TL;DR, if I reverse a 2.9% transaction fee, why do I increase my price by specifically 102.9866132%?
Background
In many provinces/states/countries, it is illegal to charge a convenience fee or surcharge for a credit card transaction. This is in dispute and in fact being reversed in some places, but for this exercise, let's add a nominal price to our product prices, building in the prospective transaction fee into the sale price, so our original sale price (and desired revenue) is recovered.
Think of this as what Walmart, McDonalds and other retail stores already do to cover salaries, electricity, rent, transportation, packaging, etc. but in this case we're only recovering one variable, credit card transaction fees.
My equation
The transaction fee in question is 2.9% or 0.029. So I came up with this equation, trying to recover the original price of the good,
$$original\_price * fee = money\_received$$ $$x * y = z$$ $$z = 0.971x$$
trying to recover the original price $x$ as money received $z$ creates,
$$z = x = 0.971xk$$ $$x = 0.971xk$$
when I ran the numbers manually using trial & error (also plugging this into Wolfram Alpha to verify) my answer for $k$ was,
$$k = 1.02987$$
This means I would have to increase my prices by 102.987% to recover the transaction fee before it occurs (assuming credit card fees occur 100% of the time).
I then tried to bring this from 5 decimal points to 10 and $k$ became,
$$k = 1.029866132$$
Why THAT solution?
If I began with trying to recover a 2.9% transaction fee and have to increase my sale prices by 102.9866132%, I'm curious as to why it's THAT number. Why is it not 102.9%? Is there a mathematical theory about how these numbers work? Should I just leave it as is and accept the answer and not over think it?
My spreadsheet
Explanation in laymans terms: Think of a percentage as "percent of something". It is the "something" that is different when the credit card company calculates the fee, on one hand, and you calculate the coverage of that cost, on the other.
The credit card company takes a fee which is 2.9% of the payment transaction (e.g. 100$).
To cover that fee, you calculate a price increase as a percentage of the amount you want to receive (97.1$)
See formula in comment of gt6989b (I need to learn how to enter formulae)