Two vessels contain milk : water as follows — A: 7 : 5 and B: 17 : 7. In what ratio should mixtures from A and B be combined to obtain milk : water = 5 : 3 in the final mixture?

Introduction / Context:
We target a milk fraction of 5/8 in the final mixture. Use alligation on milk fractions of the two sources to determine the required combining ratio.

Given Data / Assumptions:

• A milk fraction = 7/12 ≈ 0.5833.
• B milk fraction = 17/24 ≈ 0.7083.
• Target milk fraction = 5/8 = 0.625.

Concept / Approach:
For fractions fA and fB, A : B = (fB − m) : (m − fA).

Step-by-Step Solution:
A : B = (17/24 − 5/8) : (5/8 − 7/12)= (17/24 − 15/24) : (15/24 − 14/24) = (2/24) : (1/24) = 2 : 1

Verification / Alternative check:
Take 2 L from A and 1 L from B: milk = 2*(7/12) + 1*(17/24) = 14/12 + 17/24 = 28/24 + 17/24 = 45/24 = 15/8; total 3 L; milk fraction = (15/8)/3 = 5/8 → milk : water = 5 : 3.

Why Other Options Are Wrong:
Other ratios produce milk fractions above or below 5/8. Only 2 : 1 meets the target exactly.

Common Pitfalls:
Reversing the differences in alligation and getting 1 : 2.

Final Answer:
2 : 1