# Water in the wine

We have two equal glasses, one full of water and one of wine. We take a full spoon of the glass of water, pour it into the glass of wine and mix well. Now we take a full spoon of the glass of wine (which has mixed water) and pour it into the glass of water.

The question is: Is there more water in the glass of wine or is there more wine in the glass of water?

#### Solution

The amount of water in the wine is the same as the amount of wine in the water.

We know that the quantities of liquid in each glass before starting the problem were the same and that the quantities of liquid at the end are also the same since the amount of liquid transferred is always the same.

Now, it is clear that some wine remained in glass A (the one with water) and some water remained in glass V (the one with wine). That “something” of water that is missing in glass A, is in V and that “something” of wine that is missing in glass V, is in A. If those quantities were not equal, that would mean that in one of the two glasses there is more liquid than in the other and that cannot be because the amount of liquid transferred from one glass to another has been the same. As the final amounts in each glass are the same, then, this implies that what is lacking in the water in glass A is equal to what is missing in the glass V and therefore there is the same amount of wine in the glass of water than the amount of water in the wine glass.