- The effective annual interest rate is calculated as \(T_{\text{eff}} = (1 + \frac{0.043}{4})^4 - 1 \approx 0.0436984\).
- The final amount is calculated as \(A = 1300 \times (1 + \frac{0.043}{4})^{4 \times 7} \approx \$1,753.75\).
- Final answers are \(T_{\text{eff}} = \boxed{0.0436984}\) and \(A = \boxed{\$1,753.75}\).
Explanation
- Analysis of the Financial Problem
The problem asks us to calculate the effective annual interest rate and the final accumulated amount in a savings account. We are given the following information:
- Principal amount deposited: \(P = \$1,300\)
- Annual interest rate: \(r = 4.3\% = 0.043\)
- Compounding frequency: Quarterly, meaning it compounds \(n = 4\) times per year.
- Time period: \(t = 7\) years.
The effective annual rate \(T_{\text{eff}}\) represents the actual interest earned in one year, taking into account the compounding frequency. It is calculated using the formula:
\[T_{\text{eff}} = \left(1 + \frac{r}{n}\right)^n - 1\]
The final amount \(A\) in the account after \(t\) years with compound interest is calculated using the formula:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
- Calculating the Effective Annual Interest Rate
Using the formula for the effective annual interest rate with \(r = 0.043\) and \(n = 4\):
\[T_{\text{eff}} = \left(1 + \frac{0.043}{4}\right)^4 - 1\]
\[T_{\text{eff}} = (1 + 0.01075)^4 - 1\]
\[T_{\text{eff}} = (1.01075)^4 - 1\]
Calculating \((1.01075)^4\):
\[(1.01075)^4 \approx 1.04369835754...\]
Subtracting 1 gives:
\[T_{\text{eff}} \approx 0.0436983575...\]
Rounding to 7 decimal places, we get:
\[T_{\text{eff}} = 0.0436984\]
- Calculating the Final Amount
Now we calculate the final amount \(A\) using the formula for compound interest with \(P = 1300\), \(r = 0.043\), \(n = 4\), and \(t = 7\):
\[A = 1300 \left(1 + \frac{0.043}{4}\right)^{4 \times 7}\]
\[A = 1300 (1.01075)^{28}\]
Calculating \((1.01075)^{28}\):
\[(1.01075)^{28} \approx 1.34904053786...\]
Multiplying by the principal \(P = 1300\):
\[A \approx 1300 \times 1.34904053786...\]
\[A \approx 1753.752699...\]
Rounding to 2 decimal places for currency, we get:
\[A = \$1,753.75\]
Examples
Understanding compound interest is crucial for personal finance. For instance, if you save for a future goal—such as a college fund or a vacation—knowing how your interest compounds quarterly helps you predict exactly how your savings will grow over time. By utilizing the effective interest rate, you can compare different bank offers more accurately, ensuring your money works as hard as possible for you.