These questions ask how time, speed, workforce or a dimension changes while one total stays fixed. The first decision is whether the setting really keeps the same job, route or area; only then should you use inverse proportion.
Find what remains fixed
Write the known pair side by side and multiply it. That product is your fixed total. Divide it by the new value to find the missing partner. Equivalently, if one quantity is multiplied by a factor, divide the other by that factor. Finish with a direction check: more equal workers should mean less time, and greater speed over the same route should mean less travel time.
| Fixed quantity | Form | Tiny example |
|---|---|---|
| One fixed job | worker count times time equals total worker-time | Seven identical cutters take 33 minutes. Twenty-one cutters take 231 divided by 21, or 11 minutes. |
| One fixed route | speed times time equals route length | At 45 kilometres per hour for 7 hours, the route is 315 kilometres. At 63 kilometres per hour, the time is 5 hours. |
| One fixed rectangle area | length times width equals area | An area of 150 square metres has width 15 metres at length 10 metres, but width 6 metres at length 25 metres. |
When the missing value is the number of workers rather than the time, the method does not change. Find the fixed worker-time first, then divide by the required time. The assumptions matter: workers or machines must be identical, independent and working at a constant rate.
Reject the tempting wrong routes
| Tempting route | How to reject it |
|---|---|
| Give a larger team a longer completion time. | That scales both quantities in the same direction. For the same fixed work, multiply the old pair and divide by the larger team. |
| Halve the time when the number of workers is halved. | The product would become one quarter as large. Halving one factor requires doubling the other to preserve the work. |
| Increase journey time when speed increases on the same route. | Compute the fixed distance first. Dividing that distance by a greater speed must give a shorter time. |
| Choose a shorter time when fewer equal machines are working. | Fewer machines supply less work per unit of time. The fixed product shows that the missing time must rise. |
Try it
Eight identical pumps operating independently can empty a tank in 9 hours. How many hours would 12 such pumps take to empty the same tank? Enter the number of hours.
A rectangle has a fixed area of 84 square metres. Its length is changed from 12 metres to 21 metres. What must its new width be? Enter the number of metres.
Eighteen identical machines working independently finish one order in 5 hours. How many machines are needed to finish that same order in 3 hours? Enter the number of machines.