Dishonest Butler

Classic Problem
A dishonest butler removes $3$ pints of wine from a barrel, replacing them with water.

He repeats this theft twice, removing in total $9$ pints, each time replacing with water.

As a result, the wine in the barrel is half its normal strength, the rest being water.

How much wine was there in the barrel to start with?

Solution
$14.54$ pints.

Proof
Let $x$ pints be the total quantity of wine in the barrel to start with.

After the first theft, there are $x - 3$ pints left.

The concentration of wine is now $\dfrac {x - 3} x$.

So the second theft removes $3 \dfrac {x - 3} x$ pints of wine from the cask.

The concentration of wine is now $\dfrac {x - 3 - 3 \frac {x - 3} x} x$.

So the third theft removes $3 \dfrac {x - 3 - 3 \frac {x - 3} x} x$ pints of wine.

The total removed is now $\dfrac x 2$, and so: