Brent Pickett borrowed \( \$ 8000 \) from his brother Dave. He agreed to repay the money at the end of 2 years, giving Dave the same amount of interest that he would have received if the money had been invested at \( 1.25 \% \) compounded quarterly. How much money did Brent repay his brother? Brent repaid his brother \( \$ \square \). (Round to the nearest cent as needed.)

To calculate how much money Brent repaid his brother, we first need to find out how much interest Dave would earn if he invested the \$8000 at 1.25% interest compounded quarterly for 2 years. The formula for compound interest is: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] Where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed. In this case: - \( P = 8000 \) - \( r = 0.0125 \) (1.25% as a decimal) - \( n = 4 \) (quarterly) - \( t = 2 \) Plugging in the values, we have: \[ A = 8000 \left(1 + \frac{0.0125}{4}\right)^{4 \times 2} \] Calculating it step by step: 1. Calculate \( \frac{0.0125}{4} = 0.003125 \). 2. So, \( A = 8000 \left(1 + 0.003125\right)^{8} \). 3. Then, calculate \( 1 + 0.003125 = 1.003125 \). 4. Now, \( A = 8000 \left(1.003125\right)^{8} \). 5. Calculating \( (1.003125)^{8} \approx 1.025089 \). 6. Finally, \( A \approx 8000 \times 1.025089 \approx 8200.71 \). The total amount (including the principal) that Brent repaid his brother is approximately \$8200.71. Brent repaid his brother \( \$ 8200.71 \).