Final answer:
To find the outstanding balance of Kerri James's car loan after 18 payments, we use the present value annuity formula to first calculate the monthly payment and then apply the remaining balance formula. After calculating, option b ($18,858.19) is the correct answer for the remaining balance immediately after the 18th payment.
Step-by-step explanation:
The question regarding Kerri James considering the purchase of a car is a typical problem in finance mathematics, where one needs to calculate the outstanding loan balance after a certain number of payments. We need to calculate the remaining balance on the car loan immediately following the 18th payment using the loan details provided: a total loan of $24,600, an interest rate of 3.0% per year, with payments over six years (72 months), with the first payment due next month.
To solve this, we need the formula for calculating the monthly payment for an installment loan and then use the amortization formula to find the remaining balance after 18 payments. The monthly payment can be calculated using the present value annuity formula, which is:
PMT = P × [i / (1 - (1 + i)^{-n})]
Where PMT is the monthly payment, P is the principal amount of the loan, i is the monthly interest rate (annual rate divided by 12), and n is the total number of payments.
After computing the monthly payment, we use the formula for the remaining balance of an annuity after a certain number of payments, which is:
Remaining Balance = PMT × [(1 - (1 + i)^{-m}) / i]
Where m is the number of payments remaining. Substituting the 18th payment leaves us with 72 - 18 = 54 payments remaining. We can then input this into the remaining balance formula to find the answer.
After performing these calculations, the remaining balance immediately following Kerri's 18th payment is option b. $18,858.19.