A tank of blood which is 50 gallons in volume is full of this red delicious substance. The man who owns this tanks want to replace the blood with ethanol. Since the tank is heavy he cannot pour all the water out. Rather he has a 1 gallon pail. Each time he pours the ethanol in the mixture displaces 1 gallon.
He had 50 gallons of blood.
He pours 1 gallon of ethanol and the blood displaces 1 gallon.
Now he has 49 gallons of blood and 1 gallon of ethanol.
He pours 1 gallon of ethanol now 1 gallon of the mixture displaces (not the blood, otherwise this be too easy).
And so on.....
1)Show that the man can never fully (ideally) clean the tank to ehtanol only.
2)But he would be satisfied with if the concetration of blood in the tank is at least .01%
How many pouring are required?
This problem seems awesomely similar to your differencial equations rate in/rate out mixture equation. But it is not. Because this is not a continous time time. The amout of concetration is not based on the time and continuity of it going in and out but rather on the number of pouring which is a natural number and hence a discrete model.