A certain litre of milk is drawn off from a vessel that contains 80 litre of pure milk initially and the same amount of water is added. Again same quantity of mixture is drawn off and the same quantity of water is added. As a result, the mixture contains 51.2 litre water. How much of the milk was drawn off initially?

Correct option is B

​$$\textbf{Let } x \text{ litres be drawn off each time.} \\\text{Initial quantity of milk} = 80 \text{ litres} \\[10pt]\text{After first replacement:} \\\text{Milk left} = 80 - x \\\text{Water added} = x \\\text{Total volume remains} = 80 \text{ litres} \\[10pt]\text{After second replacement:} \\\text{Milk removed} = \frac{(80 - x)}{80} \times x = \frac{(80 - x)x}{80} \\\text{Milk left} = (80 - x) - \frac{(80 - x)x}{80} = (80 - x)\left(1 - \frac{x}{80}\right) \\[10pt]\text{So, final milk left:} \\= 80 \left(1 - \frac{x}{80}\right)^2 \\[10pt]\text{Given final mixture has 51.2 litres of water} \Rightarrow \text{Milk left} = 80 - 51.2 = 28.8 \text{ litres} \\[10pt]\Rightarrow 80 \left(1 - \frac{x}{80}\right)^2 = 28.8 \\\Rightarrow \left(1 - \frac{x}{80}\right)^2 = \frac{28.8}{80} = 0.36 \\\Rightarrow 1 - \frac{x}{80} = \sqrt{0.36} = 0.6 \\\Rightarrow \frac{x}{80} = 0.4 \Rightarrow x = 32 \\[10pt]\text{Since the operation was done twice, this is total replacement:} \\\text{So, per step: } \frac{32}{2} = \boxed{16 \text{ litres}}$$​​