If y is inversely proportional to x, then x × y is the same number for every pair, and you can write y = k/x. Workers and days, speed and time, and number of cups and size of cup are everyday cases, and this lesson belongs with ratio and proportional reasoning.
What is the idea behind it?
Some totals are fixed. A job needs a set amount of work, a journey covers a set distance, and a jug holds a set volume. When one quantity goes up, the other must go down so the total stays the same.
That fixed total is the product x × y. Doubling x means y must halve, and tripling x means y is divided by 3.
How to recognise and use it, step by step
- Multiply x by y for every pair in the table.
- Compare the products. If they are all the same, the relationship is inverse proportion.
- Write the rule y = k/x, where k is the product.
- Use the rule: a missing y is k ÷ x, and a missing x is k ÷ y.
Worked example
| Workers (x) | 2 | 3 | 5 |
|---|---|---|---|
| Days (y) | 30 | 20 | 12 |
Step 1, multiply: 2 × 30 = 60, 3 × 20 = 60 and 5 × 12 = 60.
Step 2, compare: all three products are 60, so days are inversely proportional to workers. The fixed total is 60 worker-days.
Step 3, rule: y = 60/x.
Step 4, use it: with 4 workers, y = 60 ÷ 4 = 15 days. With 12 workers, y = 60 ÷ 12 = 5 days.
Check: 4 × 15 = 60 and 12 × 5 = 60. More workers give fewer days, as expected.
The mistake to watch for
A common slip is to treat the situation as direct proportion.
Mistaken working: 5 workers take 12 days, so 10 workers take 12 × 2 = 24 days.
Doubling the workers cannot double the time taken.
The correct reasoning is that the total is 5 × 12 = 60 worker-days. With 10 workers, the time is 60 ÷ 10 = 6 days. If the answer moves the opposite way from common sense, the method is wrong, so always ask whether more should give less.
Check yourself
1. 8 pumps empty a tank in 9 hours. How long will 12 pumps take, at the same rate?
Show answer
Total: 8 × 9 = 72 pump-hours. With 12 pumps: 72 ÷ 12 = 6 hours.
Check: 12 × 6 = 72.
2. Is y inversely proportional to x? If so, find y when x = 6.
| x | 2 | 4 | 8 |
|---|---|---|---|
| y | 12 | 6 | 3 |
Show answer
2 × 12 = 24, 4 × 6 = 24 and 8 × 3 = 24. All equal, so yes, y = 24/x.
When x = 6, y = 24 ÷ 6 = 4.
3. A car travelling at 60 km/h takes 3 hours for a journey. How long would it take at 90 km/h?
Show answer
Distance: 60 × 3 = 180 km. Time at 90 km/h: 180 ÷ 90 = 2 hours.
Check: 90 × 2 = 180.
Where this leads next
Go back to modelling direct proportion from a table and compare the two tests side by side, then try the mixed practice set. The non-calculator working trainer supports the division steps.
If you sometimes pick the wrong model in a word problem, online one-to-one Mathematics tuition lets a teacher hear your reasoning and correct it while you work.