Inverse proportion

In inverse proportion, as one quantity increases, the other decreases at the same rate. Their product is constant. The graph of inverse proportion is a reciprocal curve.

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Key facts to remember

1If y is inversely proportional to x: y = k/x (where k is the constant of proportionality).
2The product xy = k is constant.
3As x doubles, y halves; as x triples, y is divided by 3.
4The graph of y against x is a reciprocal curve (hyperbola).
5To find k: substitute a known pair of values.

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Formulas

Inverse proportion

y = k/x or xy = k

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Worked examples

Example 1

y is inversely proportional to x. When x = 4, y = 15. Find y when x = 12.

Working

y = k/x → k = xy = 4 × 15 = 60
When x = 12: y = 60 ÷ 12 = 5

Answery = 5

Example 2

6 workers take 10 days to complete a job. How long would 4 workers take?

Working

Inverse proportion: workers × days = constant
k = 6 × 10 = 60
Days for 4 workers = 60 ÷ 4 = 15 days

Answer15 days

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Common mistakes

✗Treating inverse proportion as direct proportion — in inverse proportion, the product is constant, not the ratio.

✗Increasing y when x increases — in inverse proportion, y decreases as x increases.

✗Forgetting to find k before answering the question.

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Exam tips

✓Find the constant k = xy first using the given pair, then use it to answer the question.

✓Check: in inverse proportion, as one doubles, the other halves — use this to verify your answer.

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