A vessel is filled with liquid, 3 parts of which are water and 5 parts of 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?

Suppose the vessel initially contains 8 litres of liquid.

Let x litres of this liquid be replaced with water.
Quantity of water in new mixture =   $3-\frac{3x}{8}+x$litres.

Quantity of syrup in new mixture =   $5-\frac{5x}{8}$ litres.

   $3-\frac{3x}{8}+x=5-\frac{5x}{8}$

=>   5x + 24 = 40 - 5x 

=>   10x = 16    => x = 8/5

So, part of the mixture replaced =  $\frac{8}{5}\times \frac{1}{8}$ = 1/5.