Question
A car travelling at 60 km/h covers a distance in 4 hours. How long will it take if the speed is increased to 80 km/h?
Also: 12 workers can build a wall in 10 days. How many days will 15 workers take?
Two problems, one concept — we’ll see exactly why they behave differently.
Solution — Step by Step
First, ask yourself: as one quantity increases, does the other increase or decrease?
- Speed ↑, Time ↓ → they move in opposite directions → Inverse proportion
- Workers ↑, Days ↓ → same pattern → also Inverse proportion
This single check saves you from setting up the ratio upside-down.
In inverse proportion, the product stays constant. So:
This works because distance is fixed. More speed means the same distance is covered in less time.
Substituting:
Answer: The car takes 3 hours at 80 km/h.
More workers → fewer days, so again inverse proportion:
Answer: 15 workers will finish the wall in 8 days.
Why This Works
The underlying idea is that the total work or total distance stays constant. In Part 1, distance = speed × time. When speed goes up, time must come down by the same factor — otherwise the product (distance) would change.
In Part 2, think of “total work” as worker-days. 12 workers × 10 days = 120 worker-days of effort. If you hire 15 workers, you still need 120 worker-days of total effort, so each worker works fewer days.
This is why the product rule — — is the go-to tool for inverse proportion. Write it down first, fill in three values, solve for the fourth.
Use when: more of one quantity means less of the other.
Alternative Method — Unitary Method
Some students prefer this approach, and it’s great for checking your answer.
Part 1 (Speed and Time):
At 60 km/h → time = 4 hours At 1 km/h → time = 4 × 60 = 240 hours (to cover the same distance at walking pace!) At 80 km/h → time = 240 ÷ 80 = 3 hours ✓
Part 2 (Workers and Days):
12 workers → 10 days 1 worker → 10 × 12 = 120 days (one person doing everything alone) 15 workers → 120 ÷ 15 = 8 days ✓
The logic: first bring one quantity to 1, then scale back up. It’s slower than the product rule but makes the reasoning very visible.
Common Mistake
The most common error is treating all proportion problems as direct proportion and writing:
For the speed problem, this gives , which gives hours — meaning a faster car takes longer. That’s physically absurd, but many students don’t pause to check.
Always ask: “If this quantity goes up, does the other go up or down?” If down → invert one ratio (or use the product rule).
Quick check after solving: Does your answer make physical sense?
Faster speed → should give fewer hours. More workers → should give fewer days. If your answer goes the wrong way, you’ve used the wrong proportion type.