Two vessels contain milk and water mixtures.\n• Vessel 1 has milk : water = 3 : 1.\n• Vessel 2 has milk : water = 7 : 11.\nIn what ratio should the liquids from vessel 1 and vessel 2 be mixed so that the new mixture contains exactly half milk and half water (i.e., 50% milk)?

Introduction:
This problem tests alligation using milk fractions (concentrations). We first convert each vessel's ratio into milk fraction, then find how to mix a higher milk fraction with a lower milk fraction to achieve exactly 50% milk.

Given Data / Assumptions:

• Vessel 1: milk : water = 3 : 1
• Vessel 2: milk : water = 7 : 11
• Target mixture: 50% milk (milk fraction = 1/2)

Concept / Approach:
Convert ratios to milk fractions:\nMilk fraction = milk / (milk + water).\nThen apply alligation with fractions:\nRatio (higher : lower) = (target - lower) : (higher - target).

Step-by-Step Solution:
Vessel 1 milk fraction = 3 / (3 + 1) = 3/4Vessel 2 milk fraction = 7 / (7 + 11) = 7/18Target milk fraction = 1/2Compute (target - lower) = 1/2 - 7/18 = (9/18 - 7/18) = 2/18 = 1/9Compute (higher - target) = 3/4 - 1/2 = 1/4Required ratio (Vessel 1 : Vessel 2) = (target - lower) : (higher - target) = (1/9) : (1/4)Convert ratio: (1/9) : (1/4) = 4 : 9

Verification / Alternative Check:
Take 4 parts from vessel 1 and 9 parts from vessel 2.\nMilk amount proportional = 4*(3/4) + 9*(7/18) = 3 + 3.5 = 6.5 parts.\nTotal parts = 4 + 9 = 13.\nMilk fraction = 6.5/13 = 0.5 = 50%, correct.

Why Other Options Are Wrong:
1:1 or 4:7: does not balance the milk fraction to exactly 1/2.5:7: gives milk fraction higher than 1/2 because vessel 1 is very rich in milk.9:4: reversed ratio, would push milk fraction above 50%.

Common Pitfalls:
Using milk:water ratios directly in alligation without converting to milk fractions.Mixing up which vessel is higher and which is lower in milk content.Forgetting that 50% milk means milk fraction = 1/2 exactly.

Final Answer:
4 : 9