Solving for Variable

Date: 2/7/96 at 22:19:14
From: Anonymous
Subject: Solving for Variable

If;  Pt = PV(R) / (1-(1+R)EXP(-N)) is used to determine a Pt (monthly 
payment), if PV (present value), R (rate) and N (term) are known, how 
is this simplified to solve for R?

Date: 7/22/96 at 22:16:2
From: Doctor Jerry
Subject: Re: Solving for Variable

I'd like to change your notation a little.  

Let P stand for Pt, V for PV, and leave the remaining variables the 
same.  Your equation is (I think)

P=V*R/(1-(1+R)e^(-N)).

Start by multiplying both sides of this equation by (1-(1+R)e^(-N)).  
This gives

P*(1-(1+R)e^(-N))=V*R.

Expand the left side to get

P-P*e^(-N)-P*R*e^(-N)=V*R.

Move all terms involving R to the left and all other terms to the 
right.

-V*R-P*R*e^(-N)=-P+P*e^(-N).

Factor R from the two terms on the left and change some signs

R(V+P*e^(-N))=P(1-e^(-N)).

Finally,

R=P(1-e^(-N))/(V+P*e^(-N))

or, multiplying top and bottom of the fraction on the right by e^N,

R=P(e^N-1)/(V*e^N-P).

I hope this is clear.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/