Angela Montery has a five-year car loan for a Jeep Wrangler at an annual interest rate of \( 6.9 \% \) and a monthly payment of \( \$ 601.50 \). After 3 years, Angela decides to purchase a new car. What is the payoff on Angela's loan? (Round your answer to two decimal places.) \( \$ 1 \). 395.29 Need Help?

To calculate the payoff amount on Angela's car loan after 3 years, we need to determine how much of the loan principal remains after making monthly payments for 36 months (3 years) at an annual interest rate of 6.9%. 1. **Convert the annual interest rate to a monthly interest rate:** \[ \text{Monthly interest rate} = \frac{6.9\%}{12} = \frac{0.069}{12} \approx 0.00575 \] 2. **Calculate the total number of payments:** Since the loan is for 5 years and payments are made monthly: \[ \text{Total payments} = 5 \times 12 = 60 \text{ payments} \] 3. **Calculate the remaining balance after 36 payments:** We can use the formula for the remaining balance on an amortizing loan: \[ B = P \times \frac{(1 + r)^n - (1 + r)^p}{(1 + r)^n - 1} \] Where: - \( B \) = remaining balance - \( P \) = monthly payment (\$601.50) - \( r \) = monthly interest rate (0.00575) - \( n \) = total number of payments (60) - \( p \) = number of payments made (36) Plugging in the values: \[ B = 601.50 \times \frac{(1 + 0.00575)^{60} - (1 + 0.00575)^{36}}{(1 + 0.00575)^{60} - 1} \] 4. **Calculate \( (1 + r)^{60} \) and \( (1 + r)^{36} \):** \[ (1 + 0.00575)^{60} \approx 1.432364654 \] \[ (1 + 0.00575)^{36} \approx 1.233202 \] 5. **Substituting back into the formula:** \[ B = 601.50 \times \frac{1.432364654 - 1.233202}{1.432364654 - 1} \] \[ B = 601.50 \times \frac{0.199162654}{0.432364654} \] \[ B \approx 601.50 \times 0.459 \] \[ B \approx 276.56 \] 6. **Calculate the total amount paid after 36 months:** \[ \text{Total paid} = 36 \times 601.50 = 21654 \] 7. **Calculate the original loan amount using the formula for the present value of an annuity:** \[ L = P \times \frac{1 - (1 + r)^{-n}}{r} \] \[ L = 601.50 \times \frac{1 - (1 + 0.00575)^{-60}}{0.00575} \] \[ L \approx 601.50 \times 51.093 \] \[ L \approx 30763.50 \] 8. **Calculate the payoff amount:** \[ \text{Payoff} = L - \text{Total paid} \] \[ \text{Payoff} = 30763.50 - 21654 \] \[ \text{Payoff} \approx 9109.50 \] Thus, the payoff amount on Angela's loan after 3 years is approximately **$9,109.50**.