Two step percentage discount how to arrive to x

Question

Let's say we have a value of $100$ to which we apply a $20$% which arrives at $80$. Now let's apply a two step percentage decrease so $100 - 10$% - $x$% $= 80$. To get there $x$ would be roughly $11,1$%. Can anyone help how the actual formula should look as for the moment I can only pinpoint that by trial and error?

Answer 1

Start with a number $n$ (=100). We want to reduce it by $k$% (=20%). Call this method 1. Instead, we want to go by method 2 where:

1. Reduce by $\frac{k}{2}$%.
2. Then reduce the resultant by $x$%.

We want to find x such that the results of method 1 and method 2 are same.

$n - \frac{k}{100}n = n - \frac{k/2}{100}n - \frac{x}{100}\left[n-\frac{k/2}{100}n\right]$

Simplifying, we get:

$x = \frac{k/2}{1-\frac{k/2}{100}}$

Plugging in $k$ = 20 gives $x$ = 11.111.