0580

Proportion

Algebra and Sequences

The symbol and the constant

  • Two quantities are in proportion when a change in one forces a matching change in the other by a fixed rule
  • The symbol ∝ is read "is proportional to", and knowing it is a stated requirement of the syllabus
  • A statement using ∝ is not an equation, so it cannot be calculated with until it is turned into one
  • Turning it into an equation introduces a fixed number, written k, that stays the same for every pair of values in the question
  • Once k is known the relationship is completely determined, and any further part of the question is a substitution

The four steps that answer every question

  • Write the equation with k in it, taken straight from the wording
  • Substitute the pair of values you are given and solve for k
  • Rewrite the equation with that number in place of k
  • Use the finished equation to answer whatever the question asks next
  • Marks are awarded step by step, so the equation containing k scores before any number is worked out
Worked example

The four steps in order

y is proportional to x. When x = 3, y = 12. Find y when x = 7.

Solution:

  • Write the proportion statement as an equation with k in it — this line scores on its own
  • y = kx
  • Substitute the given pair to find k: 12 = k × 3
  • k = 4
  • Write the full equation: y = 4x
  • Substitute the new value: y = 4 × 7 = 28
  • Finding the ratio 12 ÷ 3 in your head gives the same 4, but writing the k line down is what earns the mark