From the 50 litres of milk, 5 litres of milk is taken out and after it 5 litres of water is added to the rest amount of milk. Again 5 litres of mixture of milk and water is drawn out and it was replaced by 5 litres of water. If this process is continued similarly for the third time, the amount of milk left after the third replacement :

Options

A. 45 L
B. 36.45 L
C. 40.5 L
D. 42.5 L

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Initial Setup
We start with 50 liters of pure milk.

### Step 2: First Replacement
- We take out 5 liters of milk.
- Remaining milk = 50 liters - 5 liters = 45 liters.
- We then add 5 liters of water.
- Total mixture now = 45 liters of milk + 5 liters of water = 50 liters of mixture.

### Step 3: Calculate Milk Concentration After First Replacement
- The concentration of milk after the first replacement is:
\[ \text{Concentration of milk} = \frac{\text{Remaining milk}}{\text{Total mixture}} = \frac{45}{50} = 0.9 \]

### Step 4: Second Replacement
- Now we take out 5 liters of this mixture (which is 90% milk and 10% water).
- Amount of milk taken out = 5 liters × 0.9 = 4.5 liters.
- Amount of water taken out = 5 liters × 0.1 = 0.5 liters.
- Remaining milk after second replacement = 45 liters - 4.5 liters = 40.5 liters.
- We add 5 liters of water again.
- Total mixture now = 40.5 liters of milk + 5 liters of water = 45.5 liters of mixture.

### Step 5: Calculate Milk Concentration After Second Replacement
- The concentration of milk after the second replacement is:
\[ \text{Concentration of milk} = \frac{40.5}{50} = 0.81 \]

### Step 6: Third Replacement
- Again, we take out 5 liters of this mixture (which is 81% milk and 19% water).
- Amount of milk taken out = 5 liters × 0.81 = 4.05 liters.
- Amount of water taken out = 5 liters × 0.19 = 0.95 liters.
- Remaining milk after third replacement = 40.5 liters - 4.05 liters = 36.45 liters.
- We add 5 liters of water again.
- Total mixture now = 36.45 liters of milk + 5 liters of water = 41.45 liters of mixture.

### Step 7: Final Calculation of Milk Left
After performing the operation three times, the amount of milk left is:
\[ \text{Amount of milk left} = 36.45 \text{ liters} \]

### Summary
The amount of milk left after the third replacement is **36.45 liters**.

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