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Say I want to pay someone $100.00 but the service I use keeps 3% of the payment I make. Meaning they would only receive $97.00 since they would keep $3. If I increase my payment to $103 they would keep $3.09 meaning they would only receive $99.91. Is there a formula to calculate the exact extra amount I'd need to pay in order to offset the percentage fee?

(I know this is tagged incorrectly but I can't create tags yet and I didn't see any that fit.)

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4 Answers 4

up vote 2 down vote accepted

If $x$ is the quantity you pay, then they keep $0.97x$. Since you want $0.97x$ to equal $100$, you have \begin{align*} 0.97 x &= 100\\ x & = & \frac{100}{0.97}\\ x & \approx & 103.0928 \end{align*} so you would want to give them something between $$103.09$ and $$103.10$ (as it happens, 97% of 103.09 is 99.9973, which would be rounded up to 100 by most, and 97% of 103.10 is 100.007, which would be rounded up to 100.01 by most).

In general, if you want them to get $T$ and they only get $r$% of what you give them, then the formula is that you want to give them $x$, where $$x = \frac{100T}{r}.$$

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Oh duh. I didn't think of it like that. –  Spencer Ruport Mar 1 '11 at 20:47

You want to pay x to someone, but services take r%, you have to pay them y.

Since

x = y * (1 - (r/100))

Then

y = x * / (1 - (r/100))
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If you are going to get 97% of x, and you want to get $100, then:

100 = .97x

(divide each side by .97)

100/.97 = x

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In general suppose $FV$ is the future value of an asset and $PV$ is the present value of an asset. Then $FV = \frac{PV}{(1-d)^n}$ where $d = i/(i+1)$ (where $i$ is the annual effective interest rate) is the discount rate. In this example, $d = 0.03$, $n=1$ and $FV = 100$.

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