To determine how much Derek must pay to pay off his car loan after 16 months, we need to first calculate the monthly payment and then determine the remaining balance of the loan after 16 months of payments.
Step 1: Calculate the monthly payment
The loan amount \( P \) is $43,312.00, the annual interest rate \( r \) is 6.11%, and the loan term \( n \) is 6 years (or 72 months).
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Convert the annual interest rate to a monthly interest rate
\[ r_{monthly} = \frac{r}{12} = \frac{6.11%}{12} = \frac{0.0611}{12} \approx 0.00509167 \]
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Calculate the monthly payment using the formula for an amortizing loan:
\[ M = P \frac{r_{monthly}(1 + r_{monthly})^n}{(1 + r_{monthly})^n - 1} \]
Plugging in the values:
\[ M = 43312 \cdot \frac{0.00509167(1 + 0.00509167)^{72}}{(1 + 0.00509167)^{72} - 1} \]
First, calculate \((1 + 0.00509167)^{72}\):
\[ (1 + 0.00509167)^{72} \approx 1.489851 \]
Now substitute back into the formula:
\[ M = 43312 \cdot \frac{0.00509167 \cdot 1.489851}{1.489851 - 1} \] \[ = 43312 \cdot \frac{0.0075825156}{0.489851} \] \[ = 43312 \cdot 0.015449612 \] \[ \approx 670.27 \]
Step 2: Calculate the remaining balance after 16 months
We can use the formula for the remaining balance after \( k \) payments:
\[ B_k = P \frac{(1 + r_{monthly})^n - (1 + r_{monthly})^k}{(1 + r_{monthly})^n - 1} \]
Where:
- \( k = 16 \)
Substituting in values, we need \( n = 72 \):
\[ B_{16} = 43312 \cdot \frac{(1 + 0.00509167)^{72} - (1 + 0.00509167)^{16}}{(1 + 0.00509167)^{72} - 1} \]
Calculating \((1 + 0.00509167)^{16}\):
\[ (1 + 0.00509167)^{16} \approx 1.083260 \]
Now substitute back into the balance formula:
\[ B_{16} = 43312 \cdot \frac{1.489851 - 1.083260}{1.489851 - 1} \] \[ = 43312 \cdot \frac{0.406591}{0.489851} \] \[ \approx 43312 \cdot 0.828978 \] \[ \approx 35911.81 \]
Conclusion
Derek must pay approximately $35,911.81 to pay off his car loan after 16 months.
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