Growth and decay
- Repeated percentage change: amount = starting amount × multiplierⁿ, where n is the number of times it changes.
- Growth uses a multiplier greater than 1. Decay uses a multiplier between 0 and 1.
- 2000 bacteria grow by 15% an hour. After 4 hours there are 2000 × 1.15⁴ = 3498 (to the nearest whole number).
Finding the number of years
- Keep multiplying until you pass the target. £1000 at 6% compound interest is worth £1898.30 after 11 years and £2012.20 after 12 years, so it first doubles after 12 years.
Finding the rate
- A value goes from 400 to 484 in 2 years, changing by the same percentage each year. The multiplier squared is 484 ÷ 400 = 1.21, so the multiplier is 1.1 and the rate is 10% a year.
Other repeated processes
- Some processes follow a rule such as Pₙ₊₁ = 1.2Pₙ − 50. Start with P₀ and apply the rule again and again.
- A lake has 500 fish: P₁ = 1.2 × 500 − 50 = 550, and P₂ = 1.2 × 550 − 50 = 610.
- Something that halves every hour has a multiplier of 0.5 each hour: after 3 hours, 1/8 is left.
Key terms
- Growth
- An increase by the same percentage each time period.
- Decay
- A decrease by the same percentage each time period.
- Multiplier
- The number you multiply by for each change, such as 1.15 or 0.88.
- Exponential growth
- Growth where the amount is multiplied by the same number each time.
- Half-life
- The time taken for an amount to halve.