To calculate the interest due at the end of the first month for the car purchase, we first need to find the total amount after taxes, the loan amount after the down payment, and then apply the interest rate for a fair credit score.
Step 1: Calculate the total cost including sales tax
- Sales tax rate: 5.65%
- Car price: $12,465.00
\[ \text{Sales tax} = \text{Car price} \times \text{Sales tax rate} = 12,465.00 \times 0.0565 = 705.97 \]
- Total cost including tax:
\[ \text{Total cost} = \text{Car price} + \text{Sales tax} = 12,465.00 + 705.97 = 13,170.97 \]
Step 2: Subtract the down payment to find the loan amount
- Down payment: $1,300.00
\[ \text{Loan amount} = \text{Total cost} - \text{Down payment} = 13,170.97 - 1,300.00 = 11,870.97 \]
Step 3: Find the monthly interest rate
- APR for fair credit: 6.40%
- Monthly interest rate:
\[ \text{Monthly interest rate} = \frac{\text{APR}}{12} = \frac{6.40%}{12} = 0.5333% = \frac{6.40}{100} \div 12 = 0.005333 \]
Step 4: Calculate the interest for the first month
- Interest for the first month:
\[ \text{Interest} = \text{Loan amount} \times \text{Monthly interest rate} = 11,870.97 \times 0.005333 = 63.38 \]
Conclusion
The interest due at the end of the first month is:
\[ \boxed{63.38} \]
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