There are 81 liters of pure milk in a container. One-third of the milk is removed and replaced with an equal quantity of water. Then, one-third of the resulting mixture is taken out and replaced again with the same amount of water. What is the ratio of milk to water in the final mixture?

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Accepted Answer

**Solution:**

1. Initially, the container has 81 liters of pure milk.

2. One-third of the milk is removed, so the quantity removed is:
   $$\frac{1}{3} 	imes 81 = 27 	ext{ liters}$$

3. This 27 liters is replaced by water. Now, the container has:
   - Milk: $$81 - 27 = 54 	ext{ liters}$$
   - Water: $$27 	ext{ liters}$$

4. Next, one-third of the mixture is taken out. The total mixture is still 81 liters, so one-third is 27 liters.

5. The mixture ratio before removal is milk : water = 54 : 27 = 2 : 1.

6. Amount of milk removed in 27 liters:
   $$\frac{2}{3} 	imes 27 = 18 	ext{ liters}$$

7. Amount of water removed in 27 liters:
   $$\frac{1}{3} 	imes 27 = 9 	ext{ liters}$$

8. After removal, remaining quantities:
   - Milk: $$54 - 18 = 36 	ext{ liters}$$
   - Water: $$27 - 9 = 18 	ext{ liters}$$

9. Now, 27 liters of water is added to the container.

10. Final quantities:
    - Milk: 36 liters
    - Water: $$18 + 27 = 45 	ext{ liters}$$

11. The ratio of milk to water in the final mixture is:
    $$\frac{36}{45} = \frac{4}{5}$$

So, milk : water = 4 : 5

**Answer:** 4 : 5

**Note:** The correct option is (d) 4 : 5.

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