Inverse proportion is a Higher tier topic where one quantity increases as the other decreases. The product of the two quantities remains constant — this is the key relationship that distinguishes inverse proportion from direct proportion.
What Is Inverse Proportion?
Two quantities are inversely proportional when one multiplied by the other gives a constant value. If y is inversely proportional to x, then y = k/x, where k is the constant of proportionality. Doubling x will halve y, tripling x will divide y by three, and so on.
You can also have y inversely proportional to x² (y = k/x²) or other powers — these appear on Higher papers and require the same approach but with a different relationship.
On a graph, inverse proportion produces a curved line (a hyperbola) that approaches both axes but never touches them. In a table of values, you can confirm inverse proportion by checking that xy is the same for every pair.
Real-world examples include: the more workers on a job, the less time it takes; the wider a pipe, the lower the water level for a fixed volume.
Key Formulas
Step-by-Step Method
- Write the proportionality relationship (e.g., y ∝ 1/x).
- Convert to an equation with a constant: y = k/x.
- Use a known pair of values to find k.
- Substitute k back into the equation and use it to find the unknown value.
Worked Example 1 — Foundation Level
Question: y is inversely proportional to x. When x = 4, y = 15. Find y when x = 10.
Working: y = k/x 15 = k/4 k = 15 × 4 = 60 When x = 10: y = 60/10 = 6
Answer: y = 6
Worked Example 2 — Higher Level
Question: y is inversely proportional to x². When x = 3, y = 8. Find y when x = 6.
Working: y = k/x² 8 = k/3² = k/9 k = 8 × 9 = 72 When x = 6: y = 72/6² = 72/36 = 2
Answer: y = 2
Worked Example 3 — Exam Style
Question: It takes 6 workers 10 hours to build a wall. How long would it take 15 workers, assuming they work at the same rate?
Working: Workers and time are inversely proportional. k = workers × time = 6 × 10 = 60 Time = k ÷ workers = 60 ÷ 15 = 4
Answer: 4 hours
Common Mistakes
- Confusing direct and inverse proportion. In direct proportion, both quantities increase together. In inverse proportion, one goes up while the other goes down.
- Forgetting to square x when y ∝ 1/x². If the question says "inversely proportional to the square of x," you must square x before dividing.
- Using subtraction instead of division. Inverse proportion means multiplying gives a constant, not that the difference is constant.
- Not recognising inverse proportion in context. Clues include: more workers = less time, faster speed = shorter journey, wider pipe = lower height of water.
Exam Tips
- The product xy = k is constant for simple inverse proportion — use this as a quick check.
- Always write the proportionality statement first, then convert to an equation before substituting.
- For graphs: a reciprocal curve (hyperbola) indicates inverse proportion; a straight line through the origin indicates direct proportion.
- If you are given a table of values, multiply x by y for each pair — if all products are equal, it is inverse proportion.
Practice Questions
Q1 (Foundation): y is inversely proportional to x. When x = 5, y = 12. Find y when x = 20.
Q2 (Foundation): 8 machines can produce an order in 3 hours. How long would 12 machines take?
Q3 (Higher): y is inversely proportional to x². When x = 2, y = 50. Find x when y = 2.
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Related Topics
Summary
- Inverse proportion means one quantity increases as the other decreases, with a constant product.
- y ∝ 1/x gives y = k/x, where k = xy.
- y ∝ 1/x² gives y = k/x², where k = yx².
- Find k using a known pair of values, then use the equation to find unknowns.
- The graph of inverse proportion is a hyperbola that never touches the axes.
Test your understanding
5 quick MCQs to identify any misconceptions on this topic.
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