Calculate compound interest

Compound interest models repeated percentage growth, in which each period’s interest is added to the balance and subsequently earns interest itself. The learner calculates the accumulated amount and total interest using $A=P(1+\frac{r}{n})^{nt}$ , interpreting principal, annual rate, compounding frequency, and time consistently and distinguishing this growth from simple interest; the scope excludes continuous compounding, changing rates, and logarithmic methods for solving unknown exponents.

Detailed Explanation: Calculate compound interest

Compound interest means that interest is added to the account after each compounding period. The new, larger balance then earns interest in the next period.

Use the formula

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

where:

$A$ is the accumulated amount
$P$ is the principal, or starting amount
$r$ is the annual interest rate written as a decimal
$n$ is the number of compounding periods per year
$t$ is the time in years

Example

A student deposits 2{,}000 $into an account earning$ 6% $annual interest, compounded quarterly, for$ 3$ years. Find the accumulated amount and the total interest earned.

Step 1: Identify the values

P=2000,\qquad r=0.06,\qquad n=4,\qquad t=3

The rate is written as $0.06$ because $6\%=0.06$ . Quarterly compounding means interest is added $4$ times per year.

Step 2: Substitute into the formula

A=2000\left(1+\frac{0.06}{4}\right)^{4(3)}

Step 3: Simplify

The interest rate per quarter is

\frac{0.06}{4}=0.015

There are

4(3)=12

compounding periods in $3$ years. Therefore,

A=2000(1.015)^{12}

Step 4: Calculate the accumulated amount

A\approx 2000(1.195618)

A\approx \$2{,}391.24

So, the account will contain approximately 2{,}391.24}$.

Step 5: Find the total interest

Subtract the original principal from the accumulated amount:

\text{Interest}=A-P

\text{Interest}=\$2{,}391.24-\$2{,}000

\boxed{\text{Interest}\approx \$391.24}

This is different from simple interest, where interest is calculated only on the original principal. Compound interest also earns interest on previously added interest.

Learn by doing: Calculate compound interest

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