# 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?

• You don't have to divide it evenly. You don't even have to divide it at all; you could run home screaming "mine, mine, mine" and keeping all four if the apples to yourself. But division is defined what will each person get if we divide it evenly. (You could just as easily ask "If you do multiplication why does everyone get the same amount" what if big loud sue insists she must have one more apple the timid joe no matter what and horrible hugh insists he must have twice as many as sue.) Mar 21, 2017 at 20:47

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: $4+4+4=12$

• Aha. So, when we have 10/5 = 2, what we're really saying is: " 10 divided in 5 EQUAL parts is 5 parts of 'lenght' 2"?. Mar 21, 2017 at 21:08
• @J.Doe You can say it like that, yes. Mar 21, 2017 at 21:14
• "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." I understand that no matter what, initially, we have to think in terms of equally division. Then if we want to create a non-equal division, this new division is based on the first (which is eaqually division). I almost understand it. But I don't understand what you mean by ratio x:y. Can you give a real example, with constants? Mar 21, 2017 at 21:27
• @J.Doe I added an example. Mar 21, 2017 at 21:39
• @J.Doe highlighted the non-equal and equal examples explained over ratios. Mar 21, 2017 at 21:57

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.