Verified Answer :
- Identify the principal \(P = \$1,300\), annual rate \(r = 0.043\), compounding frequency \(n = 4\), and time \(t = 7\) years.
- Calculate the effective annual interest rate using \(T_{\text{eff}} = (1 + \frac{r}{n})^n - 1 \approx 0.0436984\).
- Calculate the final account balance using the compound interest formula \(A = P(1 + \frac{r}{n})^{nt} \approx \$1753.75\).
- Final results: \(T_{\text{eff}} = \boxed{0.0436984}\), \(A = \boxed{\$1753.75}\).
Explanation
- Problem Analysis and Parameters
To determine the effective interest rate and the final account balance, we analyze the given parameters:
- Principal amount (P): \(\$1,300\)
- Annual interest rate (r): \(4.3\%\) or \(0.043\)
- Compounding frequency (n): Quarterly, which means \(n = 4\) times per year
- Time period (t): \(7\) years
The effective interest rate, \(T_{\text{eff}}\), represents the actual interest earned in one year, considering the effect of compounding. It is calculated using the formula \(T_{\text{eff}} = (1 + \frac{r}{n})^n - 1\). The final amount \(A\) is calculated using the compound interest formula: \(A = P(1 + \frac{r}{n})^{nt}\).
- Calculating Effective Interest Rate
We first calculate the effective annual interest rate \(T_{\text{eff}}\). Substituting the values \(r = 0.043\) and \(n = 4\) into the formula:
\[T_{\text{eff}} = \left(1 + \frac{0.043}{4}\right)^4 - 1\]
\[T_{\text{eff}} = \left(1 + 0.01075\right)^4 - 1 = (1.01075)^4 - 1\]
Calculating \((1.01075)^4\) yields approximately \(1.0436983575\). Subtracting 1 gives \(0.0436983575\).
Rounding this value to 7 decimal places, we get:
\[T_{\text{eff}} \approx 0.0436984\]
- Calculating Final Account Balance
Next, we calculate the final amount \(A\) in the account after 7 years using the compound interest formula \(A = P(1 + \frac{r}{n})^{nt}\):
\[A = 1300 \left(1 + \frac{0.043}{4}\right)^{4 \times 7}\]
\[A = 1300 \left(1.01075\right)^{28}\]
Evaluating \((1.01075)^{28}\) gives approximately \(1.3490405378\). Multiplying this by the principal \(P = 1300\):
\[A = 1300 \times 1.3490405378 \approx 1753.7527\]
Rounding to 2 decimal places as is standard for currency, we find the final amount to be \(\$1753.75\).
- Conclusion and Final Answers
Based on our calculations, the effective annual interest rate is \(0.0436984\) and the final amount in the account after 7 years is \(\$1753.75\).
The effective rate of interest is:
\[T_{\text{eff}} = \boxed{0.0436984}\]
The final amount is \(\$\boxed{1753.75}\)
Examples
Understanding effective interest rates is essential for personal finance and investment decisions. Imagine you are comparing two savings accounts: one offers 4.3% compounded quarterly and another offers 4.25% compounded monthly. By calculating the effective annual rate for both, you can determine which account actually provides a higher annual return on your money. This simple calculation helps you maximize your earnings and make better-informed financial choices.