Gloria purchased a $45,000 vehicle by financing it through a 3.8% rate of interest compounded monthly over 5 years. At the end of the 3^rd year, Nancy received an inheritance that allowed her to pay off the balance. Determine the balance of the vehicle at the end of year 3.

A) $19,030.11
B) $20,030.11
C) $21,030.11
D) $22,030.11
E) $23,030.11

To find the balance of the vehicle at the end of year 3, we use the formula for the future value of an annuity (since the car payments are essentially an annuity) and the remaining balance formula. The formula for the future value of an annuity is $A=P\frac{(1+r{)}^n-1}{r}$ , where $A$ is the future value of the annuity, $P$ is the payment amount, $r$ is the monthly interest rate, and $n$ is the total number of payments. However, since we're solving for the balance, we need to adjust our approach to account for the remaining balance after 3 years.First, we calculate the monthly payment using the loan formula, which is a rearrangement of the annuity formula to solve for $P$ . The monthly interest rate is $3.8\%/12=0.0031667$ , and the total number of payments over 5 years is $5\times 12=60$ .The formula for the monthly payment is $P=\frac{rA}{(1-(1+r{)}^{-n})}$ , where $A$ is the loan amount ($45,000), $r$ is the monthly interest rate, and $n$ is the total number of payments.After calculating the monthly payment, we find the balance after 3 years (36 payments) by considering the remaining term of the loan (24 months) and using the formula for the future value of the remaining payments. This essentially tells us how much is left to pay off the loan.The correct balance, after performing these calculations, aligns with option A, $19,030.11. This involves calculating the monthly payment, then determining the remaining balance after 36 months have passed, using the formulas mentioned and the specifics of the loan terms (3.8% annual interest rate, compounded monthly, over a 5-year period).