A vessel contains 20 litres of a mixture of milk and water in the ratio 3 : 2. First, 10 litres of this mixture are removed and replaced with 10 litres of pure milk. This process of removing 10 litres of the mixture and replacing it with 10 litres of pure milk is then repeated once more. What is the ratio of milk to water in the final mixture obtained in the vessel?

Introduction:
This problem is about successive replacement in a milk water mixture, where mixture is removed and replaced with pure milk twice. It tests your understanding of how the composition of a mixture changes step by step and how to compute the final ratio after repeated operations.

Given Data / Assumptions:

• A vessel initially contains 20 litres of a mixture of milk and water.
• The initial ratio of milk to water is 3 : 2.
• First, 10 litres of this mixture are removed.
• These 10 litres are replaced with 10 litres of pure milk.
• The same operation is repeated a second time.
• We must find the final ratio of milk to water.

Concept / Approach:
We can track the exact quantities of milk and water after each step rather than relying on a general formula. Starting from the initial ratio, we compute the amounts removed in the same ratio, then update the mixture after adding pure milk. Repeating the process once more gives us the final amounts, from which we can obtain the final ratio of milk to water.

Step-by-Step Solution:
Step 1: Initial mixture = 20 litres with milk : water = 3 : 2. Step 2: Total parts = 3 + 2 = 5. So milk = 20 * 3 / 5 = 12 litres, water = 20 * 2 / 5 = 8 litres. Step 3: First operation: 10 litres of mixture are removed. Step 4: Milk removed = 10 * (12 / 20) = 6 litres, water removed = 10 * (8 / 20) = 4 litres. Step 5: After removal, milk = 12 - 6 = 6 litres, water = 8 - 4 = 4 litres. Step 6: Add 10 litres of pure milk. New milk = 6 + 10 = 16 litres, water remains 4 litres. Step 7: Second operation: again 10 litres of current mixture are removed. Step 8: Current total = 20 litres, milk = 16 litres, water = 4 litres. Step 9: Milk removed = 10 * (16 / 20) = 8 litres, water removed = 10 * (4 / 20) = 2 litres. Step 10: After removal, milk = 16 - 8 = 8 litres, water = 4 - 2 = 2 litres. Step 11: Add 10 litres of pure milk again, so final milk = 8 + 10 = 18 litres. Step 12: Final water remains 2 litres, so final ratio milk : water = 18 : 2 = 9 : 1.

Verification / Alternative check:
We can also reason that each replacement operation increases the proportion of milk since only milk is added and mixture is removed. After the first replacement, milk is 16 litres and water 4 litres, ratio 4 : 1. After the second replacement, milk increases further relative to water, ending at 18 : 2 which simplifies to 9 : 1. Both the stepwise computation and this reasoning are consistent.

Why Other Options Are Wrong:
5 : 3 and 1 : 4 do not reflect the fact that milk proportion is rising due to addition of pure milk each time.
4 : 1 is the ratio after the first replacement only, not after the second replacement.
7 : 3 does not match the exact calculated final amounts of 18 litres milk and 2 litres water.

Common Pitfalls:
A typical error is to treat each 10 litre removal as affecting only water or only milk, rather than both in the existing ratio. Some students also mistakenly adjust the total volume after replacements, forgetting it stays at 20 litres. Another mistake is to stop after the first replacement instead of repeating the process as described.

Final Answer:
The ratio of milk to water in the final mixture is 9 : 1.