UPSC CDS II 2024 Math Paper

Concept:Use the alligation or component mixing method to find the ratio $m:n$ from the given mixture conditions.Explanation:Container A has milk and water in ratio $1:3$.Container B has milk and water in ratio $m:n$.The two mixtures are combined in ratio $2:3$ to form $20$ litres of final mixture with milk and water in ratio $3:7$.Volume from container A = $\frac{2}{2+3} \times 20 = 8$ litres.Volume from container B = $\frac{3}{2+3} \times 20 = 12$ litres.In container A, milk fraction = $\frac{1}{4}$, so milk from A = $\frac{1}{4} \times 8 = 2$ litres.Water from A = $8 - 2 = 6$ litres.In container B, milk fraction = $\frac{m}{m+n}$, so milk from B = $\frac{m}{m+n} \times 12 = \frac{12m}{m+n}$ litres.Water from B = $\frac{n}{m+n} \times 12 = \frac{12n}{m+n}$ litres.Total milk = $2 + \frac{12m}{m+n}$ litres.Total water = $6 + \frac{12n}{m+n}$ litres.Given the final ratio of milk to water is $3:7$, so$$\frac{2 + \frac{12m}{m+n}}{6 + \frac{12n}{m+n}} = \frac{3}{7}$$Multiply numerator and denominator by $(m+n)$:$$\frac{2(m+n) + 12m}{6(m+n) + 12n} = \frac{3}{7}$$Simplify: $\frac{14m + 2n}{6m + 18n} = \frac{3}{7}$Cross multiply: $7(14m + 2n) = 3(6m + 18n)$$98m + 14n = 18m + 54n$$80m = 40n$$\frac{m}{n} = \frac{40}{80} = \frac{1}{2}$Answer:$\frac{m}{n} = \frac{1}{2}$ (Option A)