Alligation or mixture A vessel is filled with liquid,3 parts of which are water and 5 parts syrup.How much of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup? What is the best possible way to solve this question?

The following is the manner in which I typically go about solving such problems. It might not be the quickest way to get to the solution, but I would still like to present my approach here.

Suppose the vessel is initially filled with $L$ litres of the liquid. Then, as per the question, the vessel contains $\frac{3L}{8}$ litres of water and $\frac{5L}{8}$ litres of syrup.

Now, suppose that $l$ litres of the liquid is replaced with water. Then, the quantity of water remaining in the vessel would be $\frac{3(L-l)}{8}+l$, and that of the syrup would be $\frac{5(L-l)}{8}$, and as per the question, these should be the same, i.e., \begin{eqnarray} \frac{3(L-l)}{8}+l=\frac{5(L-l)}{8}. \end{eqnarray}

Upon solving the above equation, we get $l=\frac{L}{5}$. Thus, one-fifth of the mixture (liquid) must be drawn out and replaced with water in order that water and syrup be in equal proportions thereafter.