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?

For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — Henrik supports the community, May 23 '23 at 14:41
From what I think you meant... Since $X$ is first multiplied by $85%$ (substracting $15%$) and then by $110%$ (adding $10%$), $Y=110%\times85%\times X$. Maybe you can make your wording a bit clearer — TheMather - or rather AMather, May 23 '23 at 14:43
Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. — Community, May 23 '23 at 14:45
@TheMather-orratherAMather I want to take the result found in your equation as the value and get the initial value. I added an example. — Joseph, May 23 '23 at 14:51
@Joseph oh I see... Since $Y=\frac{110\cdot85}{10000}\cdot X$ (or $\frac{187}{200}\cdot X$), multiply both sides with $\frac{200}{187}$ and we get $X=\frac{200}{187}\cdot Y$. — TheMather - or rather AMather, May 23 '23 at 15:09

MPW · Accepted Answer · 2023-05-23T15:17:32.200

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: $$X = \frac{93.5}{(1+0.10)(1-0.15)} = \frac{93.5}{(1.10)(0.85)} = 100$$

score 1 · Answer 2 · edited May 23 '23 at 15:04

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.

How can I reverse the percentage of a number?

2 Answers2