Inverse Proportion
Two quantities are in inverse proportion when one increases at exactly the rate the other decreases, so their product stays constant: if one quantity doubles, the other halves, the opposite behaviour to direct proportion. Typical inverse proportion situations include a fixed job shared between workers or machines, where more workers means less time each, and a fixed distance travelled at different speeds, where a faster speed means less time taken.
Before you start
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Method
- Decide whether the two quantities are in direct or inverse proportion: if one increasing causes the other to increase at the same rate, it is direct; if one increasing causes the other to decrease so their product stays the same, it is inverse.
- For inverse proportion, multiply each known pair of matching values together to find the constant product.
- To find a missing value, divide the constant product by the known value of the other quantity.
- Use a quick direction check: if the given quantity has gone up, the value being found should have gone down, and vice versa - if the answer moves the wrong way, the wrong operation has been used.
- Common inverse proportion situations include workers or machines completing a fixed job (at a constant rate each), and speed and time for a fixed distance.
- Set out the working clearly as 'constant = quantity 1 x quantity 2', so it can be reused to find any missing value in the same problem.
- Remember that inverse proportion only holds while every worker or machine works at the same, unchanging rate throughout the problem.
Worked example
It takes 5 identical pumps 12 hours to empty a flooded basement, all pumps working at the same constant rate. How many hours would it take 8 of the same pumps to empty the same basement?
- Multiply the known pair of values to find the constant: 5 x 12 = 60 (this represents the total 'pump-hours' needed).
- Divide this constant by the new number of pumps: 60 / 8.
- Calculate: 60 / 8 = 7.5 hours.
- Check the direction: more pumps were used (8 instead of 5), so the time should be shorter - 7.5 hours is indeed shorter than 12 hours, so the answer makes sense.
Practice questions
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Q13 identical taps fill a water tank in 20 minutes, all running at the same rate. How long would it take 5 of the same taps to fill the same tank?Show answer
Answer: 12 minutes (constant = 3 x 20 = 60, then 60 / 5)
Q2A factory job can be completed by 12 machines in 9 hours, all working at the same rate. How many machines, working at the same rate, would be needed to complete the same job in 4 hours?Show answer
Answer: 27 machines (constant = 12 x 9 = 108, then 108 / 4)
Q3A car travels between two towns at an average speed of 60 mph, taking 2.5 hours. Travelling the same distance at an average speed of 75 mph, how long would the journey take?Show answer
Answer: 2 hours (distance = 60 x 2.5 = 150 miles, then 150 / 75)
Q46 identical printers can print a batch of exam papers in 40 minutes, all printing at the same rate. How many minutes would it take 15 of the same printers to print the same batch?Show answer
Answer: 16 minutes (constant = 6 x 40 = 240, then 240 / 15)
Q5It takes 4 painters 15 days to paint a large mural, all working at the same rate. How many days would it take 10 painters, working at the same rate, to paint the same mural?Show answer
Answer: 6 days (constant = 4 x 15 = 60, then 60 / 10)
Q6A charity is packing 900 identical food parcels. Working alone at a constant rate, 1 volunteer would take 180 hours to pack them all. Working together, all at the same rate, how many hours would 9 volunteers take to pack all 900 parcels?Show answer
Answer: 20 hours (constant = 1 x 180 = 180, then 180 / 9)
Q7y is inversely proportional to x. When x = 8, y = 15. Work out the value of y when x = 24.Show answer
Answer: 5 (constant = 8 x 15 = 120, then 120 / 24)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
8 identical bottling machines can fill a batch of juice bottles in 21 minutes, all working at the same rate. (a) Work out how many minutes it would take 6 of the same machines to fill the same batch. (b) One of the 6 machines then breaks down after 10 minutes and cannot be repaired in time. Assuming the remaining machines keep working at the same rate, work out how many more minutes the remaining machines need to finish the batch.
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The time, T hours, taken to drain a swimming pool is inversely proportional to the number of pumps used, n. Using 3 pumps, it takes 10 hours to drain the pool. (a) Find the constant of proportionality, and write a formula for T in terms of n. (b) Use your formula to work out how many pumps would be needed to drain the pool in 2 hours.
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Free printable worksheet
Want more practice on paper? Download the inverse proportion worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 41 of KS3 Maths Workbook 1, the whole course as one free printable PDF.
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