Question

Derek borrows $37,151.00 to buy a car. He will make monthly payments for 6 years. The car loan has an interest rate of 5.40%. After a 13.00 months Derek decides to pay off his car loan. How much must he give the bank?

Answer 1

To determine how much Derek must pay off his car loan after 13 months, we can follow these steps:

1. **Calculate the monthly payment** \((PMT)\).
2. **Determine the remaining balance after 13 months**.

### Step 1: Calculate the Monthly Payment

To calculate the monthly payment, we can use the formula for the monthly payment on an installment loan, which is given by:

\[
PMT = \frac{P \cdot r(1+r)^n}{(1+r)^n - 1}
\]

Where:
- \(P\) = principal (loan amount) = $37,151.00
- \(r\) = monthly interest rate = annual interest rate / 12 = \(5.40\% / 100 / 12 = 0.0045\)
- \(n\) = total number of payments = 6 years \(\times\) 12 months/year = 72 payments

Plugging in the values:

\[
PMT = \frac{37,151 \cdot 0.0045(1+0.0045)^{72}}{(1+0.0045)^{72} - 1}
\]

1. Calculate \( (1 + r)^n \):

\[
(1 + 0.0045)^{72} \approx 1.34883
\]

2. Calculate \( PMT \):

\[
PMT \approx \frac{37,151 \cdot 0.0045 \cdot 1.34883}{1.34883 - 1}
\]

Calculate the numerator:

\[
= 37,151 \cdot 0.0045 \cdot 1.34883 \approx 225.225
\]

Calculate the denominator:

\[
1.34883 - 1 = 0.34883
\]

Finally, calculate the total payment:

\[
PMT \approx \frac{225.225}{0.34883} \approx 645.10
\]

### Step 2: Calculate the Remaining Balance After 13 Months

To find the remaining balance after 13 months, we can use the remaining balance formula:

\[
B = P(1 + r)^t - PMT \frac{(1 + r)^t - 1}{r}
\]

Where:
- \(B\) = balance after \(t\) months
- \(t\) = 13 (months)

1. Calculate \(B\):

\[
B = 37,151(1 + 0.0045)^{13} - 645.10 \cdot \frac{(1 + 0.0045)^{13} - 1}{0.0045}
\]

Calculating \((1 + 0.0045)^{13} \approx 1.06184\):

Calculate the first part:

\[
B \approx 37,151 \cdot 1.06184 \approx 39,417.57
\]

Calculate the second part:

\[
645.10 \cdot \frac{1.06184 - 1}{0.0045} \approx 645.10 \cdot 13.7454 \approx 8,865.99
\]

Putting it all together:

\[
B \approx 39,417.57 - 8,865.99 \approx 30,551.58
\]

Thus, the amount Derek must pay to pay off the loan after 13 months is approximately **$30,551.58**.

(Note: The actual calculations might have slight variations based on rounding.)