Inverse Proportion

Not everything grows together. Some things get smaller as others get bigger. Ask one friend to paint a fence and it takes all afternoon — ask six friends and it's done before lunch. Drive faster and the journey takes less time. Pour the same juice into more cups and each cup gets less. In every case, as one quantity goes up, its partner goes down.

This is inverse proportion, and it hides a beautifully simple rule: whatever you gain on one side you lose on the other, so their product stays constant. Six painters aren't magic — it's just that the total amount of work hasn't changed, only how it's shared out.

The rule: a fixed product

Two quantities are in inverse proportion when doubling one halves the other. They are linked by

y = \frac{k}{x}

where k is a fixed number. Multiply both sides by x and the same rule reads

x \times y = k

— the product of the two quantities never changes. That single fact does all the work: find k from one pair you know, then you can find the partner of any value.

It's worth pinning this against its opposite, direct proportion, where y = kx and it's the ratio y \div x that stays fixed. Direct: rise together, ratio constant. Inverse: one up, one down, product constant.

Worked example 1 — more workers, less time

A job takes 4 workers 6 hours. How long for 8 workers?

Find the constant: k = x \times y = 4 \times 6 = 24 worker-hours. The job is "worth" 24 worker-hours no matter who does it.
Now use it: with 8 workers, y = \dfrac{k}{x} = \dfrac{24}{8} = 3 hours.

Twice the workers, half the time — exactly what inverse proportion promises. (This assumes every worker pulls their weight equally and nobody gets in the way, which is the usual tidy classroom assumption.)

Worked example 2 — speed and time

A coach travelling at 60 km/h covers a route in 5 hours. How long at 75 km/h?

The distance is fixed, and distance = speed × time, so speed and time are inversely proportional. The constant is the distance: k = 60 \times 5 = 300 km.
At 75 km/h: \text{time} = \dfrac{300}{75} = 4 hours.

Go faster, arrive sooner — but notice it's not "25% faster means 25% less time." The product rule keeps you honest: always go back to x \times y = k.

Worked example 3 — find k, then predict

y is inversely proportional to x, and y = 9 when x = 4. Find y when x = 12.

k = xy = 4 \times 9 = 36.
y = \dfrac{36}{12} = 3.

Three steps every time: find k, write y = k/x, substitute. Nothing else to remember.

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 and never quite touches zero.

The classic slip is confusing inverse with direct proportion. Keep the signals straight:

Inverse: the quantities multiply to a constant. As one doubles, the other halves. The giveaway phrase is "more of one means less of the other" — more workers, less time.
Direct: the quantities rise together and their ratio is constant. "More of one means more of the other" — more hours worked, more pay.

So if you catch yourself thinking "8 workers, so it must take longer," stop: more hands means less time. And never add or subtract to "adjust" — always return to x \times y = k.

A see-saw is inverse proportion you can sit on. To balance, each person's weight × distance from the pivot must be equal — that product is fixed. So a heavy grown-up sits close to the middle and a light child sits far out, and they balance perfectly. Double your weight and you must halve your distance. This is the ancient law of the lever, and it's why a small force on a long spanner can undo a stubborn bolt.

The same trade-off is everywhere. A camera's aperture and shutter time trade off to let in the same light (open wider, close sooner). Bicycle and car gears trade turning speed for turning force. Push down twice as hard on a smaller piston and it moves half as far. Once you spot inverse proportion, you start seeing "trade-offs" everywhere in the real world.