Direct & Inverse Variation

y=kx grows together; xy=k trades off.

The idea

Direct and inverse variation are the two simplest ways one quantity can depend on another, and a single constant pins each down completely.

Two quantities vary directly when one is a fixed multiple of the other: $y = kx.$ Doubling $x$ doubles $y$, and the ratio $y/x$ always equals $k$. Fuel cost at a fixed price per gallon varies directly with the amount: twice the fuel, twice the bill.

They vary inversely when their product is fixed: $xy = k, \qquad \text{equivalently} \qquad y = \frac{k}{x}.$ Now doubling $x$ halves $y$. A fixed sum of money split among people behaves this way: twice as many people, half as much each.

In both cases $k$ is the constant of variation, and one known pair of values determines it: divide $y$ by $x$ for direct variation, multiply them for inverse. Knowing $k$ gives the whole relationship, and any other value follows by substitution.

Ways to work on it

• Walkthrough. Direct and inverse variation, each end to end.
• Practice. Find k and predict, direct or inverse.
• Hardest. Decide the model from a scenario, then solve.

Not sure where to start? Take the ten-question placement test.