Final answer:
Sarah's monthly car loan payment is approximately $677.31. After making 40 months of payments, the amount she would need to pay to settle the remaining balance of the loan is about $13,501.79.
Step-by-step explanation:
To calculate Sarah's monthly car loan payment, we use the formula for the monthly payment on an installment loan which can be derived from the amortizing loan formula:
M = P [ i(1 + i)^n ] / [ (1 + i)^n – 1 ]
Where:
- M is the total monthly payment.
- P is the principal loan amount ($34,500).
- i is the monthly interest rate (annual interest rate/12).
- n is the number of payments (loan term in months).
First, we calculate the monthly interest rate: i = 6.7% / 12 = 0.00558333.
Since the loan term is 60 months, n = 60. Plugging the values into the formula gives us:
M = 34500 [ 0.00558333(1 + 0.00558333)^60 ] / [ (1 + 0.00558333)^60 – 1 ]
After calculating, we find that Sarah's monthly payment is approximately $677.31.
Now, to calculate the payoff amount after 40 months of payment, we will find the remaining balance. This is done by calculating the future value of the remaining payments that would have been made. The formula for the remaining balance of an installment loan is:
B = P(1 + i)^n – [ M((1 + i)^n – 1) / i ]
B = 34500(1 + 0.00558333)^40 – [ 677.31((1 + 0.00558333)^40 – 1) / 0.00558333 ]
After computation, the balance that Sarah needs to pay off the loan is approximately $13,501.79.