Exponential Growth and Decay
O Level and IGCSE Mathematics Cheat Sheet – 1.17 Exponential Growth and Decay
Key Idea
- Exponential growth: Quantity increases by the same percentage rate at regular intervals.
- Exponential decay: Quantity decreases by the same percentage rate at regular intervals.
- Formula form:
New Value = Initial Value × (multiplier)ⁿ- n = number of time periods
- Multiplier = (1 + rate/100) for growth, (1 − rate/100) for decay
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Growth and Decay Multipliers Table
| Situation | Rate (%) | Multiplier |
|---|---|---|
| Growth 5% | 5 | 1.05 |
| Growth 12% | 12 | 1.12 |
| Decay 8% | 8 | 0.92 |
| Decay 15% | 15 | 0.85 |
Worked Examples
| Example Type | Data | Steps | Answer |
|---|---|---|---|
| Population Growth | Start: 2000 people, Growth: 4% per year, Period: 6 years | Multiplier = 1.04 → 2000 × (1.04)⁶ | 2532.98 ≈ 2533 people |
| Depreciation | Car: $15,000, Decay: 20% per year, Period: 3 years | Multiplier = 0.8 → 15000 × (0.8)³ | $7680 |
| Repeated Growth | Value: 500, Growth: 10% per year, 2 years | 500 × 1.1² | 605 |
| Repeated Decay | Value: 800, Decay: 5% per month, 4 months | 800 × 0.95⁴ | 665.45 |
Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change O Level And IGCSE Mathematics Full Scale Course
Special Notes
- Growth rate over 100% still works: 120% growth = multiplier 2.20.
- Multiplier between 0 and 1 → decay, multiplier > 1 → growth.
- Works for any interval (years, months, days) — keep rate per interval consistent.
Exam Special Tips
- Always find multiplier before calculation — avoids step-by-step % increase/decrease.
- For multi-stage problems, multiply all stage multipliers together.
- Avoid rounding until the end to maintain accuracy.
- Check if question asks for exact value or rounded to certain s.f.
Common Mistakes & Confusions
| Mistake | Why It’s Wrong | Correct Approach |
|---|---|---|
| Using rate instead of multiplier | Misses exponential effect | Convert rate to multiplier first |
| Adding rate repeatedly | That’s linear change, not exponential | Use powers of multiplier |
| Mixing time units | Multiplier must match time unit | Adjust rate if unit changes |