# Reason why multiplication gives fraction of a number

1. Sep 13, 2006

### mtanti

Why is it that when you do A * b/c the answer is one (b/c)th of A? What is the operation of multiplication doing to the two numbers so that this happens? Also how does this logic work when A is also a fraction?

2. Sep 13, 2006

### DaveC426913

Multiplication is mathematically equivalent to the word "of".

So 4 of 1/3 gives you 4 1/3rds - or 4/3rds.

Think of poker chips as counters. You've got a pile of poker chips, all with the value of "1/3" printed on them. If you have a poker chip, you have 1/3. Now what if you have 4 of them? Well, you literally have four(4) of the pile of thirds don't you? The chips you have value 4/3rds.

And if you have only 1/2 of a chip (half of a third), you have chips that value 1/6.

So, when you see the symbol 'x' (times), think 'of'.

Last edited: Sep 13, 2006
3. Sep 14, 2006

### mtanti

Yes but why is it that you use multiplication to findthat (1/3)rd or 4? From what you say you are just showing that you are finding 4 thirds and not (1/3)rd of 4.

4. Sep 14, 2006

### chroot

Staff Emeritus
It isn't.

$A \cdot \frac{b} {c} = \frac{A} {{c/b}}$

In other words, it's one "(c/b)th" of A.

In more detail,

$A \cdot \frac{b} {c} = A \cdot bc^{ - 1} = \frac{{Ab}} {c} = \frac{A} {{b^{ - 1} c}} = \frac{A} {{c/b}}$

- Warren

Last edited: Sep 14, 2006
5. Sep 14, 2006

### arildno

1/3 of 4 is the same as 4 thirds, by commutativity of multiplication.

6. Sep 14, 2006

### DaveC426913

What is the difference?

Q: If I have 4 pies, each cut into three, and I eat 8 of the peices, how much pie do I have left?

a] 4 'one-third' pie slices?
b] 4/3rds of a pie?
c] 1/3 of 4 pies?

A: all of the above

Last edited: Sep 14, 2006
7. Sep 14, 2006

### mtanti

The question is why is it that when you find 4 1/2s of a pie you are also finding 1/2 of 4? Is it simple commutivity? If so explain by logic what multiplication is doing instead of just adding 1/2 to itself.

In another thread it was concluded that multiplication is just a fast way of solving this problem (finding a fraction of a number) and that there is a longer and more logical way to do it. I was wondering if you guys knew something about it...

8. Sep 14, 2006

### chroot

Staff Emeritus
Yes, multiplication is commutative. That's the only answer. If you multiply some numbers in one order, you get the same result as multiplying in any other order.

- Warren

9. Sep 15, 2006

Do you want a mathematical operation of how multiplication works?

10. Sep 15, 2006

### mtanti

I know how to multiply, but I seem to misunderstand what it does in practice to physical quantities. Why is it that when you multiply 2 apples by 1/2 you get 1/2 of 2 = 1 apple.

11. Sep 15, 2006

### chroot

Staff Emeritus
Because multiplication is commutative. Could you think of any other sensible property?

- Warren

12. Sep 15, 2006

### DaveC426913

I don't know about the rest of you, and particularly mtanti, but simply repeating "multiplication is commutative" does not really provide a concrete understanding of the issue.

You're talking about mathematical theory while he's asking about apples. (OK, to be fair, he was talking theory at first too, but clearly he's looking for a more "grokable" answer.)

Last edited: Sep 15, 2006
13. Sep 15, 2006

### DaveC426913

I confess mtanti, like everyone else, I'm not sure what exactly you're having difficulty with.

What about the above apple-chopping does not make sense to you?

14. Sep 15, 2006

### chroot

Staff Emeritus
Yes, Dave.. I don't get it either. I don't see any other way multiplication could possibly work. I also don't understand what's confusing about "half of two apples = one apple" and "two halves of one apple = one apple."

- Warren

15. Sep 15, 2006

### mtanti

OK let me rephrase the question. Forget about the apples.

Why is it that to find a fraction of a quantity (say 1/2 of 100), you multiply that quantity by the fraction. What does multiplication have to do with finding fractions of quantities?

Better or worse explanation?

(I asked this question in the maths forum but they said that it was a philosophical question.)

Last edited: Sep 15, 2006
16. Sep 15, 2006

### chroot

Staff Emeritus
Halving an apple means dividing it by two, yes? Multiplying it by 1/2 means the same thing, because 2 and 1/2 are multiplicative inverses.

I really have no idea what you're asking us to explain, or what's not clear to you.

- Warren

17. Sep 15, 2006

### arildno

mtanti:
Are you at all familiar with the idea that you actually have to DEFINE what you want to talk about?

18. Sep 15, 2006

### eieio

mtanti:

Look at it like this:

1 * A = A

I think it's clear this makes sense to you concretely.

0 * A = 0

This also makes intuitive sense.

Doesn't it also make sense that if you choose a number smaller than 1 but greater than 0, that the result of multiplication should be between A and 0?

19. Sep 15, 2006

Debates based on topics such as this one really do not seem to have an end.

20. Sep 15, 2006

### arildno

Does it really?
Why?
Does that string of symbols mean anything independently from definitions you set up?

21. Sep 15, 2006

### chroot

Staff Emeritus
Go easy on him, arildno, he's a buddy of mine, and is only trying to help.

- Warren

22. Sep 15, 2006

### eieio

Based on observations of earlier posts, it seems that mtanti understands and accepts the idea and sense of those two "strings of symbols." I thought it made sense to point out the line between those points.

And no, they don't mean anything independently of the definitions. But then again, I don't think that's the issue here.

23. Sep 15, 2006

### arildno

Well, with all respect (and I do understand you wanted to start out with something mtanti might feel to be intuitively true), his main problem, as I see it, is that he goes around with all sorts of vague, unexpressed ideas of "what is multiplication" "what is a fraction", and these ideas somehow don't fit together properly.
Thus, I wanted to force him to re-think and express what he actually is thinking, and comparing these thoughts of his by the clear-cut definitions of mathematics.
Unless he is willing to speak the language of others, then it is rather futile trying to help him out.

24. Sep 15, 2006

### DaveC426913

I still think that my explanation in https://www.physicsforums.com/showpost.php?p=1083624&postcount=2" explains it.

multiplication means 'of'

So, 1 of 2 means:
Of the two apples, you have one. How much of the apples do you have? You have 1/2 of them. So, 1/2 x 2 = 1.

Likewise, 1/2 of 100 is the same as 1/2 x 100.

Last edited by a moderator: Apr 22, 2017
25. Sep 16, 2006

### mtanti

You know actually it's true, I do have problems expressing myself, even at school. Teachers just explain the surface of my questions and so the heart of the question remains unanswered. Still, there are questions which one cannot answer on his own so I need to ask for help when I need it...

I'll state what I think so that we'll straighten things up.

A * a/b = (A/b) * a
which means that you first break A into b equal pieces and then add the magnitude of 'a' of those pieces.

Is this it? b does not enter the multiplication process, it is just divided. But division is a process of multiplication. Therefore the question of what multiplication is doing when multiplicating A by 1/b to find (1/b)th of A is still present. What is happening to the 2 numbers so that the result is actually (1/b)th of A?

The part of the answer being between A and zero makes sense but how are you sure that the actual answer is the one you're expecting? I'm just trying to be certain of the basics instead of just accepting things as they are.