To determine how much Derek must pay off his car loan after 13 months, we can follow these steps:
- Calculate the monthly payment \((PMT)\).
- 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:
\[ PMT = \frac{37,151 \cdot 0.0045(1+0.0045)^{72}}{(1+0.0045)^{72} - 1} \]
- Calculate \( (1 + r)^n \):
\[ (1 + 0.0045)^{72} \approx 1.34883 \]
- 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)
- 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.)
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