Rates, ratios & proportions · Ages 13–14
Inverse proportion
Inverse proportion: x × y stays the same.
How to do it
Method
Inverse proportion: x × y stays the same
6 × 10 = 60, 60 ÷ 15 = 4
Another way
The multiplier
15 ÷ 6 = 2.5, 10 ÷ 2.5 = 4
In full
One quantity doubles and the other halves; one triples and the other becomes a third: the two are in inverse proportion, x × y is always the same, so xy = k for a constant k. 6 people take 10 days over a job, each working as fast as the others: how long do 15 take? One way, the whole job first: 6 × 10 = 60 person-days, and 60 ÷ 15 = 4 days. Another, the multiplier: there are 15 ÷ 6 = 2.5 times as many people, so it takes 12.5 of the time, 10 ÷ 2.5 = 4 days. Over the same distance, speed and time are inversely proportional too: double the speed, half the time. Careful: one rising as the other falls is not enough; the product must stay the same.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
y is in inverse proportion to x, and y = 14 when x = 4. In xy = k, k = ; when x = 8, y = .
Fill in each blank
Show the answer and working
Answer: 56, 7
k = xy = 4 × 14 = 56; y = 56 ÷ 8 = 7.
Example 2
2 people take 33 days over a job. Working as fast, 6 people take days
Show the answer and working
Answer: 11
The whole job is 2 × 33 = 66 person-days, and 66 ÷ 6 = 11 days.
Example 3
In which set of values is y in inverse proportion to x?
- A. (x, y) = (1, 14), (2, 8), (4, 5)
- B. (x, y) = (1, 12), (2, 6), (4, 3)
- C. (x, y) = (1, 3), (2, 6), (4, 12)
- D. (x, y) = (1, 15), (2, 8), (4, 4)
Choose the right answer
Show the answer and working
Answer: B. (x, y) = (1, 12), (2, 6), (4, 3)
Inverse proportion needs the same x × y for every pair: 1 × 12 = 2 × 6 = 4 × 3 = 12.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.y is in inverse proportion to x. When x = 4, y = 20, so when x = 16, y = 5.
True or false?
Answer:
2.In which set of values is y in inverse proportion to x?
- A. (x, y) = (1, 24), (2, 12), (4, 6)
- B. (x, y) = (1, 33), (2, 21), (4, 15)
- C. (x, y) = (1, 6), (2, 12), (4, 24)
- D. (x, y) = (1, 27), (2, 14), (4, 7)
Choose the right answer
Answer:
3.y is in inverse proportion to x, and y = 9 when x = 2. In xy = k, k = ; when x = 3, y = .
Fill in each blank
Answer:
4.At 60 km/h the trip takes 2 hours. To cover the same distance in 3 hours, the speed must be km/h
Answer:
5.y is in inverse proportion to x. If x goes up 100%, what happens to y?
- A. It goes down 100%
- B. It stays the same
- C. It goes up 100%
- D. It goes down 50%
Choose the right answer
Answer:
All the answers
- 1. True
- 2. A. (x, y) = (1, 24), (2, 12), (4, 6)
- 3. 18, 6
- 4. 40
- 5. D. It goes down 50%
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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