A vessel contains 500 litre of milk. 50 litre of milk taken out from it and replaced by water. Then again from mixture 100 litre taken out and replaced by water. mixture 100 litre taken out and replaced by water.Then again from mixture 125 litre taken out and replaced by water. Find the ratio of milk and water in the resultant mixture.

Answer

To solve the problem step by step, we will calculate the amount of milk remaining in the vessel after each operation of removing milk and replacing it with water.

### Step 1: Initial Setup
- The vessel initially contains 500 liters of milk.

### Step 2: First Operation
- **Milk Removed**: 50 liters
- **Milk Remaining**:
\[ 500 - 50 = 450 \text{ liters} \]
- **Water Added**: 50 liters
- **Total Mixture**: 450 liters of milk + 50 liters of water = 500 liters

### Step 3: Second Operation
- **Mixture Before Second Operation**: 450 liters of milk and 50 liters of water.
- **Total Mixture**: 500 liters
- **Milk Concentration**:
\[ \frac{450}{500} = 0.9 \]
- **Milk Removed**: 100 liters
- **Milk Removed from Mixture**:
\[ 100 \times 0.9 = 90 \text{ liters of milk} \]
- **Milk Remaining**:
\[ 450 - 90 = 360 \text{ liters} \]
- **Water Added**: 100 liters
- **Total Mixture**: 360 liters of milk + 140 liters of water = 500 liters

### Step 4: Third Operation
- **Mixture Before Third Operation**: 360 liters of milk and 140 liters of water.
- **Total Mixture**: 500 liters
- **Milk Concentration**:
\[ \frac{360}{500} = 0.72 \]
- **Milk Removed**: 100 liters
- **Milk Removed from Mixture**:
\[ 100 \times 0.72 = 72 \text{ liters of milk} \]
- **Milk Remaining**:
\[ 360 - 72 = 288 \text{ liters} \]
- **Water Added**: 100 liters
- **Total Mixture**: 288 liters of milk + 212 liters of water = 500 liters

### Step 5: Fourth Operation
- **Mixture Before Fourth Operation**: 288 liters of milk and 212 liters of water.
- **Total Mixture**: 500 liters
- **Milk Concentration**:
\[ \frac{288}{500} = 0.576 \]
- **Milk Removed**: 125 liters
- **Milk Removed from Mixture**:
\[ 125 \times 0.576 = 72 \text{ liters of milk} \]
- **Milk Remaining**:
\[ 288 - 72 = 216 \text{ liters} \]
- **Water Added**: 125 liters
- **Total Mixture**: 216 liters of milk + 337 liters of water = 500 liters

### Step 6: Final Calculation of Ratio
- **Final Amount of Milk**: 216 liters
- **Final Amount of Water**: 500 - 216 = 284 liters
- **Ratio of Milk to Water**:
\[ \text{Ratio} = \frac{216}{284} = \frac{54}{71} \]

### Final Answer
The ratio of milk to water in the resultant mixture is **54:71**.