A vessel contains 400 ltof milk. 20 ltof milk taken out from it and replaced by water. Then again from mixture 40 lt are taken out and replaced by water. Then again from mixture 60 ltare taken out and replaced by water. Find the ratio of milk and water in the resultant mixture.

Options

A. 15:5
B. 17:3
C. 7:3
D. 1:4

Answer

To solve the problem step by step, we will track the amount of milk and water in the vessel after each operation.

### Step 1: Initial Setup
- The vessel initially contains **400 liters of milk** and **0 liters of water**.
- Therefore, the initial ratio of milk to water is **400:0**.

### Step 2: First Operation
- **20 liters of milk** is taken out and replaced with **20 liters of water**.
- After removing 20 liters of milk, the amount of milk left is:
\[ 400 - 20 = 380 \text{ liters of milk} \]
- The amount of water added is 20 liters, so now we have:
\[ 0 + 20 = 20 \text{ liters of water} \]
- The new ratio of milk to water is:
\[ 380:20 \]
- Simplifying this ratio gives:
\[ 19:1 \]

### Step 3: Second Operation
- Now, we take out **40 liters of the mixture** (which consists of both milk and water).
- The total volume of the mixture is now **400 liters** (380 liters of milk + 20 liters of water).
- The proportion of milk in the mixture is:
\[ \frac{380}{400} = 0.95 \text{ (or 95% milk)} \]
- The proportion of water in the mixture is:
\[ \frac{20}{400} = 0.05 \text{ (or 5% water)} \]
- When we take out 40 liters of the mixture, the amount of milk removed is:
\[ 40 \times 0.95 = 38 \text{ liters of milk} \]
- The amount of water removed is:
\[ 40 \times 0.05 = 2 \text{ liters of water} \]
- After removing these amounts, the new quantities are:
- Milk:
\[ 380 - 38 = 342 \text{ liters of milk} \]
- Water:
\[ 20 - 2 = 18 \text{ liters of water} \]
- We then add **40 liters of water** back to the mixture:
- New amount of water:
\[ 18 + 40 = 58 \text{ liters of water} \]
- The new ratio of milk to water is:
\[ 342:58 \]

### Step 4: Third Operation
- Now, we take out **60 liters of the mixture**.
- The total volume of the mixture is now **400 liters** (342 liters of milk + 58 liters of water).
- The proportion of milk in the mixture is:
\[ \frac{342}{400} = 0.855 \text{ (or 85.5% milk)} \]
- The proportion of water in the mixture is:
\[ \frac{58}{400} = 0.145 \text{ (or 14.5% water)} \]
- When we take out 60 liters of the mixture, the amount of milk removed is:
\[ 60 \times 0.855 = 51.3 \text{ liters of milk} \]
- The amount of water removed is:
\[ 60 \times 0.145 = 8.7 \text{ liters of water} \]
- After removing these amounts, the new quantities are:
- Milk:
\[ 342 - 51.3 = 290.7 \text{ liters of milk} \]
- Water:
\[ 58 - 8.7 = 49.3 \text{ liters of water} \]
- We then add **60 liters of water** back to the mixture:
- New amount of water:
\[ 49.3 + 60 = 109.3 \text{ liters of water} \]
- The final ratio of milk to water is:
\[ 290.7:109.3 \]

### Final Answer
The ratio of milk to water in the resultant mixture is:
\[ \text{Milk:Water} = 290.7:109.3 \]