There are two containers, the first contains, 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 containers. 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 lt B
B. A=B
C. `A gt B`
D. can't be determined

Answer

To solve the problem, we need to determine the proportions of milk in the first container (A) and the proportion of water in the second container (B) after the mixing process. Let's break it down step by step.

### Step 1: Initial Setup
- Container 1: 1 liter of pure water.
- Container 2: 1 liter of pure milk.

### Step 2: Transfer Water to Milk
- From Container 1, we take out 5 cups of water. Let's assume each cup is a standard size (for simplicity, we can assume each cup is 100 ml, but the actual size does not affect the final ratio since we are looking at proportions).
- Amount of water taken out: 5 cups = 500 ml = 0.5 liters.
- After this transfer, Container 1 has 0.5 liters of water left, and Container 2 now has 1 liter of milk + 0.5 liters of water = 1.5 liters of mixture.

### Step 3: Mixing the Mixture
- In Container 2, we now have a mixture of 1 liter of milk and 0.5 liters of water.
- The proportion of milk in Container 2 is \( \frac{1 \text{ liter}}{1.5 \text{ liters}} = \frac{2}{3} \).
- The proportion of water in Container 2 is \( \frac{0.5 \text{ liters}}{1.5 \text{ liters}} = \frac{1}{3} \).

### Step 4: Transfer Mixture Back to Water
- Now, we take out 5 cups of this mixture from Container 2. Again, assuming each cup is 100 ml, we take out 500 ml of the mixture.
- The amount of milk in the 500 ml taken out from Container 2 is \( 500 \text{ ml} \times \frac{2}{3} = 333.33 \text{ ml} \).
- The amount of water in the 500 ml taken out from Container 2 is \( 500 \text{ ml} \times \frac{1}{3} = 166.67 \text{ ml} \).

### Step 5: Final Composition in Container 1
- After transferring back to Container 1, the new amounts in Container 1 are:
- Water: \( 0.5 \text{ liters} + 166.67 \text{ ml} = 666.67 \text{ ml} \) or \( 0.66667 \text{ liters} \).
- Milk: \( 333.33 \text{ ml} \) or \( 0.33333 \text{ liters} \).

### Step 6: Calculate Proportions
- Total volume in Container 1: \( 0.66667 \text{ liters} + 0.33333 \text{ liters} = 1 \text{ liter} \).
- Proportion of milk (A) in Container 1: \( A = \frac{0.33333 \text{ liters}}{1 \text{ liter}} = 0.33333 \) or \( \frac{1}{3} \).
- Proportion of water (B) in Container 2: \( B = \frac{0.5 \text{ liters}}{1.5 \text{ liters}} = \frac{1}{3} \).

### Conclusion
- A (proportion of milk in Container 1) = \( \frac{1}{3} \)
- B (proportion of water in Container 2) = \( \frac{1}{3} \)

Thus, we have A = B.