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Reason why multiplication gives fraction of a number

  1. Sep 13, 2006 #1
    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. jcsd
  3. Sep 13, 2006 #2

    DaveC426913

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    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
  4. Sep 14, 2006 #3
    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.
     
  5. Sep 14, 2006 #4

    chroot

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    It isn't.

    [itex]
    A \cdot \frac{b}
    {c} = \frac{A}
    {{c/b}}
    [/itex]

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

    In more detail,

    [itex]
    A \cdot \frac{b}
    {c} = A \cdot bc^{ - 1} = \frac{{Ab}}
    {c} = \frac{A}
    {{b^{ - 1} c}} = \frac{A}
    {{c/b}}
    [/itex]

    - Warren
     
    Last edited: Sep 14, 2006
  6. Sep 14, 2006 #5

    arildno

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    1/3 of 4 is the same as 4 thirds, by commutativity of multiplication.
     
  7. Sep 14, 2006 #6

    DaveC426913

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    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
  8. Sep 14, 2006 #7
    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...
     
  9. Sep 14, 2006 #8

    chroot

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    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
     
  10. Sep 15, 2006 #9
    Do you want a mathematical operation of how multiplication works?
     
  11. Sep 15, 2006 #10
    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.
     
  12. Sep 15, 2006 #11

    chroot

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

    - Warren
     
  13. Sep 15, 2006 #12

    DaveC426913

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    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
  14. Sep 15, 2006 #13

    DaveC426913

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    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?
     
  15. Sep 15, 2006 #14

    chroot

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    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
     
  16. Sep 15, 2006 #15
    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
  17. Sep 15, 2006 #16

    chroot

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    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
     
  18. Sep 15, 2006 #17

    arildno

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    mtanti:
    Are you at all familiar with the idea that you actually have to DEFINE what you want to talk about?
     
  19. Sep 15, 2006 #18
    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?
     
  20. Sep 15, 2006 #19

    radou

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    Debates based on topics such as this one really do not seem to have an end.
     
  21. Sep 15, 2006 #20

    arildno

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