Using the formula for an annuity, what are the monthly payments on a 5-year fixed-rate car loan for $20,000 if the effective annual rate is .035 (3.5 percent)? Assume the first payment is exactly one month (1/12th of a year) from now. (The effectively monthly rate is then (1.035)(1/12) -1.

After 2 years, when there are 3*12 = 36 monthly payments left (with the next payment being exactly 1 month in the future), how much will the borrower still owe in remaining principle?

Next consider a loan where you can make the payments twice a month (still for 5 years), with the first payment in exactly half a month (1/24th of a year). The effective annual rate on this loan is also .035 (3.5%). What are the twice-monthly payments? How does 2 times the twice-monthly payment compare to a single monthly payment? Why is it bigger/smaller?   

In 2 years (after 2*12*2 = 48 twice-monthly payments have been made), when there are 3*12*2= 72 remaining payments to be made (with the next payment being exactly half a month from then), how much will the borrower still owe in remaining principle? Is the answer different between the monthly-payment loan and the twice monthly-payment loan? Why?