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.