A flask initially contains pure milk. Each time, 8 litres of milk (or milk-water mixture) is drawn from the flask and the flask is refilled with 8 litres of water. This operation is done a total of 4 times (the first time plus 3 more times).\nAfter the 4 operations, the ratio of milk left to the total solution is 81/625.\nHow many litres of milk did the flask initially contain?

Introduction:
This is a repeated replacement (successive dilution) problem. When a fixed quantity is removed from a container and replaced with water, the fraction of the original liquid remaining gets multiplied by the same factor each time. The final fraction of milk is given, allowing us to solve for the initial capacity.

Given Data / Assumptions:

• Initial liquid = pure milk
• Each operation removes 8 litres and refills 8 litres water
• Total operations = 4
• Final milk fraction = 81/625 of total solution
• Let initial volume (capacity) = V litres

Concept / Approach:
After each operation, the fraction of milk remaining = (1 - 8/V).\nAfter 4 operations, milk fraction = (1 - 8/V)^4.\nSet (1 - 8/V)^4 = 81/625 and solve for V.

Step-by-Step Solution:
Final milk fraction = (1 - 8/V)^4 = 81/625Note that 81/625 = (3^4)/(5^4)So the fourth root is: (81/625)^(1/4) = 3/5Therefore, 1 - 8/V = 3/58/V = 1 - 3/5 = 2/5V = 8 * (5/2) = 20 litres

Verification / Alternative Check:
If V = 20, then remaining factor each time = 1 - 8/20 = 12/20 = 3/5. After 4 operations, milk fraction = (3/5)^4 = 81/625, exactly matching the given condition. So the capacity is confirmed.

Why Other Options Are Wrong:
10 litres: impossible because you cannot remove 8 litres repeatedly from such a small initial volume without violating the model.30 or 40 litres: would make remaining factor larger, giving a larger final milk fraction than 81/625.25 litres: does not yield the clean (3/5)^4 fraction required.

Common Pitfalls:
Using n = 3 operations instead of 4 due to wording.Subtracting 8 litres directly each time instead of using fractional reduction.Forgetting that the final ratio given is milk/total, i.e., the milk fraction.

Final Answer:
20 litres