Anything that grows or decays by the same percentage each period is modelled with a repeated multiplier: final amount = starting amount × multiplierⁿ, where n is the number of periods. Growth uses a multiplier above 1 and decay a multiplier below 1. If the rate changes, multiply by each period's multiplier in turn, and to find how many periods it takes, apply the multiplier repeatedly until the target is passed.
Compound growth and decay
- Final = start × multiplierⁿ
- Growth of r% per period: multiplier 1 + r/100
- Decay of r% per period: multiplier 1 − r/100
- Different rates: multiply by each multiplier in turn
Compound growth
Populations, bacteria and investments often grow by a fixed percentage each period. A colony of 500 bacteria growing by 20% an hour is multiplied by 1.2 each hour, so after 3 hours there are 500 × 1.2³ = 864. Each period's increase is bigger than the last, because the percentage is of a larger amount.
Compound decay
Decay uses a multiplier less than 1. A radioactive sample of 240 g losing 20% a year keeps 80% each year, so after 3 years 240 × 0.8³ = 122.88 g remains. The amount gets closer and closer to zero but never reaches it.
When the rate changes
If the percentage is different in different periods, use a different multiplier for each. A car worth £15 000 that loses 20% in its first year and 10% in each later year is worth 15 000 × 0.8 × 0.9² after three years. Write each period's multiplier explicitly so the structure is clear.
Finding the number of periods
When you know the target but not the number of periods, apply the multiplier repeatedly until you pass the target. No logarithms are needed at GCSE: use the calculator's answer key to multiply again and again, recording each result, and state the first whole period that meets the condition.
Interpreting the model
Questions sometimes ask whether a model is realistic, or what happens in the long term. A population cannot grow by the same percentage for ever, and a decay model never reaches exactly zero. Give a reason in context.
Worked examples
Example 1. A town's population of 8000 grows by 3% each year. What is the population after 2 years, to the nearest whole number?
- The multiplier is 1.03.
- 8000 × 1.03² = 8487.2.
Answer: 8487 people.
Example 2. A radioactive sample of 240 g decays by 20% each year. How much remains after 3 years?
- The multiplier is 0.8.
- 240 × 0.8³ = 240 × 0.512.
Answer: 122.88 g.
Example 3. A colony of 500 bacteria grows by 20% each hour. How many bacteria are there after 3 hours?
- The multiplier is 1.2.
- 500 × 1.2³ = 500 × 1.728.
Answer: 864 bacteria.
Example 4. A car worth £15 000 loses 20% of its value in the first year and 10% in each year after that. What is it worth after 3 years?
- Year 1: × 0.8. Years 2 and 3: × 0.9 each.
- 15 000 × 0.8 × 0.9².
Answer: £9720.
Example 5. A population of 6000 falls by 15% each year. After how many whole years does it first drop below 3000?
- Multiplier 0.85: after 4 years, 6000 × 0.85⁴ = 3132.0…
- After 5 years, 6000 × 0.85⁵ = 2662.2…
Answer: After 5 years.
Common mistakes
- Adding the percentage each period. Growth of 20% over 3 hours is × 1.2³, not + 60%.
- Using one multiplier when the rate changes. Use a separate multiplier for each period with a different rate.
- Stopping before the target is passed. Find the first period that actually crosses the target.
- Rounding at each step. Keep full accuracy until the end.
- Forgetting the context. Populations must be whole numbers; say whether to round up or down.
How compound growth and decay are examined
Compound growth and decay questions appear on Higher calculator papers for the 1MA1 GCSE and on the 4MA1 International GCSE, and they are often multi-step. Expect to:
- work out an amount after several periods of growth or decay
- handle a rate that changes partway through
- find how many periods it takes to pass a target
- comment on whether the model is realistic
Practise with the growth and decay worksheet, which includes answers, and the number revision engine.
For money contexts, see compound interest and depreciation. At A-Level, continuous growth is modelled in exponential growth and decay.
Frequently asked questions
How does compound growth work?
Growth by the same percentage each period, so each increase is calculated on the new, larger amount.
How do you work out compound decay?
Multiply the starting amount by (1 − r/100)ⁿ, where r is the percentage decrease and n is the number of periods.
How do you find how many years it takes to reach an amount?
Apply the multiplier year by year until the amount first passes the target.