proportionality constant

The proportionality constant is the fixed number, usually written k, that pins down exactly how two quantities are linked when one varies with the other. Two things might both grow together, but k tells you the rate — how much of one you get per unit of the other. It is the single number that turns a vague “they go together” into a precise rule.

In direct variation it appears as the multiplier: y = kx, so k = y/x is the unchanging ratio. In inverse variation it appears as the product: y = k/x, so k = xy is the unchanging product. In both cases k is found from a single known pair of values, after which the whole relationship is determined.

The constant k is genuinely constant across all the data of one relationship, but it changes from problem to problem and carries units in applications (such as cost per item, or force times distance). Identifying k correctly — as a ratio for direct variation, a product for inverse — is usually the key first step in any variation problem.