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Compound interest is calculated using the compound interest formula:

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

where:
\( A \) is the future value of the investment/loan, including interest.
\( P \) is the principal amount (the initial amount of money).
\( r \) is the annual interest rate (decimal).
\( n \) is the number of times that interest is compounded per unit \( t \) (usually per year).
\( t \) is the time the money is invested or borrowed for, in years.

Here's a step-by-step breakdown of how to calculate compound interest:

1. *Convert the annual interest rate to decimal form:*
\[ r = \frac{\text{annual interest rate}}{100} \]

2. *Divide the annual interest rate by the number of compounding periods per year:*
\[ \text{rate per period} = \frac{r}{n} \]

3. *Add 1 to the rate per period:*
\[ 1 + \text{rate per period} \]

4. *Raise the result to the power of the total number of compounding periods:*
\[ \left(1 + \text{rate per period}\right)^{nt} \]

5. *Multiply the principal amount (\( P \)) by the result from step 4:*
\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \]

Now, you have the future value (\( A \)), which includes both the principal and compound interest.

Here's a simple example:

Let's say you invest $1,000 at an annual interest rate of 5%, compounded quarterly over 3 years.

\( P = $1,000 \)
\( r = \frac{5}{100} = 0.05 \)
\( n = 4 \) (quarterly compounding)
\( t = 3 \) years

Plug these values into the formula:

\[ A = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3} \]

Calculate this expression to find the future value (\( A \)).
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