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I'm working with a website that can be used to pay contractors on my behalf, instead of requiring them to submit to me their W9 for taxes. The website takes $2.75\%$ in processing fees.

If I'm paying someone $\$22$ per hour, and the website requires $2.75\%$, I believe that would be $\$0.60$ of each hour that would be paid to the website.

That would mean if I still wanted to pay the developer $\$22$/hr including the fees, I would effectively be paying him $\$21.40$ per hour.

My problem is with checking my math. I was trying to figure out how to take the $\$21.40$ and multiply it some value to reach the $\$22$, but I don't know how to do that.

What value times $\$21.40$ equals $\$22$?

[I also could not figure out why the dollar sign caused the post to lose its formatting so surrounded it in preformatted tags.]

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  • $\begingroup$ For formatting: if you want to display a dollar sign you need to put a backslash in front of it. As to the question...what's wrong with taking $\frac {22}{21.4}\sim 1.028$? $\endgroup$
    – lulu
    Commented Jul 9, 2016 at 15:04
  • $\begingroup$ Note: I reformatted your question. If you click on 'edit' you can see the syntax I used to display the dollar sign and the percent sign. $\endgroup$
    – lulu
    Commented Jul 9, 2016 at 15:10
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    $\begingroup$ As you see from the posted solutions, there is some confusion as to your question. Both solutions posit that you are asking "what nominal wage should I pay if I want my contractor to receive $\$22$ per hour?". That is a sensible question, but it isn't what you actually ask. Indeed, you ask "what value times $\$21.40$ equals $\$22$?" which is a different question. Perhaps you could clarify which one you intended to ask? $\endgroup$
    – lulu
    Commented Jul 9, 2016 at 15:25
  • $\begingroup$ I'm sorry for the confusion. I really meant this to be an academic question. I intended to ask: If \$22 - (\$22/hr * 2.75%) = \$21.40, how do I go the other way around? In other words, if I was given that the rate after the 2.75% was applied was \$21.40, how do I get back to the \$22/hr? I'd also of course like to confirm that if I want to pay the developer \$22/hr including fees, his hourly rate would indeed be \$21.40/hr. $\endgroup$
    – Alex Regan
    Commented Jul 10, 2016 at 0:06
  • $\begingroup$ Ok. So...let $X$ be the nominal wage. We know that $X-.0275X = 21.4$ . But $X-.0275X = .9725X$ so we have $.9725X = 21.4\implies X=\frac {21.4}{.9725}\sim 22.00514139$. That's your $\$22$. The slight difference comes from the fact that you rounded initially ($22-.0275\times 22=.9725\times 22=21.395$ which you rounded to $21.4$). $\endgroup$
    – lulu
    Commented Jul 10, 2016 at 1:30

1 Answer 1

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If the processing fee is $2.75\%$ of the amount processed, and you want to have $\$22$ after the fee is taken out, then you have the following equation: $$x-x\times2.75\%=22,$$ where $x$ is the initial amount (i.e. before the processing fee is taken).

Read the equation as:

  1. From the initial amount $x$
  2. take out $2.75\%$ of the initial amount $x$,
  3. and that should be equal to $22$.

Factorization of the left-hand side (further referred to as LHS) gives $$x\left(1-2.75\%\right)=22$$ (multiply out to check); then notice that a percent is exactly one hundredth of the unity: $$x\left(1-2.75\frac1{100}\right)=22;$$ now rewrite the unity as $100/100$ and multiply the $2.75$ by the fraction, which in this case just moves $2.75$ into the numerator: $$x\left(\frac{100}{100}-\frac{2.75}{100}\right)=22;$$ denominators are now equal, so we can bring the numerators over one fraction bar: $$x\left(\frac{100-2.75}{100}\right)=22;$$ perform the subtraction in the numerator: $$x\frac{97.25}{100}=22;$$ divide both sides by the fraction: $$x\frac{97.25}{100}/\frac{97.25}{100}=22/\frac{97.25}{100};$$ that gets rid of the fraction on the LHS: $$x=22/\frac{97.25}{100};$$ division by a fraction is equivalent to multiplication by the same fraction but with numerator and denominator swapped: $$x=22\frac{100}{97.25};$$ multiply the integer $22$ by the fraction, which brings it into the numerator as a multiple: $$x=\frac{22\times100}{97.25};$$ perform the multiplication: $$x=\frac{2200}{97.25};$$ we arrived at the desired answer; the fraction may be further simplified, or a decimal approximation up to four decimals after the decimal point may be obtain by division on a calculator: $$x\approx22.6221;$$ round the value up (need to explain why?), which gives

$$22\text{ dollars and }63\text{ cents.}$$

P. S. Notice that this StackExchange site employs MathJax to enable $\LaTeX$ typesetting of mathematical formulae. For a basic tutorial and reference on the language, please refer to this link.

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