Why are fractions the same as divisions? Ever since I learned about fractions in Elementary School, I've known how to work with them. The problem is, although I remember the mathematical rules, I don't remember how I assimilated the equality between fractions and divisions when I was younger. This is really troubling to my mind because it feels like I partly understand this concept. I know that the denominator represents the amount of parts a whole was divided into and that the numerator represents the amount of parts I'm working with. Let's say, I want to divide 3 by 4. The result of this division is 3/4. But, in this case, I'm dividing 3 wholes by 4. Doesn't this come into conflict with the definition of the denominator, which is the amount of parts in which 1 whole is divided into?
  This confusion is really bugging me and I'd really appreciate if someone could clear it up. Thanks for the attention.
You are right that there are two things going on here, and it doesn't seem obvious that they are the same.


On the one hand you have $3$ units and you take a fourth of that (that's dividing $3$ by $4$).
On the other hand you have a unit, cut it in four parts and keep only three of these parts (that's the fraction "three fourths").


The reason why they are the same is that you can achieve the first operation by cutting each of your three units into four parts, and take one part from each. By doing that, you took a fourth of your three units, and at the same time what you have in your hands is three "fourths of a unit".
Three over four is literally another way of writing "thre divided by four", which is 0.75. 

When you see three over four "on its own" that means "0.75 'of one' ", which is of course 0.75.

Three over four, times four is of course three.  Three over four times a hundred is of course 75.  Three over four times a New York pizza is of course six slices of pizza. Three over four of your income is what the tax man takes. And so on.

If you see three over four "on it's own", it's just a convention that it means "times one".


  " I'm dividing 3 wholes by 4. Doesn't this come into conflict with the definition of the denominator, which is the amount of parts in which 1 whole is divided into?..."


Where you wrote "1 whole" just above - near the end - you meant "the whole thing"...

That is to say "the whole thing in question"...

This is completely non-mysterious: in English it's normal to say "one whole thing" where the "one whole thing" might be "a swimming pool", "the USA", "1", "88", "the color blue," or whatever. 

(The English phrase "one whole thing" happens to have a the word "one" in it.  Much as one may say, "One must carefully brush one's hair before meeting one's grandmother." It has no connection to "1.0".)

The "whole thing" here is indeed ..... "3".  That's all there is to it.

Just as you say, the denominator ("4.0" here) is what you divide in to the "whole thing".  Here the "whole thing" is 3.

That's all you're seeing there!