Combining Proportionality Statements

[prosteve037]

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.

 

[Unit]

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!

 

[mathman]

This may be a quibble, but a=b x c means a ~ b where c is constant and a ~ c where b is constant.

 

[prosteve037]

Hmm. Is there no way to prove that "k = c" and "j = b" though?

 

[prosteve037]

This may be a quibble, but a=b x c means a ~ b where c is constant and a ~ c where b is constant.

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?

 

[prosteve037]

This may be a quibble, but a=b x c means a ~ b where c is constant and a ~ c where b is constant.

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 \frac{A}{a'} to \frac{a'}{a}). Is this the only way to reason this without paying attention to what the proportionality constants are dependent on?

 

[HallsofIvy]

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.