# 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?

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The formula isn't circular, it just contains the variable for the registration fee twice. If you bring both of those occurrences to the same side and then factor out the registration fee, it only occurs once and then you can solve for it. –  joriki Apr 21 '11 at 16:29

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.

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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.

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