Growth & decay

Exponential growth and decay describe situations where a quantity increases or decreases by a fixed percentage repeatedly. The same multiplier formula applies as in compound interest.

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Key facts to remember

1Exponential growth: y = a × bⁿ where b > 1.
2Exponential decay: y = a × bⁿ where 0 < b < 1.
3Population growth, bacterial growth and radioactive decay all follow exponential patterns.
4The multiplier b = 1 + r/100 (growth) or 1 − r/100 (decay).
5Growth and decay are the same mathematically as compound interest and depreciation.

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Formulas

Growth/decay

y = a × bⁿ

a = initial amount, b = multiplier, n = time periods

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Worked examples

Example 1

A population of bacteria doubles every 3 hours. Starting at 500, how many are there after 12 hours?

Working

Number of 3-hour periods = 12 ÷ 3 = 4
Multiplier = 2 (doubling)
Population = 500 × 2⁴ = 500 × 16 = 8000

Answer8000

Example 2

A radioactive substance decays at 20% per year. Starting mass is 400 g. Find the mass after 4 years.

Working

Multiplier = 1 − 0.20 = 0.80
Mass = 400 × 0.8⁴ = 400 × 0.4096 = 163.84 g

Answer163.84 g

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Common mistakes

✗Subtracting 20% four times separately instead of using the multiplier to the power of 4.

✗Using an additive model (adding a fixed amount) instead of a multiplicative one.

✗Not identifying the number of time periods n correctly.

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Exam tips

✓Identify: initial value (a), multiplier (b), and number of time periods (n) before substituting.

✓Growth and decay questions follow the exact same pattern as compound interest — use the same approach.

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