Compound Interest — Formula and Amount

Repeatedly multiplying the running amount by the same growth factor leads directly to a compact formula. At the end of the first year the amount is $P\left(1 + \dfrac{R}{100}\right)$; at the end of the second year this is itself multiplied by $\left(1 + \dfrac{R}{100}\right)$, and so on. After $n$ years, compounded annually:

The compound interest is then the amount minus the original principal:

Here $A$ is the amount, $P$ the principal, $R$ the rate percent per annum and $n$ the number of years (number of conversion periods). Notice the power $n$ — this is the geometric growth of §4 written as one line.

Finding the other quantities. Because the formula is a single equation, it can be turned around for any unknown:

• To find the principal from a known amount: $P = \dfrac{A}{\left(1 + \frac{R}{100}\right)^{n}}$.
• To find the rate, form $\left(1 + \dfrac{R}{100}\right)^{n} = \dfrac{A}{P}$ and take the $n$-th root of both sides.
• To find the time, solve $\left(1 + \dfrac{R}{100}\right)^{n} = \dfrac{A}{P}$ for $n$ (usually by inspection at this level). …