Deduct percentage from x to reach desired value

Question

Let's suppose I need \$200,000 to pay a debt, no more, no less. So I placed a property to sell with an agent, but I know the agent will deduct 6% for himself, which will amount to \$12,000.

Since this will result in less money than what I need, through trial and error I calculated that I will need to sell it for about \$212,766 in order to receive \$200,000 on my hands.

Which formula could I use, instead of trial and error, to arrive at \$212,766, since $200000 + 6\%$ would still amount to less than the desired value?

Answer 1

You are looking for some quantity $x$ such $x-\frac{6}{100}x=200000$ (the quantity minus the $6\%$ of it equals the quantity you need). That is $$\frac{94}{100}x=200000,$$ or equivalently, $$\frac{47}{50}x=200000.$$ So $$x=\frac{50\cdot 200000}{47},$$ which is the quantity you have got.

Answer 2

If the sale price is $S$ then you want the solution to $$S\times (100\%-6\%)= \$200000$$

Answer 3

Let us call the amount you need to sell for $x$. You know that the agent will deduce $6\%$ od that money, so he will take $\frac{6}{100}\cdot x$ for himself. The rest of the money is yours, leaving you with $$x-(\frac6{100}\cdot x) = \frac{94}{100}x$$

You want that to equal $200.000$, meaning you just need to solve the equation $$\frac{94}{100} x = 200.000.$$