Calculating a total based on a percentage I have an event that I am running and I need to figure out what to charge participants so that I break even.
I know the formula for doing this:
Registration Fee = Expenses / Participants

HOWEVER, there is also a company fee that is assessed, which is where it gets tricky. The company charges 10% of the total revenue as a support fee. So if I use the formula above, then my total expenses will be higher than my income because I just added 10% of my income to my expenses for the support fee.
I then have to increase the registration fee until I break even. Right now I do this manually, but there must be a formula that I can use.
Of course, this formula is circular and won't work
Registration Fee = (Expenses + (Registration Fee * Participants * .10)) / Participants

Does anyone know how to solve a formula I can use to solve this problem?
 A: Suppose you have $N$ participants and each participant pays $x$ dollars for the event as a registration fee. You will have to pay the company a $10$% of $Nx$ dollars, i.e., $Nx/10$ dollars. Suppose you also want to make a profit of $P$ dollars (in your case, it seems you need $P=0$).
There are two cases to consider:


*

*The expenses are a fixed quantity $E$, and $E$ is constant regardless of the number of participants. Then, your net balance at the end of the day should be $P$ dollars, and the net balance will be income ($Nx$) minus expenses ($E$), minus the company fee ($Nx/10$). Thus:
$$P=Nx - E - \frac{Nx}{10} = \frac{9Nx}{10} - E.$$
Since the only unknown quantity is $x$, we can solve for $x$ and deduce that we need a registration fee of:
$$x= \frac{10(P+E)}{9N}.$$
If $P=0$, then $x=\frac{10E}{9N}$.

*The expenses depend on the number of participants, i.e., the expenses are $e$ dollars per participant. Thus, the expenses grand total is $Ne$. Then, your net balance at the end of the day should be $P$ dollars, and the net balance will be income ($Nx$) minus expenses ($Ne$), minus the company fee ($Nx/10$). Thus:
$$P=Nx - Ne - \frac{Nx}{10} = \frac{9Nx}{10} - Ne.$$
Since the only unknown quantity is $x$, we can solve for $x$ and deduce that we need a registration fee of:
$$x= \frac{10(P+Ne)}{9N}.$$
If $P=0$, then $x=\frac{10Ne}{9N}=\frac{10e}{9}$. (Notice that here $e$ is expenses per participant, and not total expenses.)
Note that these two cases are, in truth, one single case, since $E=Ne$ (the total expenses equal the expenses per participant times the number of participants), but I thought it would be clearer to break it into these two cases. 
A: Hint: To leave 10% for the support fee, your expenses without the support fee have to be $90\%$ of your total revenue.  You don't have to sum the geometric series, but that is another approach.
