A pot contains 81 litres of pure milk. $\frac{1}{3}$ of the milk is replaced by the same amount of water. Again $\frac{1}{3}$ of the mixture is replaced by that amount of water. What is the ratio of milk and water in the new mixture ?

Question

Accepted Answer

To solve the problem and justify why Option 4 is correct, we need to follow the steps of the process described in the question and calculate the final ratio of milk to water.

### Step-by-Step Explanation:

1. **Initial Situation:**
   - The pot contains 81 litres of pure milk.

2. **First Replacement:**
   - $\frac{1}{3}$ of the milk is replaced by water.
   - $\frac{1}{3}$ of 81 litres is 27 litres.
   - After removing 27 litres of milk, 54 litres of milk remain.
   - 27 litres of water are added, making the total mixture 81 litres again.
   - Now, the mixture contains 54 litres of milk and 27 litres of water.

3. **Second Replacement:**
   - Again, $\frac{1}{3}$ of the mixture (81 litres) is replaced by water.
   - $\frac{1}{3}$ of 81 litres is 27 litres.
   - The mixture currently has 54 litres of milk and 27 litres of water.
   - The 27 litres removed will contain both milk and water in the same ratio as the mixture, which is 54:27 or 2:1.
   - Therefore, in the 27 litres removed, $\frac{2}{3}$ is milk and $\frac{1}{3}$ is water.
   - Milk removed = $\frac{2}{3} \times 27 = 18$ litres.
   - Water removed = $\frac{1}{3} \times 27 = 9$ litres.

4. **After Second Replacement:**
   - Remaining milk = 54 - 18 = 36 litres.
   - Remaining water = 27 - 9 = 18 litres.
   - 27 litres of water are added again, making the total water = 18 + 27 = 45 litres.

5. **Final Mixture:**
   - Milk = 36 litres.
   - Water = 45 litres.
   - Ratio of milk to water = 36:45.

6. **Simplifying the Ratio:**
   - Simplify 36:45 by dividing both numbers by their greatest common divisor, which is 9.
   - $36 \div 9 = 4$
   - $45 \div 9 = 5$
   - The simplified ratio is 4:5.

### Conclusion:

- The final ratio of milk to water in the mixture is 4:5, which corresponds to Option 4.

### Explanation of Incorrect Options:

1. **Option 1 (1:2):**
   - This ratio suggests that there is twice as much water as milk, which is not the case after the calculations.

2. **Option 2 (2:5):**
   - This ratio implies a much larger proportion of water compared to milk than what is calculated.

3. **Option 3 (3:7):**
   - This ratio also suggests a higher proportion of water than milk, which does not match the calculated ratio.

By following the logical steps and calculations, we confirm that Option 4 (4:5) is indeed the correct answer.