Help to solve equation with percentages to form sale prices

Question

I have some values where are applied dynamic taxes, for example:

100 - 3.60% = 96.40

200 - 7.69% = 184,62

But before subtract these taxes i need to find a second percentage, for example:

(100 + x%) - 3.60% = 100

(200 + x%) - 7.69% = 200

This is done because i need to know what percentage is necessary to add to get the original value after apply a tax. Basically, this is important to get the right product sale price, because the seller have to pay some taxes, then he needs to pass along them to the buyer and receive the expected value.

Could anyone help me with this one?

Answer 1

If you have an amount $A$ and you want to add $x$%, you can use the formula

$$A+x\%=A\cdot(1+\frac{x}{100}). $$

So if you have

$$(100 + x\%) - 3.60\% = 100,$$ you can simply use $$(100(1+\frac{x}{100}))(1-\frac{3.6}{100})=100, $$ which expands to $\frac{96.4}{100}x+96.4=100 $, which has the solution $x≈3.73444 $.

Answer 2

$(200 + ((1/.9231)-1)*100\%) - 7.69\% = 200$

$(200 + 8.33\%)-7.69\% = 200$