Henry Ernest Dudeney/Puzzles and Curious Problems/294 - The Keg of Wine/Solution

by : $294$

 * The Keg of Wine

Solution
Approximately $2.93$ gallons.

Proof
Let $J$ gallons be the volume of the jug.

At the first drawing there will be $10 - J$ gallons of wine in the keg.

After mixing it with water, the concentration of wine per unit volume is $\dfrac {10 - J} {10}$.

The second drawing takes $J \dfrac {10 - J} {10}$ gallons of wine out of the keg.

Hence there remains $10 - J - J \dfrac {10 - J} {10}$ gallons of wine in the keg.

Then the keg is refilled.

Because there are equal quantities of wine and water in the keg, that means there must be $5$ gallons of each.

That is:

This evaluates to approximately $10 - 7.07$, or about $2.93$ gallons.