- Identify the principal \(P = \$3000\), annual rate \(r = 0.025\), frequency \(n = 12\), and time \(t = 5\).
- Apply the compound interest formula \(A = P \left(1 + \frac{r}{n}\right)^{nt}\) to get \(A \approx \$3399.00\).
- Calculate the interest earned using \(I = A - P\) to get \(I \approx \$399.00\).
- Final answers: a) \(\boxed{\$3399.00}\), b) \(\boxed{\$399.00}\).
Explanation
- Analysis of the problem
We are tasked with calculating the final balance and the interest earned for a savings account. The principal amount \(P\) is \(\$3000\), the annual interest rate \(r\) is \(2.5\%\) (which is \(0.025\) in decimal form), the interest is compounded monthly (\(n = 12\)), and the time \(t\) for the investment is \(5\) years. The formula for compound interest is \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where \(A\) is the final amount. The interest earned \(I\) is then calculated as \(I = A - P\).
- Calculating the final amount A
To find the final amount in the account, we plug the values into the compound interest formula:
\[A = 3000 \left(1 + \frac{0.025}{12}\right)^{12 \times 5}\]
First, calculating the base: \(1 + \frac{0.025}{12} \approx 1 + 0.00208333333 = 1.00208333333\).
Next, calculating the exponent: \(12 \times 5 = 60\).
Now, raising the base to the power of 60: \((1.00208333333)^{60} \approx 1.13300112188\).
Finally, multiplying by the principal: \(3000 \times 1.13300112188 \approx 3399.0033656\).
Rounding to the nearest hundredth, the final amount in the account is \(\$3399.00\).
- Calculating the interest earned I
The interest earned is the difference between the final amount and the initial principal:
\[I = A - P = 3399.0033656 - 3000 = 399.0033656\]
Rounding to the nearest hundredth, the interest earned is \(\$399.00\).
Examples
Compound interest is a fundamental concept in finance, crucial for understanding how investments like savings accounts, certificates of deposit (CDs), or retirement funds grow over time. By reinvesting the interest earned back into the principal, your money generates its own returns, accelerating the growth of your capital exponentially. For instance, if you save for a college fund or a major purchase, understanding how often interest is compounded (e.g., daily, monthly, or annually) can help you choose the most effective strategy to maximize your savings. Understanding these calculations helps in making informed decisions about managing personal finances and long-term wealth planning.