# Compound Interest

Imagine you put $100$ in a savings account with a yearly interest rate of $6%$ .

After one year, you have $100+6=106$ . After two years, if the interest is simple , you will have $106+6=112$ (adding $6%$ of the original principal amount each year.) But if it is compound interest , then in the second year you will earn $6%$ of the new amount:

$1.06×106=112.36$

## Yearly Compound Interest Formula

If you put $P$ dollars in a savings account with an annual interest rate $r$ , and the interest is compounded yearly, then the amount $A$ you have after $t$ years is given by the formula:

$A=P{\left(1+r\right)}^{t}$

Example:

Suppose you invest $4000$ at $7%$ interest, compounded yearly. Find the amount you have after $5$ years.

Here, $P=4000$ , $r=0.07$ , and $t=5$ . Substituting the values in the formula, we get:

$\begin{array}{l}A=4000{\left(1+0.07\right)}^{5}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx 4000\left(1.40255\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=5610.2\end{array}$

Therefore, the amount after $5$ years would be about $5610.20$ .

## General Compound Interest Formula

If interest is compounded more frequently than once a year, you get an even better deal. In this case you have to divide the interest rate by the number of periods of compounding.

If you invest $P$ dollars at an annual interest rate $r$ , compounded $n$ times a year, then the amount $A$ you have after $t$ years is given by the formula:

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

Example:

Suppose you invest $1000$ at $9%$ interest, compounded monthly. Find the amount you have after $18$ months.

Here $P=1000$ , $r=0.09$ , $n=12$ , and $t=1.5$ (since $18$ months = one and a half years).

Substituting the values, we get:

$\begin{array}{l}A=1000{\left(1+\frac{0.09}{12}\right)}^{12\left(1.5\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\approx 1000\left(1.143960\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=1143.960\end{array}$

Rounding to the nearest cent, you have $1143.96$ .