You can't get an analytical solution for r
. You need to solve it numerically, via some iterative method.
Here is the simplest iterative method I can think of. First, rearrange your equation to solve for one of the r
's:
r = (1000/7253.93)*(1 - (1 + r)^(-9))
Now, since we can only solve for one r
at a time, we rename the r
on the right side to r_0
and the one on the left side to r_1
:
r_1 = (1000/7253.93)*(1 - (1 + r_0)^(-9))
Now, take a geuss of r_0
. Try r_0 = 0.5
, since r
should be between 0%
an 100%
. Plug it in and solve for r_1
:
r_1 = (1000/7253.93)*(1 - (1 + r_0)^(-9))
r_1 = 0.1343
Now, do it again, with r_1
on the right-side, and solve for the next `r_2 on the left side:
r_2 = (1000/7253.93)*(1 - (1 + r_1)^(-9))
r_2 = 0.09350
Repeat the pattern:
r_3 = (1000/7253.93)*(1 - (1 + r_2)^(-9))
r_3 = 0.07619
and keep going:
r_4 = 0.06667
r_5 = 0.06073
r_6 = 0.05677
...
r_34 = 0.04547
and after 34 iterations you have a pretty close solution. There are better iterative methods out there, but this is the simplest.
Alternatively, you can plug it in to
Wolfram Alpha which gives the expected answer of r=0.04545...