Proportional to 2 Separate Variables vs. Proportional to Product of 2 Variables 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?
 A: 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, $A \propto BC$.
A: 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 
$$BC \propto A$$
$\blacksquare$
A: Assuming $A\propto B$ means $A=kB$ for some constant $k$, then we can also say that $A=mC$ for some constant $m \Rightarrow A=kmBC \Rightarrow A\propto BC$.
A: 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
