I'm looking at an online service which I give it a dollar amount and it returns a transaction fee on top of that, I want to know what the maths is behind the transaction fee based off this data.
If I enter.. - 1 it charges a total of 1.63 - 100 dollars it charges a total of 102.40 - 45 dollars it charges a total 46.42
Any idea how to calculate how the differences are calculated?
Yes. We will assume it is linear and can be expressed in the form of $y=mx+b$.
In this case, $\text{final charge}=m\times\text{initial amount}+\text{flat fee}$
You basically have a graph of three points and want to find the equation for the line that goes through them.
There are several methods to solve for m and b. You can look up how to solve a linear equation given two points. In this case we can use the points $(1,1.63)$ and $(45,46.42)$ to get the equation and use the third point as a check to see if the equation is correct.
This gives the equation $$\text{final charge}\approx1.01795\times\text{initial amount}+0.61205$$