Two vessels $A$ and $B$ contain milk and water mixed in the ratio $5 : 3$ and $2 : 3$. When these mixtures are mixed to form a new mixture containing half milk and half water, they must be taken in the ratio

### Concept & Rule of Alligation We apply the rule of alligation to the concentration of one of the ingredients (e.g., milk) to determine the volumetric ratio required to achieve the target concentration. ### Step-by-Step Solution * **Given:** - Vessel A (Milk:Water) = $5 : 3 \Rightarrow$ Fraction of milk = $\frac{5}{8}$ - Vessel B (Milk:Water) = $2 : 3 \Rightarrow$ Fraction of milk = $\frac{2}{5}$ - Target mixture (Milk:Water) = $1 : 1$ (half milk, half water) $\Rightarrow$ Fraction of milk = $\frac{1}{2}$ * **Calculation:** 1. Set up the alligation cross with the milk fractions: - Left (Vessel A): $\frac{5}{8}$ - Right (Vessel B): $\frac{2}{5}$ - Center (Target): $\frac{1}{2}$ 2. Calculate the differences along the diagonals: - Right diagonal difference: $|\frac{1}{2} - \frac{2}{5}| = |\frac{5 - 4}{10}| = \frac{1}{10}$ - Left diagonal difference: $|\frac{5}{8} - \frac{1}{2}| = |\frac{5 - 4}{8}| = \frac{1}{8}$ 3. The required ratio of Vessel A to Vessel B is the ratio of these differences: $\frac{1}{10} : \frac{1}{8}$. 4. Simplify the ratio by multiplying by $40$ (the LCM of $10$ and $8$): $(\frac{1}{10} \times 40) : (\frac{1}{8} \times 40) = 4 : 5$. ### Exam Strategy & Shortcut Use integers to bypass fraction alligation. The denominators are $8, 5, 2$. Their LCM is $40$. Milk in A = $\frac{5}{8}$ of $40 = 25$. Milk in B = $\frac{2}{5}$ of $40 = 16$. Target Milk = $\frac{1}{2}$ of $40 = 20$. Alligation on integers $25, 16$ with target $20$: $|16 - 20| : |25 - 20| = 4 : 5$. ### Common Pitfall A common mistake is placing the differences on the wrong side. Remember that the difference between the mean and the right-side value gives the ratio component for the left side, and vice versa. ### Final Answer Therefore, the correct answer is **$4 : 5$**.