Reverse percentage reduction formula

Question

Consider i need to reduce a value by percentage

100 - 10% = x
100 - (100 * 10 / 100) = 90

So we know x is 90
Now how do i reverse it? consider this question:

x - 10% = 90

How i can find x? What will be the formula?

Answer 1

The problem is that you are throwing around symbols without rigor.

The "equation" $$100-10\%=x$$

is meaningless. Specifically, the "$-10\%$" doesn't make sense. You aren't subtracting ten percent, you are subtracting ten percent of something (even though people usually say it that way).

Your question is "If I reduce a number by $10\%$ and the result it $90$, what was the original number?". So be more rigorous in using mathematical notation to represent the facts:

$$x - \tfrac{10}{100}x = 90$$ Solve for $x$: $$\tfrac{100}{100}x - \tfrac{10}{100}x = 90$$ $$\left(\tfrac{100}{100} - \tfrac{10}{100}\right)x = 90$$ $$\left(\tfrac{90}{100}\right)x = 90$$ $$\left(\tfrac{100}{90}\right)\left(\tfrac{90}{100}\right)x = \left(\tfrac{100}{90}\right)90$$ $$x = 100$$

Answer 2

Using your notation you would do:$$x-10\%=90$$$$x-x*\frac{10}{100}=90$$$$x\left(1-\frac{10}{100}\right)=90$$$$x*\frac{90}{100}=90$$$$\therefore x=100$$