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This question is a general one about problems that you often see that are about water in a container in salt dissolved in the water. Let's set it up like this:

$S(t)$ is the amount of salt in the container
$V(t)$ is the volume of water in the container
$V_{in}(t)$ is the volume of salt solution flowing into the container
$C_{in}(t)$ is the salt concentration of water flowing into the container
$V_{out}(t)$ is the volume of water flowing out of the container

(Assuming the container is never full or empty) Is it correct to say that
$$\frac{dV}{dt}=V_{in}(t)-V_{out}(t)$$
(hence you can solve for V), and $$\frac{dS}{dt}=C_{in}(t)V_{in}(t)-\frac{S}{V(t)}V_{out}(t)$$
or is there some weird chain rule type thing in there (in the $C_{in}(t)V_{in}(t)$ term)? I feel like there should be.

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    $\begingroup$ No, that's right. I think part of what is confusing you is that the variables are labeled in a way that isn't completely consistent with their units. The specific confusing thing is that $V_{in}$ and $V_{out}$ aren't in volume units, but rather are in volume/time units. $\endgroup$
    – Ian
    Oct 6, 2021 at 0:46
  • $\begingroup$ @Ian Wouldn't $C_{in}$ be in units of mass per volume? Then $C_{in}V_{in}$ would be in units of mass per time instead of mass per (time)^2? $\endgroup$
    – Ivan
    Oct 6, 2021 at 0:51
  • $\begingroup$ $C_{in}$ is mass per volume, so $C_{in} V_{in}$ is mass per time which is what it should be. Identifying $S/V$ as $C$ (the concentration in the tank right now) might help with intuition (since then both terms will look the same). $\endgroup$
    – Ian
    Oct 6, 2021 at 1:08

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