The swimming pool has water coming in @ 12m^3/min. The contents with the pool keep mixing. The contents flow out @ 10m^3/min. Sugar is added to the pool @rate 0.1kg/min. At the beginning there is 10 kilogram of salt within 100m^3 of water. What is the kilogram of salt left after 30 min?
Let $x$ be the amount of sugar. Then conc = amount/vol
We have that initially $sugar/water = 10/100 = 1/10$. Concentration should be $x/(100 + 2t)$. We know that $dx/dt = 0.1$ and thus we want to multiply by $2t$. Thus
$dx/dt = (x*2t/[100 + 2t])*2t$?
No, $dx/dt$ is not $0.1$. Rather $dx/dt$ is the sum of all the changes in sugar. One change is $0.1$ kg/m. Another changes is how much sugar is leaving the pool. If $x$ is the kilograms of sugar in the pool at time $t$ then the concentration of sugar is $x/(100+2t)$. The amount of sugar leaving each minute is $10$ times that. So the differential equation is
$$\frac{dx}{dt} = 0.1 -\frac{10x}{100+2t}$$ with initial condition $x(0) = 10.$
This isn't really a "related rate" problem. You need to solve the (linear) DE.