A barrel contains a mixture of wine and water in the ratio 3 : 1. How much fraction of the mixture must be drawn off and substituted by water so that the ratio of wine and water in the resultant mixture in the barrel becomes 1 : 1 ?

Let the quantity of liquid drawn out = x
$$\eqalign{ & \Rightarrow \frac{{3 - \frac{3}{4}x}}{{1 - \frac{1}{4}x + x}} = \frac{1}{1} \cr & \Rightarrow 12 - 3x = 4 - x + 4x \cr & \Rightarrow 8 = 6x \cr & \Rightarrow x = \frac{4}{3} \cr} $$
Hence, required part of quantity
$$\eqalign{ & {\text{ = }}\frac{{\frac{4}{3}}}{4} \cr & {\text{ = }}\frac{1}{3} \cr} $$