There are two containers the first contain, 1 litre pure water and the second contains 1 litre of pure milk. Now 5 cups of water from the first container is taken out is mixed well in the second container. Then 5 cups of this mixture is taken out and is mixed in the first container. Let A denote the proportion of milk in the first container and B denote the proportion of water in the second container then:

Options

A. a. `AltB`
B. b. `A=B`
C. c. `AgtB`
D. d. can't be determined

Answer

To solve the problem step by step, we will analyze the situation involving the two containers and the transfers of liquid between them.

### Step 1: Initial Setup
- **Container A**: Contains 1 litre of pure water.
- **Container B**: Contains 1 litre of pure milk.

### Step 2: Transfer from Container A to Container B
- We take out 5 cups of water from Container A and transfer it to Container B.
- Since 1 cup is approximately 0.236 litres, 5 cups is approximately 1.18 litres. However, since we are only dealing with 1 litre of water in Container A, we can assume that we are transferring 0.5 litres (which is equivalent to 5 cups) of water to Container B.

After this transfer:
- **Container A**: 0.5 litres of water (1 litre - 0.5 litres)
- **Container B**: 1 litre of milk + 0.5 litres of water = 1.5 litres of mixture (which is 1 litre of milk and 0.5 litres of water).

### Step 3: Mixing in Container B
- The mixture in Container B now consists of 1 litre of milk and 0.5 litres of water.
- The proportion of milk in Container B is \( \frac{1}{1.5} = \frac{2}{3} \) and the proportion of water is \( \frac{0.5}{1.5} = \frac{1}{3} \).

### Step 4: Transfer from Container B back to Container A
- Now we take out 5 cups (0.5 litres) of the mixture from Container B and transfer it back to Container A.
- The mixture taken out contains \( \frac{2}{3} \) litre of milk and \( \frac{1}{3} \) litre of water.

### Step 5: Final Contents of the Containers
- **Container A**:
- Initially had 0.5 litres of water.
- After adding back the mixture (0.5 litres), it now has:
- Water: \( 0.5 + \frac{1}{3} \) litres = \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \) litres of water.
- Milk: \( \frac{2}{3} \) litres of milk.

- **Container B**:
- Initially had 1 litre of milk and 0.5 litres of water.
- After removing 0.5 litres of the mixture, it now has:
- Water: \( 0.5 - \frac{1}{3} \) litres = \( \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \) litres of water.
- Milk: \( 1 - \frac{2}{3} \) litres = \( \frac{3}{3} - \frac{2}{3} = \frac{1}{3} \) litres of milk.

### Step 6: Proportions
- **Proportion of Milk in Container A (A)**:
- Total liquid in Container A = \( \frac{5}{6} + \frac{2}{3} = \frac{5}{6} + \frac{4}{6} = \frac{9}{6} = 1.5 \) litres.
- Proportion of milk \( A = \frac{\frac{2}{3}}{1.5} = \frac{2}{3} \times \frac{2}{3} = \frac{4}{9} \).

- **Proportion of Water in Container B (B)**:
- Total liquid in Container B = \( \frac{1}{6} + \frac{1}{3} = \frac{1}{6} + \frac{2}{6} = \frac{3}{6} = \frac{1}{2} \) litres.
- Proportion of water \( B = \frac{\frac{1}{6}}{\frac{1}{2}} = \frac{1}{6} \times \frac{2}{1} = \frac{1}{3} \).

### Conclusion
- The final proportions are:
- Proportion of milk in Container A \( A = \frac{4}{9} \)
- Proportion of water in Container B \( B = \frac{1}{3} \)