Why division gives equal parts? I'm ashamed to ask this question. For a while, I can't pull out of my mind really basic kindergarten questions like: what is actually a number, what is a function and what is the connection between the name (function) and its purpose; what is association; what is what I actually learned so far, etc.
I probably shouldn't do math. But, anyway, what is division? 
2 x 3 = 2 + 2 + 2 = 3 + 3 = 6.
Now, 2b = 6, who is b? We divide both sides by 2 and the answer is: b = 3.
For multiplication is easy. We define multiplication of 2 integers a x b to be the addition of a with itself b times, or vice versa.
   But what I don't understand is when it comes to consider just and only just this: 1/2 = 0.5. When I was little I was told: if you give one apple to two persons, you must divide it equally. That's the problem for me: why a number, when divided by another, it's divided in equal parts? Why 4 apples distributed to 2 persons is 2 apples for each one (4/2 = 2)? Why not 3 for one and 1 for another one? Or why not 1.589348 for one and the rest for the other one?
I guess, when we see a fraction like: a/b we could read it like: " What number multiplied by b gives a?". Ok, that makes sense. But when I say: " a divided by b is c" it does not makes sense for me anymore. It again does if I say: " a divided in b equal parts gives c".
So it's because division must be made fairly? Or it's a logical thing and it also happens that it's a fair thing?
I appreaciate your answer.
 A: If you would define non-equal division, you would need to specify in what way you would like to divide. 
It would mean something like divide $n$ apples to Tom and Jerry in some non-equal ratio, say ratio $x:y$, then the $x$ and $y$ fraction assumes equal division.
The non-equal division is naturally explained through equal division.
Hence instead of saying divide $n$ apples among Tom and Jerry in the $1:1$ ratio, we would say divide $n$ apples among Tom and Jerry equally. 
But we do not include the word "equally" since equal division is naturally assumed unless specified otherwise. This might also be something we humans agreed on mutually. Thus, we just say divide $n$ apples among Tom and Jerry.
Writing that as $n/2$.
Where on the other hand, non-equal division needs you to define how you would like to divide. Can't say give all to Tom (no division takes place) or something like "give a lot to Tom" because you need to specify what is "a lot" in some way. 
If you do not specify the way to divide, there is only one way to divide it, something we all agreed on, and that's to divide it equally. And that's because it makes sense mathematically when referencing multiplication, as seen in Peter's answer.
If it would have been a specified division in a $x:y$ ratio, we would write that Tom and Jerry get respectively:
$$ \frac{n}{x+y}\times x,  \frac{n}{x+y}\times y$$
Where you again see that these fractions assume equal division. Non-equal division is noted by the equal division, and equal division is, well, just division as we defined it.

I guess I need to explain ratios to you now.
Story example using non-equal division could be something like divide $12$ apples among the two brothers, where the older brother gets twice as much as the younger brother. 
Our ratio is $1$ to $2$, which we write $1:2$ or $1/2$.
That's because for each one that the younger brother gets, older one gets twice as much, and that's two.
Then when using $12$ apples, for each $4$ the younger one gets, older one gets $4\times 2 = 8$, it's divided in ratio of $4:8$, divide both numbers by $4$ to get $1:2$.
$4$ to $8$ is actually $1:2$ but stacked $4$ times to reach $12$.

When dividing one to twice as much, 1:2, then $\frac{12}{1+2}=4$ gives
  us the number of times we need to stack, and we just stack the ratio
  then, $(1\times4 ): (2\times4) = 4:8$.

You can have a third brother which gets $3$ times as much as the youngest. If we give the smallest number of items to the youngest, he gets $1$. second one twice as much, $2$, third one three times as much as the first one, $3$. We have a $1:2:3$ ratio that we can stack. 
Notice that $1+2+3=6$, we can divide $6$ apples at a time.
Thus, you can stack up these ratios and divide them if you have $6,12,18,...$ apples. 
Stacking it twice gives $2:4:6$, which is the number of apples each brother gets.


Dividing just $12$ apples among two brothers equally, is like dividing
  $2$ at a time to make sure everyone has the same amount at each step.
  We stack the $1:1$ ratio $6$ times because $\frac{12}{1+1}=6$, to get
  $(1\times6):(1\times6) = 6:6$, each one gets $6$.

For three brothers, it's $3$ at a time in $1:1:1$ ratio, which you can stack $4$ times, hence all get $4$ since we have $4:4:4$ and all apples are used: $4+4+4=12$
A: In the case the denominator is a positive integer, we alway get equal parts, that is because of 
$$\underbrace{\frac{a}{b}+\frac{a}{b}+\frac{a}{b}+\cdots+\frac{a}{b}+\frac{a}{b}}_{b\ \ times\ \frac{a}{b}}=b\cdot\frac{a}{b}=a$$
So, if we denote $u=\frac{a}{b}$, $a$ can be divided in $b$ parts $u$, so the parts are always equal.
