Water and Wine Problem

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Problem

You are presented with two glasses of similar size.

One glass contains wine.

The other contains water in like volume to the wine.

One spoonful of wine is taken from the wine glass and mixed into the water.

Then an identically-sized spoonful of the water-wine mixture is transferred from the water glass into the wine glass.

Which mixture is purer: the one in the water glass, or the one in the wine glass?

Solution

Let $x$ be the volume of liquid held by a glass.

Let $y$ be the volume of liquid held by the spoon.

After the first operation during which a spoonful of wine is transferred to the water glass:

the volume of wine in the wine glass will be $x - y$

the volume of liquid in the water glass will be $x + y$, of which $x$ is water.

During the course of the second operation, where a spoonful of wine-water mixture is transferred back to the wine glass, let $z$ be the volume of wine in the spoon.

The volume of water in the spoon is therefore $y - z$.

Thus, while $z$ is the volume of wine being added to the wine glass, $y - z$ is the volume of water being removed from the water glass.

After the second operation:

the volume of wine in the wine glass is $\left({x - y}\right) + z$

the volume of water in the water glass is $x - \left({y - z}\right) = \left({x - y}\right) + z$.

The mixtures are therefore of equal purity.

$\blacksquare$