UPSC CDS I 2024 Math Paper

Concept:Alligation or weighted average: The overall ratio of spirit to water is obtained by mixing two mixtures in a certain proportion.Explanation:Let the quantity taken from the first bottle be $x$ and from the second bottle be $y$.In the first bottle, spirit $= \frac{1}{5}x$ and water $= \frac{4}{5}x$.In the second bottle, spirit $= \frac{4}{5}y$ and water $= \frac{1}{5}y$.In the final mixture, total spirit $= \frac{x}{5} + \frac{4y}{5}$ and total water $= \frac{4x}{5} + \frac{y}{5}$.The required ratio of spirit to water is $1 : 3$, so:$$\frac{\frac{x}{5} + \frac{4y}{5}}{\frac{4x}{5} + \frac{y}{5}} = \frac{1}{3}$$Multiply numerator and denominator by $5$:$$\frac{x + 4y}{4x + y} = \frac{1}{3}$$Cross-multiply:$$3(x + 4y) = 4x + y$$$$3x + 12y = 4x + y$$Rearrange:$$12y - y = 4x - 3x$$$$11y = x$$Thus $x : y = 11 : 1$.Answer:The two mixtures should be mixed in the ratio $11 : 1$.