Mathematics · Ratio, proportion and percentages

Inverse proportion

Why does a job finish sooner when more people work on it? Learn why the product of inversely proportional quantities stays constant, solve speed-time and worker-day problems, and see why not every decreasing relationship is inverse proportion.

20 min
1 simulation
Mini quiz

Updated: 21 October 2026

Preview

The question

Volunteer pupils are getting together to paint the wall in the school garden. 6 pupils can paint the wall in 8 hours. But only 4 pupils can come this weekend. How many hours will the wall take? Fewer people means more time, but how much more?

Try it yourself

No account needed: change the values and check your answer. Some simulations ask for a prediction before they start; it doesn't matter whether it's right or wrong.

Number line

Garden clean-up

Drag the point or use the buttons to move in steps of 1 hour. When you reach the right value, press Check.

Predict first

What happens to the time if the number of people cleaning the garden goes from 8 to 12?

It doesn’t matter whether you’re right; the simulation opens once you choose.

Lab

The gearbox

Open full page in the Lab →

Fig. 1The gearbox

Change the teeth on the right gear, watch both speeds and the product

n₂ × z₂ = 720 teeth/min

Teeth z₂12 teeth

Crank speed n₁30 rpm

Your own values

z₁ (teeth)

z₂ (teeth)

n₁ (rpm)

Teeth are whole numbers from 6 to 200, crank speed 1–1,000 rpm. A value applies when you press Enter or leave the box.

Product stays the same

k = 720 teeth/min

n₁ × z₁: 30 × 24 = 720
n₂ × z₂: 60 × 12 = 720

The ratios are inverted: z₂ : z₁ = 1/2 → n₂ : n₁ = 2

n₂ – z₂ graph

Shaded area = n₂ × z₂

Learn

Progress is kept on this page only; no points are given.

1Predict
2Try
3Explain

If you double the teeth on the right gear, from 12 to 24, what happens to its turns per minute?

Because the gears mesh without slipping, the same number of teeth passes the mesh point each minute on both gears: n₁ × z₁ = n₂ × z₂. With the crank speed and z₁ fixed, this number (k) does not change; double the teeth on the driven gear and its speed halves. That is why the graph is not a straight line but a curve that never reaches zero (n₂ = k ÷ z₂). The gears are drawn with a 20° involute profile and equal module.

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Frequently asked

5 questions

If one quantity grows while the other shrinks, are they inversely proportional?

Not always. As you spend from 100 TL, the money left goes down, but here the sum is constant: spent + left = 100. 20 × 80 = 1600 while 50 × 50 = 2500; the product changes, so this is not inverse proportion. In inverse proportion the product stays the same.

What stays the same in an inverse proportion?

The product of the two quantities; often it is the total amount of work. The wall needs 6 × 8 = 48 student-hours of work, and that 48 does not change however many students come. If 4 students come, the job takes 48 ÷ 4 = 12 hours.

How can I tell that I solved an inverse problem as if it were direct?

Check the answer with common sense. With fewer students, the job should take longer. Saying "4 students finish in 4 × 8 ÷ 6 ≈ 5.3 hours" means fewer people finish faster, which makes no sense. The correct answer is 12 hours.

Why does the graph of an inverse proportion never reach zero?

Since y = k ÷ x, y gets smaller as x grows, but however large the number you divide k by, the result stays above zero. That is why the graph is not a straight line but a curve that falls to the right and never reaches zero.

In real life, are the number of workers and the time always exactly inversely proportional?

Only if everyone works at the same speed and nobody gets in anyone's way. Inverse proportion is a model; the lesson's example also assumes the students work at the same speed. In reality a very crowded team can get in each other's way, and then the calculation gives only a rough estimate.