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Combining Proportionality Statements

  1. Jun 29, 2011 #1
    If you have two statements, "a ∝ b" and "a ∝ c", you would get: "a = xb" and "a = yc" (where x and y are the constants of proportionality)

    But what do you do so that it turns out to be "a = b × c"?

    I've been searching for a DETAILED MATHEMATICAL explanation but have failed in finding one that responds thoroughly to my inquiry. I mean, I understand how it would result in "a = bc", but I'm having trouble understanding how to get there in concise mathematical steps and easy-to-understand mathematical reasoning.
     
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  3. Jun 29, 2011 #2
    If anything, I would intuitively think that "a ∝ b" and "a ∝ c" imply "a ∝ bc", and thus a = kbc, where k is a constant of proportionality.

    EDIT:
    Actually, I'm wrong.

    First, some definitions:
    -to say of two variables a and b that "a ∝ b" means there exists a nonzero real constant k such that a = kb

    "a ∝ b" and "a ∝ c" imply a = bk and a = jc for some nonzero k, j in R, respectively.
    Then bc = (a/k)(a/j) = a2/(kj) and so a2 = (kj)bc, where (kj) is the product of two arbitrary constants in R and is thus itself an arbitrary constant in R. Therefore a2 ∝ bc.

    EDIT 2: And in fact, a = bc would imply that a2 = (kj)bc = (kj)a, implying a = kj = constant. But a is a variable!
     
    Last edited: Jun 29, 2011
  4. Jun 29, 2011 #3

    mathman

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    This may be a quibble, but a=b x c means a ~ b where c is constant and a ~ c where b is constant.
     
  5. Jun 30, 2011 #4
    Hmm. Is there no way to prove that "k = c" and "j = b" though?
     
  6. Jul 10, 2011 #5
    I've been thinking about this and somehow it managed to clear things up a lot haha. Thank you.

    But now there's something bothering me about this description; if you hold one variable constant, how can you say that that same variable is the constant of proportionality?

    Looking at this now, this seems more of a general science question than it is a math question...

    EDIT: What I mean is that given this description of what proportionality statements signify about dependent and independent/constant variables, do we just ASSIGN the constant variables to constants of proportionalities?
     
    Last edited: Jul 10, 2011
  7. Aug 12, 2012 #6
    Yes, sorry, I should have included that as well.

    Here's something I found though that explains the procedure a bit better (page 243, #389). What I don't understand though is why the two cases are taken consecutively (ie. the author goes from [itex]\frac{A}{a'}[/itex] to [itex]\frac{a'}{a}[/itex]). Is this the only way to reason this without paying attention to what the proportionality constants are dependent on?
     
  8. Aug 12, 2012 #7

    HallsofIvy

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    Saying "a ∝ b" means a= kb (k constant) all other factors held constant. saying "a ∝ c" means a= hc (h constant) all other factors held contstant. That tells you that the "k" in a= kb includes c as a factor (which is being held constant) and that the "h" in a= hc includes b as a factor (which is being held contstant). The two together tell you that a= jbc where j is a constant (as long as all other factors are held constant.
     
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