Two vessels A and B contain milk and water mixed in the ratio 8 : 5 and 5 : 2 respectively. The ratio in which these two mixtures be mixed to get a new mixture containing $69 \frac{3}{13}\%$ milk, is

### Concept & Concentration Alligation Convert mixed fraction percentages into simple fractions, then apply the Rule of Alligation using the fraction of milk in each mixture. ### Step-by-Step Solution 1. Convert the target milk percentage to a fraction: - Target milk $= 69 \frac{3}{13}\% = \frac{(69 \times 13) + 3}{13}\% = \frac{897 + 3}{13}\% = \frac{900}{13}\%$ - As a fraction: $\frac{900}{13} \times \frac{1}{100} = \frac{9}{13}$ 2. Identify the fraction of milk in vessels A and B: - Vessel A (8 : 5) $\implies$ Fraction of milk = $\frac{8}{8 + 5} = \frac{8}{13}$ - Vessel B (5 : 2) $\implies$ Fraction of milk = $\frac{5}{5 + 2} = \frac{5}{7}$ 3. Apply alligation on the milk fractions: - Difference 1 (for A) $= \frac{5}{7} - \frac{9}{13} = \frac{65 - 63}{91} = \frac{2}{91}$ - Difference 2 (for B) $= \frac{9}{13} - \frac{8}{13} = \frac{1}{13}$ 4. Determine the ratio of A : B: - Ratio $= \frac{2}{91} : \frac{1}{13}$ - Multiply by 91: $2 : \left(\frac{91}{13}\right) = 2 : 7$ ### Exam Strategy & Shortcut Notice the denominators: $13$ and $7$. Scale all fractions to a common denominator of $91$. - Target: $\frac{9}{13} = \frac{63}{91}$ - A: $\frac{8}{13} = \frac{56}{91}$ - B: $\frac{5}{7} = \frac{65}{91}$ Alligate numerators: $(65 - 63) : (63 - 56) = 2 : 7$. ### Common Pitfall A calculation error when converting the complex mixed fraction $69 \frac{3}{13}\%$ into a standard improper fraction. Always double-check the multiplication step: $69 \times 13$. ### Final Answer Therefore, the correct answer is **2 : 7**.