The logic behind the rule of three on this calculation First,
to understand my question, checkout this one:
Calculating percentages for taxes
Second, 
consider that I'm a layman in math.
So, after trying to understand the logic used to get the final result. I was wondering:
Why multiply $20,000 by 100 and then divide by 83? I know this is the rule of three, but, I can't understand the "internals"  of this approach. It isn't intuitive as think in this way:
Say 100% of one value, is the same of divide this value by 100. In other words: I have 100 separeted parts of this integer.
It's intuitive think about the taxes like this:
$$X - 17\% = \$20.000$$
So: 
$$\$20.000 = 83\%$$
For me, the most easy and compreensive way to solve this is:
$$\$20.000 / 83 = 240.96$$
It's the same as think, if 100% is 100 parts of one integer, 83% of that integer is the same of divide this integer by 83.
And finally to get the result:
$$\$20.000 + 17 * 240.96$$
My final question is:
How can I think intuitively like this using the Rule of Three?
In other words, why multiply 20.000 by 100 and then divide by 83 is a shorcut to get the result? 

The "rule of three" is an ancient ad-hoc mindless rote rule of inference that is best ignored. Instead, you should strive to learn the general principles behind it - namely, the laws of fraction arithmetic. Let's consider the example at hand. You  seek the number of dollars $\rm\:X\:$ such that when decremented by $\:17\%\:$ yields $\rm\:N\:$ dollars. First, recall that $\rm\  17\%\:$ of $\rm\:X\:$ means $\rm\displaystyle \frac{17}{100}\ X\:.\ $ Thus your equation is:
$$\rm N\ =\ X - \frac{17}{100}\ X\ =\ \bigg(1 - \frac{17}{100}\bigg)\ X\ =\ \bigg(\frac{100}{100}-\frac{17}{100}\bigg)\ X\ =\ \frac{83}{100}\ X $$
Thus $\rm\displaystyle\ \frac{83}{100}\ X\ =\ N\ \ \Rightarrow\ \ X\ =\ \frac{100}{83}\ N\ $ follows by multiplying both sides by $\rm\displaystyle\ \frac{100}{83}$
Note that we applied no ad-hoc rules above - just the basic laws of the arithmetic of fractions. These are the laws that are worthy of mastering.
It's interesting to look at the decline of the use of the "rule of three" over the last two centuries as the knowledge of general (abstract) algebra evolved. This is very easy using the recently-released Google Books Ngram viewer - which searches for phrases over 5 million books back to 1500. Browsing one of the earliest textbooks in the Google corpus containing the rule of three I noticed that it is immediately followed by a section titled "method of making taxes". So it seems this was a big application in the old days. Also notice how "fraction arithmetic" really ramped up circa 1960 (perhaps due to "new math" programs?).


