Charlie cannot remember how much he financed to buy his car. he does remember that his monthly payment is $200. his add-on interest rate was 9% and he made a total of 30 payments. find the amount of his loan to the nearest penny.

First find the total payments
Total paid
200×30=6,000 (this is the future value)

Second use the formula of the future value of annuity ordinary to find the monthly payment.
The formula is
Fv=pmt [(1+r/k)^(n)-1)÷(r/k)]
We need to solve for pmt
PMT=Fv÷[(1+r/k)^(n)-1)÷(r/k)]
PMT monthly payment?
Fv future value 6000
R interest rate 0.09
K compounded monthly 12
N=kt=12×(30months/12months)=30
PMT=6000÷(((1+0.09÷12)^(30)
−1)÷(0.09÷12))
=179.09 (this is the monthly payment)

Now use the formula of the present value of annuity ordinary to find the amount of his loan.
The formula is
Pv=pmt [(1-(1+r/k)^(-n))÷(r/k)]
Pv present value or the amount of his loan?
PMT monthly payment 179.09
R interest rate 0.09
N 30
K compounded monthly 12
Pv=179.09×((1−(1+0.09÷12)^(
−30))÷(0.09÷12))
=4,795.15

The answer is 4795.15

Let's write equations to reflect this situation:

d = 2n + 5, and

n-2     1
= ---
d-2     3

Let's subst. 2n+5 for d, obtaining:

n-2              1
=   Cross multiplying, 3n - 6 = 2n + 3
2n+5-2        3       Then n = 9.  d=2n+5 becomes d=2(9)+5 = 23.

The fraction in question is   9/23.

Note that (9-2) / (23-2) = 7/21 = 1/3 (as specified in the problem statement)