Solution
The mixtures should be visualised as separated into their water and wine components. Conservation of substance then implies that the volume of wine in the barrel holding mostly water has to be equal to the volume of water in the barrel holding mostly wine. To help in grasping this, the wine and water may be represented by, say, 100 red and 100 white marbles, respectively. If 20, say, red marbles are mixed in with the white marbles, and 20 marbles of any color are returned to the red container, then there will again be 100 marbles in each container. If there are now x white marbles in the red container, then there must be x red marbles in the white container. The mixtures will therefore be of equal purity. An example is shown below.
| Time Step | Red Marble Container | Action | White Marble Container |
|---|---|---|---|
| 0 | 100 (all red) | 100 (all white) | |
| 1 | 20 (all red) → | ||
| 2 | 80 (all red) | 120 (100 white, 20 red) | |
| 3 | ← 20 (16 white, 4 red) | ||
| 4 | 100 (84 red, 16 white) | 100 (84 white, 16 red) |
Another solution is to simply imagine two extremes: mix none, mix all. Mix None: Both stay the same, both have equal purity. Mix All: Pour all of one barrel into the other. Then pour one barrel of the mixture back into the empty container. Each barrel now contains a 50:50 mixture, i.e. both have the same purity.
So it does not make a difference; mix none or mix all, they both end up with the same purity. This is a "linear" problem with equal endpoints so all the cases from none to all will have the result: Equal Purity.
Read more about this topic: Wine/water Mixing Problem
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