Inverse proportion

Why "inversely proportional" means the product stays fixed, how worker-time problems really work, and how to tell inverse proportion apart from direct.

Ratio, proportion & rates of change · Direct & inverse proportion

Inverse proportion

Why "inversely proportional" means the product stays fixed, how worker-time problems really work, and how to tell inverse proportion apart from direct.

Why it works

Two quantities are inversely proportional when one goes up by the same factor the other goes down by: double one and the other halves, triple one and the other drops to a third. In symbols,

y \propto \frac{1}{x}

means

y = \frac{k}{x}, \qquad \text{equivalently} \qquad xy = k.

The second form is the useful secret: in inverse proportion the product of the two quantities never changes. That fixed product

k

is the "total amount of stuff" in the problem.

Worker-time problems are a fixed product. "8 workers take 9 days" — the wall needs

8 \times 9 = 72

worker-days of labour, and that total is the same however you split it. With 6 workers:

72 \div 6 = 12

days. More workers, less time — but the product (the work itself) is untouched. Never scale the same way as direct proportion: 12 workers do not take longer than 8.

Finding the formula: multiply, don't divide. Given "

y

is inversely proportional to

x

, and

y = 12

when

x = 10

", the constant is the product:

k = xy = 12 \times 10 = 120

, so

y = \frac{120}{x}

. (Dividing,

k = 12 \div 10

, is the direct-proportion move — the single most common slip.) Check with the formula's own logic:

x = 20

(doubled) gives

y = 6

(halved). ✓

How to recognise inverse proportion. In a table,

x \times y

gives the same answer in every column — that answer is

k

. (Direct proportion:

y \div x

constant. Inverse:

x \times y

constant.) The graph is not a straight line sloping down — it is the curve

y = \frac{k}{x}

, steep near the axes and flattening out, never touching either axis: however large

x

gets,

y

is small but never zero.

It shows up wherever a total is fixed: a journey of fixed distance (speed

\times

time is constant — faster means proportionally shorter, and speed–time questions at a fixed distance are inverse proportion), sharing a fixed prize between more winners, emptying a tank with more pumps.