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IGCSE Maths Direct & Inverse Proportion Notes

In direct proportion two quantities rise together; in inverse proportion one rises as the other falls. Both are modelled with a constant k.

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Key concepts

Direct proportion
y ∝ x means y = kx; the graph is a straight line through the origin.
Inverse proportion
y ∝ 1/x means y = k/x; the graph is a reciprocal curve.
Powers
Relationships may involve x², √x, etc., e.g. y = kx².

Key formulas

Direct

y = kx

Inverse

y = k/x

Worked example

y is inversely proportional to x. y = 4 when x = 3. Find y when x = 6.

  1. 1Write y = k/x and find k: 4 = k/3, so k = 12.
  2. 2Use y = 12/x with x = 6.
  3. 3Calculate y.

Answer: y = 2

Common mistakes

✗ Using y = kx for an inverse relationship.

✓ Inverse proportion uses y = k/x.

✗ Not finding k before answering.

✓ Always calculate the constant from the given pair first.

✗ Ignoring a power in the relationship.

✓ Read whether it is proportional to x, x² or √x.

Quick check

Try these, then reveal the answers.

If y = kx and y = 15 when x = 3, find k.Reveal answer

5

y = k/x, k = 20. Find y when x = 4.Reveal answer

5

Direct-proportion graph shape?Reveal answer

Straight line through the origin

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Exam tips

  • Translate the proportion statement into an equation first.
  • Find k using the given values before answering the question.
  • Check the answer direction — inverse means larger x gives smaller y.

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Related topics

Ratio & ProportionStraight-Line Graphs

Key terms explained

Short, exam-ready definitions with a formula, a worked example and an examiner tip.

What is area under a curve?What is chain rule?What is completing the square?What is definite integral?What is differentiation?What is discriminant?Full glossary

Frequently asked questions

How do I know if it is direct or inverse proportion?+

If both quantities increase together it is direct (y = kx); if one increases as the other decreases it is inverse (y = k/x).

What does the symbol ∝ mean?+

It means 'is proportional to' — you convert it to an equation by introducing the constant k.

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