Alligation–Mixture (Combining Three Bottles):\nThree equal-capacity bottles contain milk:water in ratios 2:5, 3:4, and 4:5. After pouring all into a larger bottle, what is the overall ratio of milk to water?

Introduction / Context:
When combining mixtures with equal capacities, add component fractions directly.

Given Data / Assumptions:

• Each bottle has capacity 1 unit (for convenience).
• Ratios: 2:5 ⇒ milk 2/7; 3:4 ⇒ milk 3/7; 4:5 ⇒ milk 4/9.

Concept / Approach:
Total milk = 2/7 + 3/7 + 4/9. Total volume = 3 units; so total water = 3 − total milk. Convert to a common denominator to sum exactly.

Step-by-Step Solution:
1) Milk = 2/7 + 3/7 + 4/9 = 5/7 + 4/9 = 45/63 + 28/63 = 73/63.2) Water = 3 − 73/63 = 189/63 − 73/63 = 116/63.3) Ratio milk:water = (73/63) : (116/63) = 73 : 116.

Verification / Alternative check:
Proportional reasoning with equal capacities preserves the ratio calculation.

Why Other Options Are Wrong:
Other denominators do not match the exact fractional sums.

Common Pitfalls:
Averaging ratios directly (invalid); ignoring unequal denominators 7 and 9.

Final Answer:
73 : 116