If $A\propto BC,$ then $A\propto B$?

Question

Let $A,B,C$ be variables. If $A$ is directly proportional to $BC,$ can we always say that $A$ is directly proportional to $B$ and that $A$ is directly proportional to $C\,?$

Answer 1

Unfortunately, no: a counterexample is $\boldsymbol{A=BC}$ with $\boldsymbol{B=C},$ so that $A=B^2.$

However, if we impose the condition that $B$ and $C$ are independent of each other, then $A\propto BC$ indeed implies that $A\propto B$ and $A\propto C\;$ (proof).

Answer 2

If $A$ is directly proportional to $B\cdot C$ then $A=k\cdot B\cdot C$ for some constant $k$. If we only vary $B$ and keep $C$ constant then $A=(k\cdot C)\cdot B$ where $k\cdot C$ is a constant, so $A$ is directly proportional to $B$. Similarly, if we only vary $C$ and keep $B$ constant then $A=(k\cdot B)\cdot C$ where $k\cdot B$ is a constant, so $A$ is directly proportional to $C$.