What type of relationship intercede between fractions and division?

[gullpacha]

*QUESTION DETAILS*
Let's start with a fraction, it is just a number which represent a value by telling us how many pieces of a certain size we have.
Now, what about another way to represent this value, like considering it as a single piece of that size? Well, we can take a bigger number, and divide it into equal pieces of the size we need, to make one of that pieces equal to the value we want to represent.
So, what value, divided by another value, gives the piece we need? Exactly! The same values of a fraction, numerator and denominator, but why this relationship? Why does it work? Is there an axiom/theorem that makes it possible, or is it just a coincidence?
EDIT: e.g. What relationship makes the number 5/2 equals to what the division between the number 5 and the number 2 gives as result?
EDIT 2: What I'm looking for is, if possible, a more intuitive way to understand deeply this relationship without the use of algebraic expressions to prove it or rules that "make it true and stop".
EDIT 3: Ok so I thought this, let's say we have the fraction 1/5, it means we have one fifth of a whole and to get a fifth of a whole we can just divide the whole in 5 pieces and get the size of one.
Now, let's say we have 2/5, as a fraction, it still make sense, it still means we have two fifth of a whole, but as a division, it suddently stop working, because in a division, we're not working with a single whole anymore, we're working with two wholes.
Why does this works than? Well because division still takes only the size of a piece, while fractions can get the size of more pieces togheter, so how do we make it work? We can just double the starting integer so that when divided by the same size as before, the size of a single piece is equivalent to the size of two pieces of the previous value.
But now, why is the number (in this case the number 2), still valid for both the division and the fraction, why is it still appliable to both and so on with all the other numbers?

 

[Dr.Peterson]

It's hard to be sure exactly what you are thinking, but I'll try.

A fraction is introduced in these two ways (m pieces out of n of a whole, and m items out of n in a set), but ultimately it is better defined simply as a division that hasn't been carried out. The number 2/5 means 2 divided by 5; that is, it is a number that, when multiplied by 5, yields 2.

I used to share two apples among my family of five in exactly the way you describe. I would cut each apple into five pieces, and give two pieces to each person. That way each got 2/5 of an apple (which adds up to 10/5, which is the two apples). And you can think of this as carrying out the division of 2 by 5, by dividing 1 by 5 and doubling the result.

Now, what is it that you don't understand?

 

[Otis]

… Now, let's say we have 2/5 … we have two fifth of a whole …

… we're not working with a single whole anymore, we're working with two wholes.

Hi gullpacha. It seems like you're associating two 1/5ths with two wholes. (Would you say also that 4/5 denotes four wholes?)

When we talk about fractional parts of a whole, there is only one whole. When that whole is separated into five equal pieces, then each piece is 1/5th of the whole. If we consider two of those pieces, then we're considering 2/5ths of the whole. If we take four of the five pieces, then we have 4/5ths of the same whole.

\(\dfrac{2}{5} \; = \; 2 × \dfrac{1}{5} \; = \; \dfrac{1}{5} + \dfrac{1}{5}\)

\(\dfrac{4}{5} \; = \; 4 × \dfrac{1}{5} \; = \; \dfrac{1}{5} + \dfrac{1}{5} + \dfrac{1}{5} + \dfrac{1}{5}\)

… so how do we make it work? …

Please excuse me, but I don't understand your question. How do we make what work?

?

 

[gullpacha]

It's hard to be sure exactly what you are thinking, but I'll try.

A fraction is introduced in these two ways (m pieces out of n of a whole, and m items out of n in a set), but ultimately it is better defined simply as a division that hasn't been carried out. The number 2/5 means 2 divided by 5; that is, it is a number that, when multiplied by 5, yields 2.

I used to share two apples among my family of five in exactly the way you describe. I would cut each apple into five pieces, and give two pieces to each person. That way each got 2/5 of an apple (which adds up to 10/5, which is the two apples). And you can think of this as carrying out the division of 2 by 5, by dividing 1 by 5 and doubling the result.

Now, what is it that you don't understand?

i want to know that in division we take wholes like 2 divide by 5 means two whole apple divide between 5 persons ans in fraction mean taking 2 peices from 5 are these ideas opposite of each other so how to illustrate them logically thank tou

 

[Dr.Peterson]

Why do you think they are opposite?

Didn't I demonstrate that the result is the same, each person getting 2 of 5 pieces of an apple?

Isn't it clear that 5 times 2/5 is 2?

 

[Otis]

… wholes like 2 means two whole apple …

As I said before, gullpacha, if you're going to interpret a whole divided into equal pieces, then there is only one whole.

In your first interpretation, the two apples comprise a single whole. That whole is the number 2. We are chopping the number 2 into five equal pieces.

2/5 + 2/5 + 2/5 + 2/5 + 2/5 = 2

Each piece has size 2/5.

Now, if you want to think of 2/5ths as the fractional amount of one apple, then, yes, cut the apple into five pieces and take two.

Those two pieces represent 2/5ths of one apple, AND they also represent 1/5th of two apples.

In other words, 2/5ths of one apple is exactly the same amount as 1/5th of two apples.

2/5 = (1/5)(2)

In math, we may often view an object in different ways; that doesn't mean we're looking at different objects.

Here's something else to think about:

315 ÷ 7 = 45

What could be the meaning of the number 45?

45 is the size of each piece, after dividing 315 into seven equal pieces.
45 is the number of pieces, after dividing 315 into pieces of size seven.

There are two different ways of viewing the number 45.