- Identify the variables from the problem: Principal \(P = \$3000\), annual rate \(r = 0.025\), compounding periods per year \(n = 12\), and time \(t = 5\) years.
- Apply the compound interest formula \(A = P(1 + \frac{r}{n})^{nt}\).
- Compute the values: \(A = 3000(1 + \frac{0.025}{12})^{60}\).
- Determine the final amount, rounded to the nearest cent: \(\boxed{\$3399.00}\).
Explanation
- Analysis and Problem Setup
We are tasked with calculating the final balance in a savings account after a specific time period under compound interest. We are given the following information:
- Principal amount (\(P\)): \(\$3000\)
- Annual interest rate (\(r\)): \(2.5\% = 0.025\)
- Compounding frequency (\(n\)): monthly, which is \(12\) times per year
- Time (\(t\)): \(5\) years
The formula for the total amount (\(A\)) in an account with interest compounded \(n\) times per year is:
\[A = P\left(1 + \frac{r}{n}\right)^{nt}\]
- Substituting Values and Simplifying
Using the identified values, we substitute them into the compound interest formula:
\[A = 3000 \times \left(1 + \frac{0.025}{12}\right)^{12 \times 5}\]
First, we simplify the terms within the exponent and the parentheses:
- The exponent: \(12 \times 5 = 60\)
- The rate per period: \(\frac{0.025}{12} \approx 0.00208333333\)
- The base of the exponent: \(1 + \frac{0.025}{12} \approx 1.00208333333\)
Now, calculate the expression:
\[A = 3000 \times (1.00208333333)^{60}\]
- Final Calculation and Rounding
Calculating the power \((1.00208333333)^{60}\) gives approximately \(1.13300112\).
Multiplying this by the principal amount:
\[A = 3000 \times 1.13300112 \approx 3399.0033656\]
Rounding the result to the nearest hundredth (as is standard for financial currency calculations), we get:
\[A \approx \$3399.00\]
Examples
Understanding compound interest is a vital skill for personal finance. For instance, if you decide to start a savings account at age 20 with a modest deposit of \(\$3000\), seeing how that money grows over time due to compound interest helps you appreciate the power of starting early. Even with a relatively low interest rate, the compounding effect—where you earn interest on your previous interest—ensures your money works for you, demonstrating why long-term saving and investing are cornerstones of financial health.