A bank made a farmer a loan of \( \$ 1,200 \) at \( 15 \% \) for three years compounded annually. Find the future value and the compound interest paid on the loan. Compare the compound interest with simple interest for the same period. Future value \( =\$ \square \) (Round to the nearest cent as needed.)

To find the future value when interest is compounded annually, we can use the formula: \[ FV = P(1 + r)^n \] where \( FV \) is the future value, \( P \) is the principal amount (loan amount), \( r \) is the annual interest rate (in decimal), and \( n \) is the number of years. Given: - \( P = 1200 \) - \( r = 0.15 \) (15%) - \( n = 3 \) We plug in the values: \[ FV = 1200(1 + 0.15)^3 \\ FV = 1200(1.15)^3 \\ FV = 1200 \times 1.520875 \\ FV \approx 1824.75 \] So, the future value is approximately \( \$1824.75 \). Next, we find the compound interest paid on the loan: \[ \text{Compound Interest} = FV - P \\ \text{Compound Interest} = 1824.75 - 1200 \\ \text{Compound Interest} \approx 624.75 \] Now, for simple interest, we use the formula: \[ SI = P \times r \times n \] Calculating the simple interest: \[ SI = 1200 \times 0.15 \times 3 \\ SI = 1200 \times 0.45 \\ SI = 540 \] Comparing the two interest types, compound interest is \( \$624.75 \) while simple interest is \( \$540 \). Therefore, the farmer pays an additional \( \$84.75 \) in interest due to compounding. Future value \( =\$1824.75 \)