Henry Ernest Dudeney/Puzzles and Curious Problems/297 - Mixing the Wine/Solution

Puzzles and Curious Problems by Henry Ernest Dudeney: $297$

A glass is one-third full of wine,

and another glass, with equal capacity, is one-fourth full of wine.

Each is filled with water and their contents mixed in a jug.

Half of the mixture is poured into one of the glasses.

What proportion of this is wine and what part water?

$7$ parts wine to $17$ parts water.

Let $c$ be the capacity of one glass.

After they have been filled with water:

the first one contains $\dfrac {4 c} {12}$ of wine

the second one contains $\dfrac {3 c} {12}$ of wine.

In the jug together, there is $\dfrac {7 c} {12}$ of wine out of a total of $2 c$ of liquid.

When poured into one of the glasses, there is then $\dfrac 1 2 \times \dfrac {7 c} {12} = \dfrac {7 c} {24}$ of wine.

As this is in a container of capacity $c$, the ratio of wine to water is $\dfrac {7 c} {24} \div \dfrac {24 - 7 c} {24} = \dfrac 7 {17}$.

$\blacksquare$