Inverse Proportion

Two quantities are in inverse proportion when doubling one halves the other — as one grows, the other shrinks in step. They are linked by the rule

y = \frac{k}{x}

where k is a fixed number. Rearranged, this says the product xy = k stays constant: whatever you gain on one side you lose on the other.

For example, more workers means less time. If 4 workers take 6 hours, the job is worth 4 \times 6 = 24 worker-hours. So 8 workers — twice as many — would take 24 \div 8 = 3 hours, exactly half the time.

When y is inversely proportional to x:

they are linked by y = \dfrac{k}{x};
the product xy = k stays constant;
the graph is a curve falling towards the axes (a hyperbola);
find k = xy from one known pair, then use it for any other.

Watch the curve fall

Drag the slider to change the constant k. As x grows the curve dives towards the axis — a bigger k holds the curve higher, but it always falls away.