What is Logan's payoff

Question

Logan had seamless rain gutters installed around her home at a cost of $1,300. She financed the total amount for 18 months at an annual interest rate of 3.5% compoundedmonthly. After making 12 payments, Logan decided to repay the loan in full. What is Logan's payoff (in dollars)? (Round your answer to the nearest cent.)

$

Answer 1

Steps to Calculate the Payoff

1. Determine the monthly interest rate:
2. Annual interest rate is 3.5%, so the monthly interest rate is 3.5% divided by 12.
3. Monthly interest rate = 0.035 / 12
4. Calculate the monthly payment:
5. Use the loan amount ($1,300), the monthly interest rate, and the total number of payments (18 months) to find the monthly payment.
6. Calculate the remaining balance after 12 payments:
7. Use the formula for the remaining balance on a loan after a certain number of payments have been made.

Step-by-Step Calculation

1. Monthly interest rate:
2. Monthly interest rate = 0.035 / 12 ≈ 0.0029167
3. Monthly payment calculation:
4. Loan amount (P) = $1,300
5. Number of payments (n) = 18
6. Monthly interest rate (r) = 0.0029167
7. The formula for the monthly payment (M) is:
8. M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
9. Plugging in the values:
10. M = 1300 * (0.0029167 * (1 + 0.0029167)^18) / ((1 + 0.0029167)^18 - 1)
11. Calculate the remaining balance after 12 payments:
12. Remaining balance after 12 payments can be calculated using the formula for the remaining balance on an installment loan.
13. Remaining balance (B) after k payments is:
14. B = P * (1 + r)^n - M * ((1 + r)^k - 1) / r
15. Where:
16. k = 12 (number of payments made)

Calculations:

1. Monthly payment (M):
2. M ≈ 1300 * (0.0029167 * (1 + 0.0029167)^18) / ((1 + 0.0029167)^18 - 1)
3. M ≈ 1300 * (0.0029167 * 1.054) / (0.054)
4. M ≈ 1300 * 0.00307 / 0.054
5. M ≈ 1300 * 0.0569
6. M ≈ $73.97 (approximately)
7. Remaining balance after 12 payments:
8. B = 1300 * (1 + 0.0029167)^18 - 73.97 * ((1 + 0.0029167)^12 - 1) / 0.0029167
9. B ≈ 1300 * 1.054 - 73.97 * (1.036 - 1) / 0.0029167
10. B ≈ 1370.2 - 73.97 * 0.036 / 0.0029167
11. B ≈ 1370.2 - 73.97 * 12.34
12. B ≈ 1370.2 - 913.77
13. B ≈ $456.43 (approximately)

Result

Therefore, Logan's payoff amount after making 12 payments is approximately $456.43.