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 $\$100$.