Growth and Decay

Quantities changing by a constant percentage per period follow an exponential pattern captured by a single formula.

The Compound Formula

\[ A = P \left(1 + \frac{r}{100}\right)^n \]

\( P \) = initial amount, \( r \) = percentage rate per period, \( n \) = number of periods, \( A \) = final amount. For decay use \( \left(1-\frac{r}{100}\right) \).

Compound vs Simple Interest

\[ \text{Simple: } A = P + \frac{Prn}{100} \qquad \text{Compound: } A = P\left(1+\frac{r}{100}\right)^n \]

Compound earns more because interest is added to the principal before the next calculation.

Depreciation

Assets lose value: multiply by \( \left(1-r/100\right)^n \).

Worked Examples

£5,000 at 3.5% compound for 8 years vs simple interest.

\[ A_{\text{comp}} = 5000 \times 1.035^8 \approx £6{,}584.37 \] \[ A_{\text{simple}} = 5000 + \frac{5000 \times 3.5 \times 8}{100} = £6{,}400 \]

Compound gives £184.37 more.

Car £18,000 depreciates at 12%/yr. Value after 5 years.

\[ A = 18000 \times 0.88^5 \approx £9{,}499 \]

Population 24,000 grows at 2.5%/yr. First year exceeding 30,000?

\[ 1.025^9 \approx 1.249 \quad 1.025^{10} \approx 1.280 > 1.25 \implies \textbf{10 years} \]

Key Takeaways

Growth multiplier \( > 1 \); decay multiplier \( < 1 \). Raise to the power \( n \).
Find target year by trial: compute \( P \times m^n \) for successive integers \( n \).
The compound growth graph is exponential — never a straight line.