How to calculate compound interest

Identify the variables required for the formula: $P$ is the principal investment amount, $r$ is the annual nominal interest rate in decimal form, $n$ is the number of compounding periods per year, and $t$ is the time the money is invested in years. The total accumulated value is $A$. The compound interest $I$ is isolated by subtracting the principal from the accumulated value: $I = A - P$.

1. Identify $P$, $r$, $n$, and $t$ from the problem statement. 2. Convert the interest rate $r$ from a percentage to a decimal. 3. Calculate the periodic interest rate $\frac{r}{n}$ and the total number of compounding periods $nt$. 4. Substitute these values into the formula $A = P(1 + \frac{r}{n})^{nt}$ to find the total amount. 5. Subtract $P$ from $A$ to find the total compound interest earned.

Ensure that $A > P$ and $I > 0$ for positive interest rates. For annual compounding, you can use the Rule of 72 as a quick sanity check: the time to double the investment is approximately $72 / (r imes 100)$. If your computed $t$ aligns with this heuristic for a doubled principal, the magnitude of your calculation is likely correct.

The most frequent errors include using the percentage form instead of the decimal form for $r$ (e.g., using $5$ instead of $0.05$), failing to multiply $t$ by $n$ in the exponent, and confusing nominal annual rate with periodic rate. Finally, remember that the formula yields the total accumulated amount $A$, not the interest $I$; you must subtract $P$ to find $I$.