To calculate the amount Derek must pay to the bank after 16 months when he decides to pay off his car loan, you need to follow these steps:
- Calculate the monthly payment for the loan.
- Determine the remaining balance of the loan after 16 months of payments.
Step 1: Calculate Monthly Payment
We can use the formula for the monthly payment on an amortizing loan:
\[ M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
where:
- \(M\) = monthly payment
- \(P\) = principal (loan amount) = $43,312.00
- \(r\) = monthly interest rate = annual interest rate / 12 = 0.0611 / 12
- \(n\) = total number of payments = number of years \(\times\) 12 = 6 \(\times\) 12 = 72 months
Calculating Monthly Interest Rate:
\[ r = \frac{0.0611}{12} \approx 0.00509167 \]
Total Number of Payments:
\[ n = 72 \text{ months} \]
Using these values in the payment formula:
\[ M = 43312 \times \frac{0.00509167(1 + 0.00509167)^{72}}{(1 + 0.00509167)^{72} - 1} \]
Now calculate \( (1 + r)^{n} \):
\[ (1 + 0.00509167)^{72} \approx 1.48985 \text{ (approximately)} \] Substituting this value in: \[ M \approx 43312 \times \frac{0.00509167 \times 1.48985}{1.48985 - 1} \]
Calculating further:
\[ M \approx 43312 \times \frac{0.0075791}{0.48985} \approx 43312 \times 0.01545 \approx 669.17 \] Thus, the monthly payment is approximately $669.17.
Step 2: Calculate Remaining Balance After 16 Months
After making 16 payments, we can find the remaining balance of the loan using the following formula for loan balance:
\[ B = P \times \frac{(1 + r)^{n} - (1 + r)^{k}}{(1 + r)^{n} - 1} \]
where:
- \(B\) = remaining balance
- \(k\) = number of payments made = 16
Substituting in values:
\[ B = 43312 \times \frac{(1 + 0.00509167)^{72} - (1 + 0.00509167)^{16}}{(1 + 0.00509167)^{72} - 1} \]
Calculating \( (1 + 0.00509167)^{16} \):
\[ (1 + 0.00509167)^{16} \approx 1.08713 \text{ (approximately)} \]
Now substitute back into the formula:
\[ B \approx 43312 \times \frac{1.48985 - 1.08713}{1.48985 - 1} \] Calculating:
\[ B \approx 43312 \times \frac{0.40272}{0.48985} \approx 43312 \times 0.821 \approx 35558.76 \]
Step 3: Final Amount to Pay Off Loan
The remaining balance after 16 months is approximately $35,558.76. Therefore, this is the amount Derek must give to the bank to pay off his car loan.
Conclusion
Derek must pay approximately $35,558.76 to pay off his car loan after 16 months.