Combining Proportionality Statements

In summary, the conversation discusses how to get from "a ∝ b" and "a ∝ c" to "a = b × c". The process involves understanding the definitions of proportionality and how to manipulate equations with variables and constants. The conversation also brings up the concept of holding variables constant and how that affects the proportionality constant. Ultimately, it is important to pay attention to what the proportionality constants are dependent on in order to fully understand the mathematical reasoning behind the equation.
  • #1
prosteve037
110
3
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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  • #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!
 
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  • #3
This may be a quibble, but a=b x c means a ~ b where c is constant and a ~ c where b is constant.
 
  • #4
Hmm. Is there no way to prove that "k = c" and "j = b" though?
 
  • #5
mathman said:
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?
 
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  • #6
mathman said:
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 [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?
 
  • #7
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.
 

1. What is a proportionality statement?

A proportionality statement is a mathematical expression that shows the relationship between two quantities that change proportionally. This means that as one quantity increases or decreases, the other quantity also changes in the same ratio.

2. How do you combine proportionality statements?

To combine proportionality statements, you need to first identify the common variable in both statements. Then, you can set up a proportion using the common variable and the corresponding proportional quantities from both statements. Finally, solve the proportion to find the combined proportionality statement.

3. Can you combine more than two proportionality statements?

Yes, you can combine more than two proportionality statements by setting up multiple proportions and using the common variable and corresponding proportional quantities from each statement.

4. What is the importance of combining proportionality statements?

Combining proportionality statements allows you to simplify and solve complex mathematical problems involving multiple proportional relationships. This can be useful in various fields such as science, engineering, and economics.

5. Are there any special rules for combining proportionality statements?

Yes, when combining proportionality statements, it is important to make sure that the units of measurement are consistent. You may need to convert units if necessary. Additionally, you should also check that the resulting combined statement makes sense in the context of the problem.

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