Two vessels A and B contain spirit and water mixed in the ratio 5 : 2 and 7 : 6 respectively. Find the ratio in which these mixtures be mixed to obtain a new mixture in vessel C containing spirit and water in the ratio 8 : 5 ?

### Concept & Alligation with Fractions When mixing two compound mixtures to get a third, apply the Rule of Alligation to the fractional concentration of a single component (e.g., just the spirit). $$ \frac{\text{Quantity of A}}{\text{Quantity of B}} = \frac{\text{Fraction in B} - \text{Mean Fraction}}{\text{Mean Fraction} - \text{Fraction in A}} $$ ### Step-by-Step Solution 1. Identify the fraction of spirit in each vessel: - Vessel A: Spirit = $5$, Water = $2 \implies$ Fraction of Spirit = $\frac{5}{5 + 2} = \frac{5}{7}$ - Vessel B: Spirit = $7$, Water = $6 \implies$ Fraction of Spirit = $\frac{7}{7 + 6} = \frac{7}{13}$ - Vessel C (Target): Spirit = $8$, Water = $5 \implies$ Fraction of Spirit = $\frac{8}{8 + 5} = \frac{8}{13}$ 2. Set up the alligation cross for the spirit fractions: - Difference for A = $\left|\frac{8}{13} - \frac{7}{13}\right| = \frac{1}{13}$ - Difference for B = $\left|\frac{5}{7} - \frac{8}{13}\right| = \frac{65 - 56}{91} = \frac{9}{91}$ 3. The required ratio of A : B corresponds to the cross differences: - Ratio = $\frac{1}{13} : \frac{9}{91}$ 4. Simplify the ratio: - Multiply by the LCM of denominators ($91$): - $91 \times \left(\frac{1}{13}\right) : 91 \times \left(\frac{9}{91}\right)$ - $7 : 9$ ### Exam Strategy & Shortcut To avoid fraction subtraction, equalize all denominators first. Fractions are $\frac{5}{7}$, $\frac{7}{13}$, and $\frac{8}{13}$. LCM of $7$ and $13$ is $91$. - A = $\frac{65}{91}$ - B = $\frac{49}{91}$ - Mean = $\frac{56}{91}$ Apply alligation on numerators: $65$ and $49$ with mean $56$. Ratio = $(56 - 49) : (65 - 56) = 7 : 9$. ### Common Pitfall Mixing up the ratios of spirit and water when determining the fraction. Ensure you divide the part by the *total sum of parts* (e.g., $5 / (5+2)$), not by the other component. ### Final Answer Therefore, the correct answer is **7 : 9**.