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Puzzle:
"You are given two glass jugs. Each contains the same volume, V , of liquid. One jug contains pure alcohol, and the other jug contains pure water. A modest quantity, Q, of water is poured from the water jug into the alcohol jug, which is then thoroughly mixed. The same modest quantity, Q, of (now diluted) alcohol is then poured back into the water jug to equalize the volumes of the jugs at their initial levels. The initial concentration of alcohol in the alcohol jug equals the initial concentration of water in the water jug (at 100%). What is the relationship between the final concentrations of alcohol in the alcohol jug and water in the water jug?"
Solution:
- First, let us define what is quantity of solution:
"The liquid measurement is the amount of liquid a vessel contains and its measurement in standard units. We also refer to it as the “capacity” or the “volume” of the vessel. the quantity of Q is still in liter units."
- what is concentration:
"The concentration of a solution is a measure of the amount of solute that has been dissolved in a given amount of solvent or solution. If both are solvents, it can be decided by the ratio of volumes."
- With enough understanding of the problem, we take a look:
- at first both are 100% concentration ---> the container contains pure water / alcohol.
- after adding Q amount of water into alcohol ---> the concentration of alcohol becomes V / (Q+V).
- even though we will pour back the water into the water jug, the concentration alcohol doesn't change ---> V / (Q+V)
- Since the volume after pouring back and forth is the same, although without calculation, we don't actually know how much alcohol in alcohol container was replaced by water, we know: there must be a portion of alcohol moved to water jug, and we know: the alcohol cannot disappear. For water container, it will receive the portion of alcohol.
- And it has the same volume as alcohol container, so like alcohol container, it will lose the same ratio of water, giving it to the alcohol container. By symmetry, the concentration of water in water jug will also be V / (Q+V).
*** Note: I have written a simple code to generalize this problem. If you want to check, please click on this link:
https://github.com/Kexin996/brainteaser/blob/master/july8.py
