Question

A firefighter for the National Park Service has a 5-year car loan for which the monthly payment is $620.46 with an annual interest rate of 4.25% compounded monthly. After making 36 payments, the firefighter decides to trade in the car for a new car. Calculate the amount (in dollars) the firefighter still owes on the car. (Round your answer to the nearest cent. See Example 3 in this section.)

Answer 1

To find out how much the firefighter still owes on the car after making 36 payments, we need to calculate the remaining balance on the car loan at that point.

Given data:
- Monthly payment (\(PMT\)) = $620.46
- Annual interest rate (\(r\)) = 4.25%
- Monthly interest rate (\(i\)) = \(\frac{4.25\%}{12} = \frac{0.0425}{12} \approx 0.00354167\)
- Total number of payments (\(n\)) = 5 years × 12 = 60 payments
- Number of payments made = 36

The formula to calculate the remaining balance after making \(k\) payments is:

\[
B_k = PMT \times \frac{(1 + i)^{n} - (1 + i)^{k}}{i}
\]

where:
- \(B_k\) is the remaining balance after \(k\) payments,
- \(PMT\) is the monthly payment,
- \(i\) is the monthly interest rate,
- \(n\) is the total number of payments,
- \(k\) is the number of payments made.

Substituting the known values:

1. Calculate \((1 + i)^{n}\):
\[
(1 + 0.00354167)^{60} \approx (1.00354167)^{60} \approx 1.233032
\]

2. Calculate \((1 + i)^{k}\):
\[
(1 + 0.00354167)^{36} \approx (1.00354167)^{36} \approx 1.128097
\]

3. Substitute these values into the balance formula:
\[
B_{36} = 620.46 \times \frac{1.233032 - 1.128097}{0.00354167}
\]

4. Calculate the numerator:
\[
1.233032 - 1.128097 \approx 0.104935
\]

5. Divide by \(i\):
\[
\frac{0.104935}{0.00354167} \approx 29.64
\]

6. Finally, multiply by the monthly payment:
\[
B_{36} = 620.46 \times 29.64 \approx 18348.4144
\]

So, the remaining balance after 36 payments is approximately:

\[
B_{36} \approx 18348.41
\]

Thus, rounding to the nearest cent, the firefighter still owes **$18,348.41** on the car.