Final answer:
To find the minimum term of the loan she needs, Francesca has to use the amortizing loan formula to calculate the number of months, and then divide by 12 to determine the years.
Step-by-step explanation:
To determine the minimum term Francesca needs to keep her monthly car loan payments within her budget of $250, we must calculate the loan term using the formula for an amortizing loan payment. The loan payment formula is PV = PMT × ((1 - (1 + i)^{-n}) / i), where PV is the present value (loan amount), PMT is the monthly payment, i is the monthly interest rate, and n is the total number of payments (terms).
In Francesca's case, PV is $9500, PMT is $250, and the annual interest rate is 8.5%, which needs to be converted to a monthly rate by dividing by 12. Therefore, i = 0.085 / 12 = 0.007083. The formula now needs to be rearranged to solve for n, which represents the number of months needed to repay the loan.
To solve for n, we use the formula:
n = -log(1 - PV × i / PMT) / log(1 + i)
After calculating with the known values:
n = -log(1 - 9500 × 0.007083 / 250) / log(1 + 0.007083)
The calculation will yield the number of months Francesca will need to pay off the loan. She can then divide this number by 12 to find the minimum number of years required to keep the monthly payments within her $250 budget.