Amortization Problem

[KelvinScale]

Hi everyone, I am new here, and stuck on a question from the Mathematical Interest Theory Textbook:


A thirty-year mortgage has level payments. The amount of principal in the 82nd payment is 259.34, and the amount of principal in the 56th payment is 230.19.
The question asks what the interest payment is in her 133rd payment.

The notation we learnt was:
Given Q (monthly payment) and v (discount rate), a given time period t, and total number of period n, we have that

Payment at time t = Pt = Q v^n+1-t
Interest at time t = It = Q (1-v^n+1-t)


Hence I get the following equations:

Payment at time 82 is Q*v^279
Payment at time 56 is Q*v^305

I solved for V(1.004596489), and solved for Q(56.83), but when I subbed it into the interest equation for time period 133, Q(1-v^(228)), I did not get the right answer ($604.59)....

Does anyone have a better approach to the question, or spot where I went wrong?
Any insight would be much appreciated!

 

[tkhunny]

This little piece of the world is so much more obvious when you start form the back.

Let's start with:

P = Level Payment
i = Periodic Interest
v = 1/(1+i) = Periodic Discount Factor
n = Number of Payment

The nth payment has Pv principal in it. The second to last payment (n-1) has Pv^2 principal in it. etc.

nth = n-0 ==> Pv = Pv^1 Principal ==> P - Pv^1 = P(1-v^1) Interest
n-1th ==> Pv^2 Principal ==> P - Pv^2 = P(1-v^2) Interest
n-2th ==> Pv^3 Principal ==> P - Pv^3 = P(1-v^3) Interest
...
n-jth ==> Pv^(j+1) Principal ==> P - Pv^(j+1) = P(1-v^(j+1)) Interest
...
n-(n-1) = 1st ==> Pv^n Principal ==> P - Pv^n = P(1-v^n) Interest