Mohammed A. answered • 12/13/19
University Lecturer/PhD Candidate
So the first part is obviously the $9000 paid for the car. She borrowed to pay for the rest of the car value. So we need to find out how much she borrowed.
We know she will pay the bank 470 x 36 payments= $16920. However, parts of this are interest and the rest are principal payments. since the compounding is monthly, we convert the interest to 7/12= 0.583% or 0.00583 monthly. Lets call this r. We start from the standard monthly payment formula:
Monthly Payment = Principal x (r) x (1 + r)36 / ( (1 + r)36 -1).
--> Principal = Monthly Payment * ((1+r)36 -1) / (r x (1+r)36
Principal = (470 x (1.00583)36 -1 ) / (0.00583 x (1.00583)36 )
Principal = 470 x (1.233- 1) / (0.00719)
= 15230.9 (note I rounded up during calculations so the answer would be slightly different if we only rounded up at last step)
a little note on some of the elements of the formula: so (1+r)36 is a compounding formula. it tells you how a balance will increase if compounded 36 times by the interest rate r. so if we borrow $1 at 0.583%, it will become $1.0583 next period, then we take the 1.0583 and add another 0.583% applied to the new balance, we get 1.0583^2 which is 1.12 and so on.
The formula is derived from observing that each month, there's a payment of 470 -- a part of this goes to paying the interest accumulated on the principal, and a part goes on to pay part of the principal and reduce it. next period there will be less balance on the loan so the interest is less and even more of the 470 goes to pay off the principal. In the last period very little is left of the loan so almost all of the 470 goes on to pay the principal and reduce it to zero.
