Solve inverse proportion problems using the unitary method
Apply inverse proportion to real-life contexts (workers, speed, time)
Real-World Applications
Engineering: More machines working on a production run means each machine runs for less time to meet the same quota — a key concept in manufacturing planning.
Construction: More workers on a building site means the job is completed faster — double the workers, half the time.
Transport: Higher speed means a shorter journey time for the same distance — double the speed, half the time.
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Two quantities are in inverse proportion when one increases at the same rate as the other decreases. When one quantity doubles, the other halves. When one triples, the other becomes one third.
Notice that Workers × Days = 20 every time. The total amount of work (20 worker-days) never changes — only how it is shared out. That is the key feature of inverse proportion.
💡 In plain words: double the workers and the time halves; three times as many workers and it takes a third of the time. The job stays the same size — you just have more (or fewer) people sharing it.
✓ Quick Check 1 — What is Inverse Proportion?
Question 1 of 10
Direct vs Inverse Proportion
Direct Proportion (recap)
As one quantity increases, the other increases too.
More items bought → more money spent
Inverse Proportion
As one quantity increases, the other decreases.
More workers → less time to finish
The Unitary Method
Steps:
Find the total amount of work by multiplying the two values you are given
Divide that total by the new value to find the answer
⚠️ Common mistake: Multiplying when you should divide (or vice versa). Always ask: "As one value gets bigger, does the other get bigger or smaller?" If smaller → it's inverse proportion.
Example A: 4 workers take 6 days to complete a job. How long would 8 workers take?
Step 1 — Find the total work
Total work = 4 workers × 6 days = 24 worker-days
Step 2 — Divide by new number of workers
Time = 24 ÷ 8 = 3 days
Answer: 3 days (double the workers → half the time ✓)
Example B: 3 machines take 10 hours to produce a batch. How long would 5 machines take?
Step 1 — Total work
3 × 10 = 30 machine-hours
Step 2 — Divide by new number
30 ÷ 5 = 6 hours
Answer: 6 hours
Example C: A journey takes 3 hours at 60 mph. How long at 90 mph? (Same distance)
Step 1 — Find the distance
Distance = speed × time = 60 × 3 = 180 miles
Step 2 — Find the new time
Time = distance ÷ speed = 180 ÷ 90 = 2 hours
Answer: 2 hours
✓ Quick Check 2 — The Unitary Method
Question 1 of 10
Practice: Word Problems
💡 For each problem: multiply the two values to find the total amount of work, then divide that total by the new value.
✓ Quick Check 3 — Mixed Inverse Proportion
Question 1 of 10
Practice: Extra Questions
Work out each answer, then type it in and click Check to see if you're right.
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Question
Your answer
Result
1
5 workers take 12 days to complete a job. How many days would 3 workers take?
2
4 pipes fill a tank in 9 hours. How many hours would 6 pipes take?
3
3 machines take 20 minutes to print 600 leaflets. How long would 5 machines take?
4
6 painters take 8 days to paint a building. How many days would 4 painters take?
5
2 workers take 15 days. How many days would 5 workers take?
6
8 taps fill a bath in 3 minutes. How many minutes would 4 taps take?
7
10 machines take 6 hours to finish a batch. How many hours would 15 machines take?
8
12 workers take 5 days to finish a job. How many days would 20 workers take?
9
6 pumps drain a pool in 10 hours. How many hours would 4 pumps take?
10
9 chefs take 4 hours to prepare a banquet. How many hours would 6 chefs take?
⚠ Things to Watch Out For
Treating as direct proportion: More workers → less time. If you find "more workers → more time" you've applied direct proportion by mistake. Always check: does increasing one quantity increase or decrease the other?
Not finding the total (k) first: Use the unitary method: find what ONE unit gives, then scale. 4 workers take 15 days → 1 worker takes 60 days → 6 workers take 60÷6 = 10 days.
Inverse proportion graphs: The graph is a curved hyperbola (NOT a straight line). If you draw a straight line, it is not inverse proportion.
Applying direct proportion formula: Do not use y = kx for inverse proportion. Use y = k/x (equivalently xy = k, a constant).
Rounding mid-calculation: Calculate the total (k) with full precision before dividing. Rounding early introduces cumulative error.