Growth and decay

When something grows or shrinks by the same percentage each time, it changes exponentially. Populations, savings, cooling drinks and radioactive samples all behave this way.

Key facts

  • Growth: N = N₀ × (1 + r)ᵗ. Decay: N = N₀ × (1 − r)ᵗ.
  • Half-life: the time to halve, whatever the starting amount.
  • Use trial and improvement or logs to find how long it takes to reach a target.

Worked examples

Example 1

A town of 12 000 people grows by 2.5% a year. How many after 6 years?

  1. Multiplier: 1.025.
  2. 12000 × 1.025⁶ = 13916.3…
  3. About 13 916 people.

Answer: 13 916

Example 2

A drink cools from 80°C losing 10% of its temperature above room temperature (20°C) each minute. What is it after 3 minutes?

  1. Difference above room temperature: 60°C.
  2. 60 × 0.9³ = 43.74
  3. Add room temperature back. 20 + 43.74 = 63.74°C

Answer: 63.7°C