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Ratio & Proportion · Higher

Algebraic direct & inverse proportion

Algebraic proportion expresses relationships using equations with constants. y can be proportional to x, x², x³, √x, or their reciprocals. You find the constant k from given values and use it to answer further questions.

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Revising algebraic direct & inverse proportion the night before an exam? These are condensed revision notes — the facts, formulas and mistakes examiners actually mark, with no long explanations. Prefer a full walkthrough with step-by-step reasoning instead? Search “Algebraic direct & inverse proportion GCSE” on the site for the complete guide.

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Key facts to remember

  • 1Direct proportion: y ∝ x means y = kx.
  • 2y ∝ x² means y = kx²; y ∝ √x means y = k√x.
  • 3Inverse proportion: y ∝ 1/x means y = k/x; y ∝ 1/x² means y = k/x².
  • 4To find k: substitute the given pair of values into the equation.
  • 5Once k is found, use the equation to find any unknown value.
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Formulas

Direct proportion
y = kxⁿ

n depends on the type of proportion

Inverse proportion
y = k/xⁿ
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Worked examples

Example 1

y is directly proportional to x². When x = 3, y = 36. Find y when x = 5.

Working

  1. y = kx²
  2. 36 = k × 9 → k = 4
  3. y = 4x²
  4. When x = 5: y = 4 × 25 = 100
Answery = 100
Example 2

y is inversely proportional to √x. When x = 16, y = 3. Find y when x = 4.

Working

  1. y = k/√x
  2. 3 = k/√16 = k/4 → k = 12
  3. y = 12/√x
  4. When x = 4: y = 12/√4 = 12/2 = 6
Answery = 6
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Common mistakes

✗Writing y = kx when the question says y ∝ x² — read the type of proportion carefully.
✗Forgetting to find k before attempting to calculate the unknown.
✗Substituting into the wrong equation (e.g. using y = kx instead of y = k/x for inverse proportion).
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Exam tips

✓Read the proportion statement carefully to identify the correct equation form before finding k.
✓Always state your equation with k substituted in (e.g. y = 4x²) before finding the unknown value.

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