Let's break down the problem step by step to determine the amount of unearned interest when paying off the loan after 30 payments.
1. Initial Details
- Car price: [tex]$32,000
- Trade-in amount (down payment): $[/tex]6,000
- Loan amount: [tex]\( \$32,000 - \$6,000 = \$26,000 \)[/tex]
- APR: 10%
- Loan term: 48 months
- Monthly payment: [tex]$659.43
- Payments made: 30
2. Total Amount Paid After 30 Payments
- Total paid = \( 30 \times 659.43 = 19782.90 \)
3. Number of Payments Remaining
- Payments remaining = \( 48 - 30 = 18 \)
4. Calculate Remaining Principal
- The remaining principal (balance) after 30 payments is about \( \$[/tex] 10,980.01 \).
5. Total Interest Paid if Loan Went Full Term
- Total amount paid after 48 months = [tex]\( 48 \times 659.43 = 31652.64 \)[/tex]
- Total interest over the full term = [tex]\( 31652.64 - 26000 = 5652.64 \)[/tex]
6. Interest Paid Up to Payoff
- Interest paid up to payoff = [tex]\( 19782.90 - \\text{remaining principal part paid out of } total\_paid = 4762.91 \)[/tex]
7. Unearned Interest Calculation
- Unearned interest is the difference between the total interest over the full term and the interest paid up to the payoff point. Hence, unearned interest = [tex]\( 5652.64 - 4762.91 = 889.73 \)[/tex]
After evaluating these steps, we find the unearned interest amount to be approximately [tex]$889.41, which matches closest with option a.
Therefore, the best answer is:
a. $[/tex]889.41
