1Cademy - Direct Variation

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Proportional Quantities and Variation

Definition

Direct Variation

For any two variables $x$ and $y$ , the variable $y$ varies directly with $x$ if their relationship can be expressed as

$y = kx, \quad \text{where } k \neq 0$

The nonzero constant $k$ is called the constant of variation (sometimes called the constant of proportionality). It represents the fixed multiplier that connects the two quantities: whenever one variable is known, the other is found by multiplying by $k$ . Because $k$ is constant, doubling $x$ will always double $y$ , tripling $x$ will triple $y$ , and so on.

For example, if Lindsay earns $15 per hour, her salary $s$ and hours worked $h$ satisfy $s = 15h$ . Here $k = 15$ , and her salary varies directly with the number of hours she works — the salary is always the product of the constant $15$ and the hours.

Updated 2026-05-01

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