Combined Proportions

Question

I've been taking an ODE course and a word problem involved some quantity $x$ being directly and inversely proportional to others. I recall it said something like: $x$ is directly proportional to $y$ and $z$ and inversely proportional to $t$. I didn't really paused and ponder that the appropriate formula for $x$ was $x = k\dfrac{yz}{t}$, for some $k \in \mathbb{R}$. I assumed that this proportionality took place whenever the other variables were held constant.

However, today I wanted to, step by step, get to the formula given above. Writing all definitions: $x = a\cdot y$, $x = b \cdot z$ and $x = \dfrac{c}{t}$. But I wasn't able to get to that formula!

I'm stuck. Can anyone shed some light on this problem?

Interestingly, this is something that seems to be ignored in textbooks despite being not entirely trivial (which I initially expected), at least I don't recall having seen it discussed. See Proportional to 2 Separate Variables vs. Proportional to Product of 2 Variables AND If A $\propto$ B and A $\propto$ C while keeping each other constant, then why is A $\propto$ BC? AND Proportion of one varaiable to two different variables. — Dave L. Renfro, Sep 01 '21 at 14:06
@UriToti The theorem holds under the assumption that $y,z,t$ are independent of one another. My proof of the theorem is at Does $A$ being directly proportional to both $B$ and $C$ imply that $A$ is jointly proportional to $BC?$. — ryang, Jun 24 '23 at 13:45

Combined Proportions

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