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I've always seen the following in physics and math textbooks but never understood the process by which it was mathematically deducted:

$A \propto B$ $\space$ and $\space$ $A \propto C \space\space\space \rightarrow \space\space\space A \propto BC$

Could someone walk me through how this is done? This has been bothering me for a while now :P

Thanks

Update: Here's something I found that explains how this works. (Page 387; "Proof" section). Still, this proof takes the two statements one after the other. The author uses $x \propto y$ when $z$ is constant, and then takes care of $x \propto z$ when $y$ is constant, where it left off from the first (going from $x$ to $x'$ and then $x_1$). Is this the only way it can be done?

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    Awesome. Thanks so much for clearing this up for me! :-) You should answer my question so I can +1 you and mark it as the answer haha2012-09-02

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Putting my comments into an answer: If we say $A \propto B$ when $A$ also depends on other things, what we mean is that holding everything else fixed, $A$ increases linearly with $B$. So $A = mB$ for some $m$, and $m$ is constant relative to $B$, but may vary depending on the other things.

This is getting a bit wooly, so let's be more explicit. Let's say $A$ is a function of $B$, $C$, and $D$. Then $A \propto B$ means $A(B,C,D) = f(C,D)\cdot B$ for some function $f$. On the other hand, if $A \propto C$, then $A(B,C,D) = g(B,D)\cdot C$ for some other $g$. When you can put those together and go through some algebra, you'll find that $A(B,C,D) = h(D)\cdot BC$, that is, $A \propto BC$.

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Temperature $T$ is proportional to pressure $P$ ( Charles Law $1$);

Temperature $T$ is proportional to Volume $V$ ( Charles Law $2$);

So, Temperature $T$ is prortional to product $P V$ ( Gas Law, $P V /T = \mathrm{const}$)

Narasimham

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    Thanks for the response! I've been reading up on the Combined Gas Law after seeing this post and have come across multiple times that the constants in each of the laws are actually functions dependent on the other variable. In other words, $T \propto P \space\rightarrow\space T = c_1(V)P$ and $T \propto V \space\rightarrow\space T = c_2(P)V$ Will this always be the case though? Will the constants always depend on each other like they do here?2012-07-31
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This is based off of wikipedia's article on Combined Gas Law and some of my own inputs.

We have A $\propto B$ and $A \propto C$.

Thus we get (noting that proportionality is symmetric), $B = k_C(C)A$ $C = k_B(B)A$

where $k_C$ is a function of only C and $k_B$ that of B. The proportionality constants could depend on various other things, but for now, we focus on A, B, C only as all other parameters are considered fixed. Then we get $A = \frac{B}{k_C(C)} = \frac{C}{k_B(B)}$ $\Rightarrow Bk_B(B) = Ck_C(C)$

Now in the previous line, LHS is a function of B only, RHS is a function of C only!!!! Thus both are independent of B and C and therefore must be a constant. Call this constant K.

Then $k_B(B) = \frac{K}{B}$. Substitute back to get $BC = KA$ or $BC \propto A$ $\blacksquare$

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Assuming $A\propto B$ means $A=kB$ for some constant $k$, then we can also say that $A=mC$ for some constant $m \Rightarrow A=kmBC \Rightarrow A\propto BC$.

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    I'm also wondering if there is a geometric intuition behind this, can it be proven using lines and areas?2012-07-27