A container has 30 litres of milk, from which 3 litres of milk is taken out and replaced with water. The process is done three times. What is the final ratio of the water and the milk in the container ?

Options

A. 729 :271
B. `19:81`
C. `81 : 19`
D. 271 : 729

Answer

To solve the problem step by step, we will calculate the amount of milk left in the container after each replacement of milk with water.

### Step 1: Initial Setup
- The container initially has 30 liters of milk.
- We will take out 3 liters of milk and replace it with 3 liters of water.

### Step 2: First Replacement
1. **Amount of milk taken out**: 3 liters.
2. **Amount of milk remaining after first replacement**:
\[ \text{Remaining milk} = 30 - 3 = 27 \text{ liters} \]
3. **Total liquid in the container**: 30 liters (27 liters of milk + 3 liters of water).

### Step 3: Second Replacement
1. **Amount of milk in the container before the second replacement**: 27 liters.
2. **Proportion of milk in the container**:
\[ \text{Proportion of milk} = \frac{27}{30} \]
3. **Amount of milk taken out in the second replacement**:
\[ \text{Milk taken out} = 3 \times \frac{27}{30} = 2.7 \text{ liters} \]
4. **Amount of milk remaining after second replacement**:
\[ \text{Remaining milk} = 27 - 2.7 = 24.3 \text{ liters} \]
5. **Total liquid in the container**: 30 liters (24.3 liters of milk + 5.7 liters of water).

### Step 4: Third Replacement
1. **Amount of milk in the container before the third replacement**: 24.3 liters.
2. **Proportion of milk in the container**:
\[ \text{Proportion of milk} = \frac{24.3}{30} \]
3. **Amount of milk taken out in the third replacement**:
\[ \text{Milk taken out} = 3 \times \frac{24.3}{30} = 2.43 \text{ liters} \]
4. **Amount of milk remaining after third replacement**:
\[ \text{Remaining milk} = 24.3 - 2.43 = 21.87 \text{ liters} \]
5. **Total liquid in the container**: 30 liters (21.87 liters of milk + 8.13 liters of water).

### Step 5: Final Calculation of Ratios
1. **Total amount of water**:
\[ \text{Water} = 30 - 21.87 = 8.13 \text{ liters} \]
2. **Final ratio of water to milk**:
\[ \text{Ratio of water to milk} = \frac{8.13}{21.87} \]
This can be simplified to:
\[ \text{Ratio} \approx \frac{813}{2187} \text{ (after multiplying by 100)} \]

### Final Answer
The final ratio of water to milk in the container is approximately 813:2187.