From a container filled with milk, 6 litres of milk are drawn out and replaced with water. Again, 6 litres of the resulting mixture are drawn out and replaced with water. After these two operations, the ratio of milk to water in the container becomes 9 : 16. What is the total original quantity of the mixture in the container (in litres)?

Introduction:
This problem involves repeated replacement in a mixture of milk and water, a classic concept in alligation and mixture. You remove some mixture and replace it with another liquid, which changes the ratio step by step. Understanding the formula for repeated replacement saves a lot of time in competitive exams.

Given Data / Assumptions:

• The container is initially full of pure milk.
• First, 6 litres of milk are removed and replaced with 6 litres of water.
• Second, 6 litres of the resulting mixture are removed and replaced with 6 litres of water.
• After these two operations, the ratio of milk to water is 9 : 16.
• The total capacity of the container remains constant throughout.

Concept / Approach:
When a quantity x is removed from a container of volume V and replaced with another liquid, the fraction of the original liquid left after one operation is (1 - x / V). After n identical operations, the fraction left is (1 - x / V)^n. Here, milk is the original liquid, x is 6 litres, V is the capacity, and n = 2 operations. We will relate this fraction to the final milk fraction from the given ratio.

Step-by-Step Solution:
Step 1: Let the capacity of the container be V litres. Step 2: After two operations, the fraction of milk left is (1 - 6 / V)^2. Step 3: The final ratio of milk to water is 9 : 16. Step 4: So the fraction of milk in the final mixture is 9 / (9 + 16) = 9 / 25. Step 5: Equate the two expressions for the milk fraction: (1 - 6 / V)^2 = 9 / 25. Step 6: Take square roots on both sides: 1 - 6 / V = 3 / 5 (we discard the negative root since V must be greater than 6). Step 7: Solve for V: 6 / V = 1 - 3 / 5 = 2 / 5 V = 6 * 5 / 2 = 15 litres.

Verification / Alternative check:
If the capacity is 15 litres, after the first operation milk left is 15 - 6 = 9 litres. After the second operation, we remove 6 * (9 / 15) = 3.6 litres of milk, so milk left is 9 - 3.6 = 5.4 litres. Water then is 15 - 5.4 = 9.6 litres. The ratio 5.4 : 9.6 simplifies to 9 : 16, confirming that V = 15 litres is correct.

Why Other Options Are Wrong:
16 litres: This capacity gives a different final ratio and does not match 9 : 16.
20 litres: The remaining fraction (1 - 6 / 20)^2 does not equal 9 / 25.
25 litres: With this capacity, milk remains too large a fraction after two operations.
30 litres: This also leads to a milk fraction much larger than 9 / 25.

Common Pitfalls:
Students often try to track milk and water litres stepwise without using the repeated replacement formula, which is error prone. Another mistake is taking the negative square root when solving, which would imply a capacity smaller than the amount removed, which is impossible. Also, sometimes the final ratio is misread as water to milk rather than milk to water.

Final Answer:
The total original quantity of the mixture in the container is 15 litres.