How can I reverse the percentage of a number?

Question

For example, I have an x ​​number. 15% of x number is taken as discount and subtracted from x number. Then 10% of the remaining result is added to the result as tax and the result is found. Let's call the result Y. I want to find the number X by giving the information Y and making the necessary percentage calculations. How can I do this I need an equation. Tax and discount values ​​because they will constantly change. How can I do that?

For example, when we apply a 15% discount to the 100 amount, 15 discount is applied and the current value is 85, then 10% tax and 8.5 tax are added to the amount to be paid to 93.5. I, on the other hand, want to get 100 when I enter 93.5 and keep the tax and discount rates the same, how can I do this?

Answer 1

Starting with $X$, you calculate $Y$ as follows:

$$Y = \underbrace{(X - 0.15X)}_{\textrm{discounted amount}} + \underbrace{0.10(X - 0.15X)}_{\textrm{tax on discounted amount}}$$ which simplifies to $$Y = 0.85 X + (0.10)(0.85X)$$ $$Y = 0.85 X + 0.085X$$ $$Y = 0.935X$$

So you can recover $X$ from $Y$ as $$\boxed{X = \frac{Y}{0.935}}$$

In your example, $X = 93.5 / 0.935 = 100$ as expected.

---

Addendum: A more general formula would be

$$Y = (1+t)(1-d)X$$ and the companion formula $$X = \frac{Y}{(1+t)(1-d)}$$ where the discount rate is $d$ (in your example, $0.15$) and the tax rate is $t$ (in your example, $0.10$). To see this, start with the more cumbersome but intuitive expression $$Y = \underbrace{(X - dX)}_{\textrm{discounted amt}} + \underbrace{t(X - dX)}_{\textrm{tax on discounted amt}}$$ and simplify as we did above (try to fill in the details!).

Checking your example with the general formula: $$X = \frac{93.5}{(1+0.10)(1-0.15)} = \frac{93.5}{(1.10)(0.85)} = 100$$

Answer 2

See that a $15\%$ discount leaves you with $85\%$ of the original amount. So your first step is $\$100 \times .85 = \$85.$ Then note that when you add tax you have $100\%$ of your original value plus $10\%$ which makes $110\%$, so your second step is to multiply by $1.10$ to get $\$85\times 1.1 = \$93.5.$

You could combine the steps:

$$\$100\times(1.00-0.15)\times(1.00+0.10) = \$93.5.$$

So to get what you want, you have to do the opposite:

$$\$93.5 \times \frac{1}{1.00-0.15}\times \frac{1}{1.00+0.10} = \$100.$$

Because division is the opposite of multiplication.