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:

\(\ds \frac x 2\)

\(=\)

\(\ds 3 + 3 \dfrac {x - 3} x + 3 \dfrac {x - 3 - 3 \frac {x - 3} x} x\)

\(\ds \leadsto \ \ \)

\(\ds 2 \paren {x - 3}^3\)

\(=\)

\(\ds x^3\)

\(\ds \leadsto \ \ \)

\(\ds x\)

\(=\)

\(\ds \dfrac {3 \times \sqrt [3] 2} {\sqrt [3] 2 - 1}\)

\(\ds \)

\(=\)

\(\ds 14.54\)

$\blacksquare$

Historical Note

According to David Wells, in his Curious and Interesting Puzzles of $1992$, the Problem of the Dishonest Butler appears in one of the works of Niccolò Fontana Tartaglia, but he does not say where.

Sources

• 1992: David Wells: Curious and Interesting Puzzles ... (previous) ... (next): Exchanging the Knights: $104$