In a vessel containing 70 litres of milk, 12 litres of it is replaced with water, and after that 5 litres of the mixture is replaced with water. What is the ratio of milk and water in the final mixture?

Options

A. `377:113`
B. `351:69`
C. `279:141`
D. `293:197`

Answer

To solve the problem step by step, we will follow the operations performed on the milk and water mixture in the vessel.

### Step 1: Initial Setup
- We start with 70 liters of milk in the vessel.

### Step 2: First Replacement
- We replace 12 liters of milk with water.
- After this operation, the amount of milk left in the vessel is:
\[ \text{Milk remaining} = 70 - 12 = 58 \text{ liters} \]
- The amount of water added is 12 liters.

### Step 3: Mixture After First Replacement
- Now, the total volume in the vessel is still 70 liters, consisting of:
- Milk: 58 liters
- Water: 12 liters

### Step 4: Second Replacement
- Next, we replace 5 liters of the mixture with water.
- The mixture consists of 58 liters of milk and 12 liters of water, making a total of 70 liters.
- The fraction of milk in the mixture is:
\[ \text{Fraction of milk} = \frac{58}{70} \]
- The fraction of water in the mixture is:
\[ \text{Fraction of water} = \frac{12}{70} \]

### Step 5: Calculate Milk and Water Removed
- When we remove 5 liters of the mixture, the amount of milk removed is:
\[ \text{Milk removed} = 5 \times \frac{58}{70} = \frac{290}{70} = \frac{29}{7} \text{ liters} \]
- The amount of water removed is:
\[ \text{Water removed} = 5 \times \frac{12}{70} = \frac{60}{70} = \frac{6}{7} \text{ liters} \]

### Step 6: Update Amounts After Second Replacement
- After removing the 5 liters, the new amounts are:
- Milk remaining:
\[ \text{New milk} = 58 - \frac{29}{7} = \frac{406}{7} - \frac{29}{7} = \frac{377}{7} \text{ liters} \]
- Water remaining:
\[ \text{New water} = 12 - \frac{6}{7} + 5 = \frac{84}{7} - \frac{6}{7} + \frac{35}{7} = \frac{113}{7} \text{ liters} \]

### Step 7: Final Ratio of Milk to Water
- The final amounts of milk and water are:
- Milk: \(\frac{377}{7}\) liters
- Water: \(\frac{113}{7}\) liters
- The ratio of milk to water is:
\[ \text{Ratio} = \frac{\frac{377}{7}}{\frac{113}{7}} = \frac{377}{113} \]

### Final Answer
- The ratio of milk to water in the final mixture is:
\[ \boxed{\frac{377}{113}} \]