In order to solve this problem, we use the formula for the present value of an annuity:
PV = (p/r) * (1-(1/(1+r)^n)
Where PV=present value, p=periodic payment amount, r=rate of interest, and n =number of payments
We want to find the monthly interest rate, so we set n=60 (12 months x 5 years), and PV = 18,349, which is the amount owed after the down payment
18349 = 352/r * (1-(1/1+r)^60) Multiply both sides by r/352
52.128r=1-(1/(1+r))^60
Solving from here is a doozy, so let's try substitution to find which equality gives us the correct value
Remember that 12r is actually our APR, since r in our function is just the monthly rate of interest, not the annual rate.
Let's try 5%. 0.05 ÷ 12 = .00416
52.128(.00416) ?=? 1-(1/(1+r))^60
.2172 ?=? 1- (1/1.00416)^60
.2172 ?=? .22048
.2172 does not equal .22048, although we're close! Although they are very similar numbers, they are about 1.5% apart, which is a pretty big difference for this problem. Let's try 5.9%, the next closest option, to see if we can get closer.
Let's try an APR of 5.9%, .059 ÷ 12 = .004916
52.128(.004916) ?=? 1- (1/1.004916)^60
.2562 ?=? 1-0.745
.2562 = .2549
Our values are equal within about one thousandth of eachother, for a difference of roughly 0.5%. Based on the other options, this is as close as we're going to get! Keep in mind that there is a lot of rounding that happens in these problems, so the values won't be exactly the same.
Answer is C) 5.9%
