Why are fractions the same as divisions? [duplicate]
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.
Solution 1:
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".
Solution 2:
It should not be too difficult to convince yourself that the sum of the pink areas in the single circle in the top row is the same as the sum of the pink areas in the three circles in the bottom row. Both are three quarters: the first as three quarters of a single circle, and the second as a one quarter of three circles
