Question 16 Notyet answered Points out of 10.00 P Flag question Calculate the annual percentage yield (APY) on an investment of \( \$ 7,000 \) at \( 8 \% \) interest, compounded quarterly, for one year. (Round the annual percentage yield to the nearest hundredth percent.) Select one: o a. \( 24.73 \% \) ob. \( 8.00 \% \) oc. \( 7.75 \% \) od. \( 8.24 \% \)

To calculate the annual percentage yield (APY) for an investment compounded quarterly, we can use the formula: \[ APY = \left(1 + \frac{r}{n}\right)^{nt} - 1 \] Where: - \( r \) is the annual interest rate (as a decimal), - \( n \) is the number of compounding periods per year, - \( t \) is the number of years. Given: - \( r = 0.08 \) (8% as a decimal), - \( n = 4 \) (quarterly compounding), - \( t = 1 \) (for one year). Now, we can substitute these values into the formula step by step. 1. Calculate \( \frac{r}{n} \): \[ \frac{r}{n} = \frac{0.08}{4} \] 2. Calculate \( nt \): \[ nt = 4 \times 1 \] 3. Substitute these values into the APY formula: \[ APY = \left(1 + \frac{0.08}{4}\right)^{4 \times 1} - 1 \] Now, let's perform the calculations. Calculate the value by following steps: - step0: Calculate: \(4\times 1\) - step1: Multiply: \(4\) Calculate or simplify the expression \( 0.08/4 \). Calculate the value by following steps: - step0: Calculate: \(\frac{0.08}{4}\) - step1: Convert the expressions: \(\frac{\frac{2}{25}}{4}\) - step2: Multiply by the reciprocal: \(\frac{2}{25}\times \frac{1}{4}\) - step3: Reduce the numbers: \(\frac{1}{25}\times \frac{1}{2}\) - step4: Multiply the fractions: \(\frac{1}{25\times 2}\) - step5: Multiply: \(\frac{1}{50}\) Calculate or simplify the expression \( (1 + 0.02)^4 - 1 \). Calculate the value by following steps: - step0: Calculate: \(\left(1+0.02\right)^{4}-1\) - step1: Add the numbers: \(1.02^{4}-1\) - step2: Convert the expressions: \(\left(\frac{51}{50}\right)^{4}-1\) - step3: Rewrite the expression: \(\frac{51^{4}}{50^{4}}-1\) - step4: Reduce fractions to a common denominator: \(\frac{51^{4}}{50^{4}}-\frac{50^{4}}{50^{4}}\) - step5: Transform the expression: \(\frac{51^{4}-50^{4}}{50^{4}}\) Now that we have the necessary calculations: 1. We found that \( \frac{r}{n} = 0.02 \). 2. We confirmed that \( nt = 4 \). Now we substitute these values into the APY formula: \[ APY = \left(1 + 0.02\right)^{4} - 1 \] Calculating this gives us: \[ APY = (1.02)^{4} - 1 \approx 0.08243216 \] To express this as a percentage, we multiply by 100: \[ APY \approx 0.08243216 \times 100 \approx 8.24\% \] Rounding to the nearest hundredth percent, the annual percentage yield (APY) is approximately \( 8.24\% \). Thus, the correct answer is: **d. \( 8.24\% \)**.