Fritz Benjamin buys a car costing $23900. He agrees to make payments at the end of each monthly period for 6 years. He pays 10.8% interest, compounded monthly. What is the amount of each payment? Find the total amount of interest Fritz will pay.
Using the formula for the present value of an annuity, we can find the amount of each payment:
PV = (PMT * (1 - (1+r)^-n)) / r
where PV is the present value (i.e. the cost of the car), PMT is the amount of each payment, r is the monthly interest rate (10.8% / 12 = 0.009), and n is the total number of payments (6 years * 12 months/year = 72).
Substituting in the given values, we get:
23900 = (PMT * (1 - (1+0.009)^-72)) / 0.009
23900 * 0.009 = PMT * (1 - 0.108517)
215.1 = PMT * 0.891483
PMT = 215.1 / 0.891483
PMT = $241.62 per month
To find the total amount of interest Fritz will pay, we can subtract the original cost of the car from the total amount of payments and then subtract that amount from the cost of the car:
Total payments = PMT * n = $241.62 * 72 = $17,353.04
Total interest = Total payments - Cost of car = $17,353.04 - $23,900 = $-6,546.96
The negative result indicates that Fritz will not pay any interest on the car, because the amount of each payment includes both the principal and interest.