Direct and inverse proportion

Direct proportion keeps a ratio constant

A constant-rate machine makes 15 parts in 3 minutes and 25 in 5 minutes. In each case output divided by time is 5 parts per minute. The model is p = 5t, and its graph passes through the origin. Doubling time doubles output under the model’s assumptions.

A taxi model C = 4 + 2d is not direct proportion between cost and distance: the fixed charge means cost divided by distance changes. A straight-line graph alone does not prove direct proportion.

Inverse proportion keeps a product constant

Six equally productive workers complete a fixed job in 8 hours. The model requires 6 × 8 = 48 worker-hours. With 12 workers at the same rate and no interference, time is 48÷12 = 4 hours. Worker count and time are inversely proportional: doubling one halves the other.

The model needs the same job and productivity. It is not a universal claim that adding people halves every task’s time.

Check your understanding 1 Which quantity stays fixed?

A journey of 180 km is modelled at a constant speed. Find the time at 60 km/h and at 90 km/h. Is time directly or inversely proportional to speed?

Your answer or drawing on paper

Show answer

Times are 180/60 = 3 h and 180/90 = 2 h. The product of speed and time stays 180 km, so time is inversely proportional to speed for this fixed distance.