/x, not k − x: a 10% rise in x produces a 9.1% fall in y, not a 10% fall. Second, watch for inverse-square laws such as gravity and light intensity, where doubling the distance quarters the effect rather than halving it — those need x² in the denominator and are a different formula.