Calculating percentage to compensate for percent discount. Missing something very basic here and cannot pin point it.
We need to charge a client \$100 for a product. Let's say our payment processor charges us 10% on every transaction. We make this transparent to the client and charge them accordingly:
$\$x = \$100 / (1 - (10 / 100)) = $111.11111..$
Now, we want to offer to contribute to transaction fees such that they only have to pay half of what the payment processor charges. But clearly, $(\$100 + ((\$111.11111... - \$100) / 2))$ is not the answer.
The question is, how do we calculate the selling price such that the transaction fee is split half way between us and the client?
UPDATE: Based on the answers, here is the clarification that should have been part of the original question. $\$x=\$100$ is the pre-determined "selling price" to the client. We add whatever the payment processor charges us (10% in this case) and increase the selling price such that we receive $\$x=\$100$. Since we now want to "split" the processing fee with the client, we should lose from $\$100$ exactly how much extra the client is paying above $\$100$.
 A: As I understand it, you will charge the customer $p$ and receive $0.9p$.  You want to pay half the processing fee, so $0.95p=100$, $p=\frac {100}{0.95}\approx 105.263$  You receive about $94.737$, so you are paying $5.263$ as are they.
A: I understand you want to calculate the price $a$ such that your contribution $100-a$ equals the customer's contribution $\left(1+\frac{10}{100}\right)a-100$. 
(At least this is the case if the payment service adds ten percent to your request and charges an accordingly higher amount from the customer; see below for th eothre way around).
That is equivalent to 
$$\left(2+\frac{10}{100}\right)a=200$$
or 
$$a=\frac{2000}{21}= 95.\overline{238095} $$
(For $p$ % the answer would be $\frac{20000}{200+p} $)

If the payment service charges the original amount from the customer and subtracts ten percent from the amount and gives the rest to you, there is a slightly different siutuation:
Your contribution is $100-\left(1-\frac{10}{100}\right)a$ and the customer's contribution is $a-100$. Again we equate, but this time obtain
$$\left(2-\frac{10}{100}\right)a=200 $$
or 
$$a = \frac{2000}{19}=105.\overline{263157894736842105}$$
(For $p$ % the answer would be $\frac{20000}{200-p}$).
