Knowledge Organiser: Exponential Growth and Decay
Key Terms
- Exponential growth: Quantity increases by a fixed percentage each period
- Exponential decay: Quantity decreases by a fixed percentage each period
- Multiplier: Growth uses (1 + r); decay uses (1 − r), where r is the decimal rate
- Half-life: Time taken for a quantity to halve (decay to 50%)
- Compound interest: Interest calculated on the accumulated amount each period
- N₀: The initial (starting) amount
Must-Know Facts
- Growth multiplier for r%: (1 + r/100); decay multiplier: (1 − r/100)
- 5% growth over 4 years ≠ 20% total (it is 21.6%: compound, not linear)
- Doubling every period: N = N₀ × 2ⁿ
- Half-life: N = N₀ × (0.5)ⁿ where n = time ÷ half-life period
- n is the number of TIME PERIODS, not the total time
- Growth gives J-shaped curve; decay gives reverse J-shaped curve
Key Formulas
- Exponential growth: N = N₀ × (1 + r)ⁿ
- Exponential decay: N = N₀ × (1 − r)ⁿ
- Doubling: N = N₀ × 2ⁿ
- Half-life: N = N₀ × (0.5)ⁿ
- Compound interest: A = P × (1 + r)ⁿ
- r = rate as a decimal (e.g. 5% → r = 0.05)
Common Mistakes
- Rate as percentage not decimal: Use r = 0.05 for 5%, NOT r = 5 in the formula
- Multiplier for decay: 10% decay uses multiplier 0.9, not 1.1 or 0.1
- Simple vs compound: Simple growth adds the same amount each time; exponential multiplies: don't confuse them
- n is number of periods: n = number of years/periods, NOT the final year number
- Rounding too early: Keep full calculator value until the final step to avoid rounding errors