A vessel contains 60 litres of milk. If 12 litres of milk is replaced by water, and the process is repeated two more times, how much milk is left in the vessel after the third replacement?

This is a classical problem of successive dilution, where each time a certain amount of milk is replaced by water, the concentration of milk decreases. Here's how we can solve it:

### Step-by-Step Solution

1. **Initial Amount of Milk:**
   The vessel initially contains 60 litres of pure milk.

2. **After the First Replacement:**
   When 12 litres of milk is removed and replaced with 12 litres of water, the amount of milk left in the vessel is:
   \[
   \text{Milk remaining after first replacement} = \left(\frac{60 - 12}{60}\right) \times 60 = \left(\frac{48}{60}\right) \times 60 = 48 \, \text{litres}.
   \]
   So, 48 litres of milk remain after the first replacement.

3. **After the Second Replacement:**
   Now, we repeat the process: we remove 12 litres of the new mixture (which now contains both milk and water) and replace it with 12 litres of water. The amount of milk removed in this step is proportional to the current milk content. Thus, the amount of milk left after the second replacement is:
   \[
   \text{Milk remaining after second replacement} = \left(\frac{60 - 12}{60}\right) \times 48 = \left(\frac{48}{60}\right) \times 48 = 38.4 \, \text{litres}.
   \]

4. **After the Third Replacement:**
   Repeating the process a third time, we again remove 12 litres of the mixture and replace it with water. The amount of milk left after the third replacement is:
   \[
   \text{Milk remaining after third replacement} = \left(\frac{60 - 12}{60}\right) \times 38.4 = \left(\frac{48}{60}\right) \times 38.4 = 30.72 \, \text{litres}.
   \]

### Final Answer:
After the third replacement, the amount of milk left in the vessel is **30.72 litres**.

Would you like further details or have any questions?

Here are 5 related questions to expand on this problem:

1. What if 10 litres of milk were replaced each time instead of 12 litres?
2. How much milk would be left after 5 replacements?
3. What would happen if we replaced 15 litres of milk each time?
4. How does the process change if instead of replacing, we simply remove 12 litres of milk each time?
5. Can you generalize a formula to find the amount of milk left after \( n \) replacements?

**Tip:** The concept used here is based on exponential decay, where the remaining quantity reduces by a constant fraction in each step.

Solve the successive dilution problem where 12 litres of milk are replaced with water in a 60-litre vessel three times. Learn how to calculate the remaining milk using exponential decay. Suitable for Grades 10-12.

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