There are three containers A, B and C which contain water, milk and acid respectively in equal quantities. `10%` of the content of A is taken out and poured into B. Then, the same amount from B is transferred to C, after then the same amount from C is transferred to A. What is the proportion of milk in container A at the end of the process?

Answer

To solve the problem step by step, we will analyze the transfers of liquid between the containers A, B, and C, and calculate the proportion of milk in container A at the end of the process.

### Step 1: Initial Setup
- Container A contains water.
- Container B contains milk.
- Container C contains acid.
- Let's assume each container has a total volume of 100 liters for simplicity.

### Step 2: Transfer from A to B
- We take out 10% of the content from A, which is 10 liters (10% of 100 liters).
- Since A contains only water, we pour 10 liters of water into B.
- After this transfer:
- Container A: 90 liters of water
- Container B: 100 liters of milk + 10 liters of water = 110 liters (now contains 100 liters of milk and 10 liters of water)

### Step 3: Transfer from B to C
- Now, we take out 10% of the content from B, which is 11 liters (10% of 110 liters).
- The composition of B is 100 liters of milk and 10 liters of water. The ratio of milk to total liquid in B is 100/110 = 10/11.
- Therefore, the amount of milk transferred to C is (10/11) * 11 liters = 10 liters, and the amount of water transferred is (1/11) * 11 liters = 1 liter.
- After this transfer:
- Container B: 100 liters of milk + 10 liters of water - 10 liters of milk - 1 liter of water = 90 liters of milk and 9 liters of water.
- Container C: 0 liters of acid + 10 liters of milk + 1 liter of water = 10 liters of milk and 1 liter of water.

### Step 4: Transfer from C to A
- Now, we take out 10% of the content from C, which is 11 liters (10% of 11 liters).
- The composition of C is 10 liters of milk and 1 liter of water. The ratio of milk to total liquid in C is 10/11.
- Therefore, the amount of milk transferred to A is (10/11) * 11 liters = 10 liters, and the amount of water transferred is (1/11) * 11 liters = 1 liter.
- After this transfer:
- Container C: 10 liters of milk + 1 liter of water - 10 liters of milk - 1 liter of water = 0 liters of milk and 0 liters of water.
- Container A: 90 liters of water + 10 liters of milk + 1 liter of water = 10 liters of milk and 91 liters of water.

### Step 5: Final Calculation of Proportion of Milk in A
- The total volume in container A is now 91 liters of water + 10 liters of milk = 101 liters.
- The proportion of milk in container A is:
\[ \text{Proportion of milk} = \frac{\text{Volume of milk}}{\text{Total volume}} = \frac{10}{101} \]

### Final Answer
The proportion of milk in container A at the end of the process is \( \frac{10}{101} \).