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Mathematics 0580

Quick revision notes · Mathematics 0580 · Chapter 21 of 37

Proportionality

Direct and inverse proportion with ∝ and k, including squares, cubes and roots.

3 sectionsWorked examples Common mistakesIGCSE: whole chapter is Extended tier
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At a glance

Direct and inverse proportion with ∝ and k, including squares, cubes and roots.

  1. Direct proportion
  2. Inverse proportion
  3. The effect of a change

1Direct proportion

  • y ∝ x means y = kx: when x doubles, y doubles. The graph is a straight line through the origin.
  • Other direct forms: y ∝ x2 (y = kx2), y ∝ x3, y ∝ √x, y ∝ ∛x.
  • Find k from a given pair of values, then use the formula.

Method for every proportion question

  1. Write the formula with k.
  2. Substitute the pair you are given to find k.
  3. Write the formula again with the number for k.
  4. Substitute the new value.

Formulas

y ∝ x ⇒ y = kx
y ∝ x2 ⇒ y = kx2
y ∝ √x ⇒ y = k√x

Worked example

y is directly proportional to the square of x. y = 18 when x = 3. Find y when x = 5.

  1. y = kx2.
  2. 18 = k × 9, so k = 2.
  3. y = 2x2.
  4. y = 2 × 25.

y = 50

2Inverse proportion

  • y ∝ 1/x means y = k/x: when x doubles, y halves. xy is always the same.
  • Other inverse forms: y ∝ 1/x2, y ∝ 1/√x.
  • The graph of y = k/x is a curve that gets close to both axes.

Formulas

y ∝ 1/x ⇒ y = k/x
y ∝ 1/x2 ⇒ y = k/x2

Worked example

The force F is inversely proportional to the square of the distance d. F = 20 when d = 3. Find F when d = 6.

  1. F = k/d2.
  2. 20 = k/9, so k = 180.
  3. F = 180/36.

F = 5

3The effect of a change

  • If y ∝ xn and x is multiplied by m, then y is multiplied by mn.
  • Inverse: if y ∝ 1/xn and x is multiplied by m, y is divided by mn.
  • Percentage changes work the same way: x up 10% means x × 1.1.
Relationshipx doubled → yx tripled → y
y ∝ x× 2× 3
y ∝ x2× 4× 9
y ∝ x3× 8× 27
y ∝ 1/x÷ 2÷ 3
y ∝ 1/x2÷ 4÷ 9

Worked example

y ∝ x3. x is increased by 10%. By what percentage does y increase?

  1. x is multiplied by 1.1.
  2. y is multiplied by 1.13 = 1.331.

33.1%

✗Common mistakes

Don’t writeWrite instead
Using y = kx for 'proportional to the square' Square it: y = kx2
Inverse proportion written y = x/k It is y = k/x
Not squaring when substituting into y = kx2 18 = k × 32, not 18 = k × 3
Forgetting to write the formula with k Write y = 2x2 — it is often worth a mark

★In the exam

  • Read the wording carefully: 'square', 'square root', 'cube', 'inversely'.
  • Always write the formula with your value of k in it before the final step.
  • Check: a larger x should make y bigger (direct) or smaller (inverse).