Amanda purchased $25,000 vehicle through Mazda's Graduate Program. She would pay monthly payments over 5 years, at a rate of 2.8% compounded monthly. At the end of the third year, Amanda wished to pay off her loan outright. Determine the Balance on the loan at the end of year 3.

A) $10,421.30
B) $10,830.60
C) $11,054.60
D) $11,265.50
E) $11,480.60

To find the balance on the loan at the end of year 3, we first calculate the monthly payment and then use it to find the remaining balance. The loan amount is $25,000, the annual interest rate is 2.8%, compounded monthly, and the loan term is 5 years.1. Calculate the monthly interest rate: $2.8\%/12=\mathrm{0.002333...}$ 2. Calculate the number of monthly payments: $5\times 12=60$ 3. Use the formula for the monthly payment on an amortizing loan: $$PMT=P\times \frac{r(1+r{)}^n}{(1+r{)}^n-1}$$ Where: - $PMT$ is the monthly payment - $P$ is the principal loan amount ($25,000) - $r$ is the monthly interest rate (0.002333...) - $n$ is the total number of payments (60)Plugging in the numbers: $$PMT=25000\times \frac{0.002333(1+0.002333{)}^{60}}{(1+0.002333{)}^{60}-1}$$$$PMT\approx 443.87$$ 4. Calculate the total number of payments made by the end of year 3: $3\times 12=36$ 5. Calculate the remaining balance using the formula for the balance of an amortizing loan after $k$ payments: $$B_k=P\times \frac{(1+r{)}^n-(1+r{)}^k}{(1+r{)}^n-1}$$ Where: - $B_k$ is the balance after $k$ payments - $k$ is the number of payments made (36)Plugging in the numbers: $$B_{36}=25000\times \frac{(1+0.002333{)}^{60}-(1+0.002333{)}^{36}}{(1+0.002333{)}^{60}-1}$$$$B_{36}\approx 10421.30$$ Therefore, the balance on the loan at the end of year 3 is approximately $10,421.30, which matches choice A.