QuestionJuly 29, 2025

What interest rate compounded semiannually will yield an effective interest rate of 4% 7 A rate of square % compounded semiannually will yield an effective rate of 4% (Round to two decimal places as needed.)

What interest rate compounded semiannually will yield an effective interest rate of 4% 7 A rate of square % compounded semiannually will yield an effective rate of 4% (Round to two decimal places as needed.)
What interest rate compounded semiannually will yield an effective interest rate of 4% 7 A rate of square % compounded semiannually will yield an effective rate of 4% (Round to two decimal places as needed.)

Solution
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Answer

3.92\% Explanation 1. Use the Effective Interest Rate Formula The effective interest rate E is given by **E = \left(1 + \frac{r}{n}\right)^n - 1**, where r is the nominal rate and n is the number of compounding periods per year. Here, E = 0.04 and n = 2. 2. Set Up the Equation Set up the equation: 0.04 = \left(1 + \frac{r}{2}\right)^2 - 1. 3. Solve for r Rearrange to find r: 1. 1.04 = \left(1 + \frac{r}{2}\right)^2 2. \sqrt{1.04} = 1 + \frac{r}{2} 3. \frac{r}{2} = \sqrt{1.04} - 1 4. r = 2(\sqrt{1.04} - 1) 5. Calculate r: r \approx 0.03922 or 3.92\%.

Explanation

1. Use the Effective Interest Rate Formula<br /> The effective interest rate $E$ is given by **$E = \left(1 + \frac{r}{n}\right)^n - 1$**, where $r$ is the nominal rate and $n$ is the number of compounding periods per year. Here, $E = 0.04$ and $n = 2$.<br />2. Set Up the Equation<br /> Set up the equation: $0.04 = \left(1 + \frac{r}{2}\right)^2 - 1$.<br />3. Solve for r<br /> Rearrange to find $r$: <br />1. $1.04 = \left(1 + \frac{r}{2}\right)^2$<br />2. $\sqrt{1.04} = 1 + \frac{r}{2}$<br />3. $\frac{r}{2} = \sqrt{1.04} - 1$<br />4. $r = 2(\sqrt{1.04} - 1)$<br />5. Calculate $r$: $r \approx 0.03922$ or $3.92\%$.
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