Two vessels contain milk : water as follows — Vessel A: 5 : 3 and Vessel B: 2 : 3. In what ratio should mixtures from A and B be combined to obtain a final mixture that is exactly half milk and half water?

Introduction / Context:
We have two mixtures with different milk fractions. To achieve a 1 : 1 milk-to-water target, use alligation on milk fractions to find the mixing ratio of the two sources.

Given Data / Assumptions:

• Vessel A milk fraction = 5/8 = 0.625.
• Vessel B milk fraction = 2/5 = 0.4.
• Target milk fraction = 1/2 = 0.5.

Concept / Approach:
Alligation formula for fractions f1 (A) and f2 (B) to reach m: required A : B = (f2 − m) : (m − f1), ensuring f1 > m > f2 or vice versa with correct ordering.

Step-by-Step Solution:
A : B = (0.4 − 0.5) : (0.5 − 0.625) in magnitudeUse absolute differences with proper sides: A : B = (m − f2) : (f1 − m) = (0.5 − 0.4) : (0.625 − 0.5)= 0.1 : 0.125 = 4 : 5

Verification / Alternative check:
With 4 parts of A and 5 parts of B: milk fraction = (4*0.625 + 5*0.4) / 9 = (2.5 + 2) / 9 = 4.5 / 9 = 0.5.

Why Other Options Are Wrong:
Other ratios give milk fraction different from 0.5. Only 4 : 5 balances the fractions.

Common Pitfalls:
Reversing the alligation differences and obtaining 5 : 4 instead of 4 : 5.

Final Answer:
4 : 5