A pot contains 81 litres of pure milk. 1/3 of the milk is replaced by the same amount of water. Again of the mixture is replaced by that amount of water. The ratio of milk and water in the new mixture is

Options

A. ` 1 : 2`
B. ` 1 : 1`
C. ` 2 : 1`
D. ` 4 : 5`

Answer

To solve the problem step by step, we will follow the operations of replacing milk with water and calculating the new quantities of milk and water after each replacement.

### Step 1: Initial Quantities
Initially, the pot contains:
- Milk = 81 liters
- Water = 0 liters

### Step 2: First Replacement
We replace 1/3 of the milk with water.
- Amount of milk replaced = \( \frac{1}{3} \times 81 = 27 \) liters

After this replacement:
- Milk remaining = \( 81 - 27 = 54 \) liters
- Water added = 27 liters

Now, the new quantities are:
- Milk = 54 liters
- Water = 27 liters

### Step 3: Second Replacement
Now, we again replace 1/3 of the new mixture (which is now 54 liters of milk and 27 liters of water) with water.
- Total mixture = \( 54 + 27 = 81 \) liters
- Amount of milk replaced = \( \frac{1}{3} \times 81 = 27 \) liters

Since we are replacing milk, we need to calculate how much of the milk is left after this replacement:
- The proportion of milk in the mixture = \( \frac{54}{81} \)
- Amount of milk replaced = \( \frac{54}{81} \times 27 = 18 \) liters

After this replacement:
- Milk remaining = \( 54 - 18 = 36 \) liters
- Water added = 27 liters (previous water) + 18 liters (new water) = 45 liters

Now, the new quantities are:
- Milk = 36 liters
- Water = 45 liters

### Step 4: Final Ratio
Now, we need to find the ratio of milk to water in the final mixture.
- Ratio of Milk to Water = \( \frac{Milk}{Water} = \frac{36}{45} \)

To simplify this ratio, we can divide both numbers by their greatest common divisor (GCD), which is 9:
- \( \frac{36 \div 9}{45 \div 9} = \frac{4}{5} \)

Thus, the final ratio of milk to water in the new mixture is **4:5**.