Interest and growth: simple, compound and repeated change

Simple interest grows by a fixed amount

Simple interest is paid only on the original amount, the principal:

$$I = Prt$$

where $P$ is the principal, $r$ the yearly rate as a decimal and $t$ the time in years. $\$2,000$ at 5% simple interest earns $\$100$ every year, so $\$300$ over three years. The balance grows in a straight line.

Compound interest grows by a fixed factor

Compound interest is paid on the balance, including interest already earned, so each year's interest is larger than the last. With yearly compounding,

$$A = P(1 + r)^t$$

and with compounding $n$ times a year at a yearly rate $r$,

$$A = P\left(1 + \frac{r}{n}\right)^{nt}$$

Quarterly compounding at 8% a year means 2% per quarter, four times a year. The more often interest is compounded, the more the account ends up holding, though the gain from compounding more often is small.

Powers worth knowing by hand

Without a calculator, these products come up again and again:

Factor	Square	Cube
$1.02$	$1.0404$	$1.061208$
$1.05$	$1.1025$	$1.157625$
$1.1$	$1.21$	$1.331$
$1.2$	$1.44$	$1.728$
$0.9$	$0.81$	$0.729$

Square a factor like $1.05$ by expanding: $(1 + 0.05)^2 = 1 + 0.1 + 0.0025 = 1.1025$. The expansion also shows why compounding helps: the extra $0.0025$ is interest on interest.

Compound minus simple

Over two years at rate $r$, compound interest exceeds simple interest by

$$P(1 + r)^2 - P(1 + 2r) = Pr^2$$

the interest earned in the second year on the first year's interest. On $\$5,000$ at 6% that is $5000 \times 0.0036 = \$18$. Questions that give this difference and ask for the principal or the rate are solved with this one line.

Growth and decay in general

Interest is one case of repeated percent change. Anything that grows by a fixed percent per period is multiplied by the same factor each period: a population growing 3% a year is multiplied by $1.03$ per year; a car losing 15% of its value a year is multiplied by $0.85$. Quantities that double or triple in a fixed time work the same way: bacteria that triple every 20 minutes are multiplied by $3^6 = 729$ in two hours, since two hours contain six 20-minute periods. Count the periods carefully; that is where most errors start.

Working backwards

To find an earlier value, divide by the factor. To find a rate, divide the end value by the start value and recognise the power: growth from $\$2,500$ to $\$3,025$ over two years is a factor of $\frac{3025}{2500} = 1.21 = 1.1^2$, so the yearly rate is 10%.

Estimating

For a rough doubling time, the rule of 72 gives about $\frac{72}{\text{rate in percent}}$ years; at 8% a year, money doubles in roughly 9 years ($1.08^9$ is just under 2). Use it only to discard answer choices that are far off, never for an exact answer.