Proportion

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Express direct and inverse proportion in algebraic terms and use this form of expression to find unknown quantities.
  2. Work with different types of proportion including linear, square, square root, cube, and cube root proportion.
  3. Use the proportional symbol (∝) correctly.

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What is Proportion?

Proportion is a mathematical relationship between two quantities (variables) that describes how they change together. When two quantities are proportional, there is a consistent pattern in how one changes when the other changes.

There are two main types of proportion you need to understand:

  • Direct proportion – both quantities increase or decrease together
  • Inverse proportion – when one quantity increases, the other decreases

The Proportionality Symbol (∝)

The symbol ∝ means "is proportional to". This is an important mathematical symbol you must recognize and use.

For example:

  • "y is proportional to x" is written as: y ∝ x
  • "M is proportional to L" is written as: M ∝ L

This symbol tells you there is a proportional relationship, but it doesn't give you the exact equation yet. To create an equation, you replace the ∝ symbol with "= k", where k is a special number called the constant of proportionality.

Direct Proportion

What is Direct Proportion?

Direct proportion means that as one variable increases, the other variable increases by the same factor. Similarly, if one decreases, the other decreases by the same factor.

Real-life example: If you buy apples at £2 per kilogram, then:

  • 2 kg costs £4
  • 3 kg costs £6
  • 4 kg costs £8

The cost is directly proportional to the weight. If you double the weight, you double the cost.

Linear Direct Proportion (Basic Form)

When y is directly proportional to x, we write:

y ∝ x

This becomes the equation:

y = kx

Here, k is the constant of proportionality – it's a fixed number that stays the same for the relationship.

Key point: The graph of y = kx is a straight line passing through the origin (0, 0).

Example: If y ∝ x, and when x = 5, y = 20, find the equation connecting y and x.

Solution:

  • Start with: y = kx
  • Substitute the values: 20 = k × 5
  • Solve for k: k = 20 ÷ 5 = 4
  • Write the equation: y = 4x

Direct Proportion with Squares

Sometimes one variable is proportional to the square of another variable.

When y is directly proportional to the square of x, we write:

y ∝ x²

This becomes the equation:

y = kx²

Example: If y ∝ x², and when x = 3, y = 18, find y when x = 5.

Solution:

  • Start with: y = kx²
  • Substitute x = 3, y = 18: 18 = k × (3)² = k × 9
  • Solve for k: k = 18 ÷ 9 = 2
  • Write the equation: y = 2x²
  • Now find y when x = 5: y = 2 × (5)² = 2 × 25 = 50

Direct Proportion with Square Roots

When y is directly proportional to the square root of x, we write:

y ∝ √x

This becomes the equation:

y = k√x

Example: If y ∝ √x, and when x = 9, y = 12, find the equation.

Solution:

  • Start with: y = k√x
  • Substitute x = 9, y = 12: 12 = k × √9 = k × 3
  • Solve for k: k = 12 ÷ 3 = 4
  • Write the equation: y = 4√x

Direct Proportion with Cubes

When y is directly proportional to the cube of x, we write:

y ∝ x³

This becomes the equation:

y = kx³

Example: If M ∝ L³, and when L = 2, M = 24, find M when L = 3.

Solution:

  • Start with: M = kL³
  • Substitute L = 2, M = 24: 24 = k × (2)³ = k × 8
  • Solve for k: k = 24 ÷ 8 = 3
  • Write the equation: M = 3L³
  • Now find M when L = 3: M = 3 × (3)³ = 3 × 27 = 81

Direct Proportion with Cube Roots

When y is directly proportional to the cube root of x, we write:

y ∝ ∛x

This becomes the equation:

y = k∛x

Example: If y ∝ ∛x, and when x = 8, y = 10, find the equation.

Solution:

  • Start with: y = k∛x
  • Substitute x = 8, y = 10: 10 = k × ∛8 = k × 2
  • Solve for k: k = 10 ÷ 2 = 5
  • Write the equation: y = 5∛x

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