Exponential Growth and Decay • Topic 2 of 4

Decay Models

A quantity that shrinks by a fixed percentage each period follows A = P(1 − r)^t, the mirror of growth. The base (1 − r) lies between 0 and 1, which signals decay. For 800 decaying 20% per period, after 2 periods it is 800(0.8)² = 800 × 0.64 = 512. Subtract the rate from 1 first, then raise to the number of periods and multiply by the start. Decay appears in depreciation, cooling and half-life style problems. The most common error is forgetting to subtract from 1 (using 0.2 instead of 0.8). The SAT tests setting up (1 − r)^t and telling decay from growth by the base.

Exponential decay curve starting high and falling toward but never touching the x-axisExponential decayPtimeamountA = P(1 − r) to the power t → shrinks each period

✅ Solved examples

1. 800 decays 20% per period. Value after 2 periods?
800(0.8)² = 800 × 0.64 = 512.
2. 500 decays 10% per period. Value after 2 periods?
500(0.9)² = 405.
3. 400 decays 50% per period. Value after 2 periods?
400(0.5)² = 100.
4. Does A = 600(0.85)^t describe growth or decay?
Base 0.85 < 1, so decay.

✏️ Practice — try these, take hints as needed

1. 1,000 decays 20% per period. Value after 2 periods?
1000(0.8)².
1000 × 0.64.
—
640.
2. 200 decays 10% per period. Value after 2 periods?
200 × 0.81.
—
—
162.
3. 600 decays 50% per period. Value after 2 periods?
600 × 0.25.
—
—
150.
4. 900 decays 10% per period. Value after 2 periods?
900 × 0.81.
—
—
729.
5. Does A = 700(0.9)^t describe growth or decay?
Base < 1?
—
—
Decay.

📝 Topic test — 8 questions

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