Vessel contains a mixture of milk and water in ratio `14:3`. 25.5 litres of mixture is taken out from the vessel and 2.5 litres of pure water and 5 litres of pure milk is added of the mixture. If resultant mixture contains 20% water. What was the initial quantity of mixture in the vessel before the replacement (in litres)?

Options

A. 51 L
B. 85 L
C. 81 L
D. 68 L

Answer

To solve the problem step by step, we need to find the initial quantity of the mixture in the vessel before the replacement.

### Step 1: Understand the initial mixture ratio
The initial mixture of milk and water is in the ratio of 14:3. This means for every 14 parts of milk, there are 3 parts of water.

### Step 2: Define the total initial quantity
Let the initial quantity of the mixture be \( x \) litres. The total parts of the mixture is \( 14 + 3 = 17 \) parts.

### Step 3: Calculate the quantity of milk and water in the initial mixture
- Milk in the initial mixture = \( \frac{14}{17} \times x \)
- Water in the initial mixture = \( \frac{3}{17} \times x \)

### Step 4: Calculate the amount of mixture taken out
From the vessel, 25.5 litres of the mixture is taken out. We need to find out how much milk and water is in this 25.5 litres.

- Milk taken out = \( \frac{14}{17} \times 25.5 \)
- Water taken out = \( \frac{3}{17} \times 25.5 \)

### Step 5: Calculate the amounts taken out
Calculating the amounts:
- Milk taken out = \( \frac{14 \times 25.5}{17} = 21 \) litres
- Water taken out = \( \frac{3 \times 25.5}{17} = 4.5 \) litres

### Step 6: Add pure milk and water
After taking out the mixture, we add 2.5 litres of pure water and 5 litres of pure milk back to the vessel.

### Step 7: Calculate the new amounts of milk and water
- New amount of milk = Initial milk - Milk taken out + Milk added
- New amount of water = Initial water - Water taken out + Water added

Substituting the values:
- New amount of milk = \( \frac{14}{17}x - 21 + 5 \)
- New amount of water = \( \frac{3}{17}x - 4.5 + 2.5 \)

### Step 8: Set up the equation for the resultant mixture
The resultant mixture contains 20% water. Therefore, the total amount of water in the resultant mixture can be expressed as:
\[ \text{Total mixture} = x - 25.5 + 2.5 + 5 = x - 18 \]
\[ \text{Water in the resultant mixture} = \frac{3}{17}x - 4.5 + 2.5 \]
Setting up the equation:
\[ \frac{3}{17}x - 2 = 0.2(x - 18) \]

### Step 9: Solve the equation
Expanding the equation:
\[ \frac{3}{17}x - 2 = 0.2x - 3.6 \]
Multiplying through by 17 to eliminate the fraction:
\[ 3x - 34 = 3.4x - 61.2 \]
Rearranging gives:
\[ 3x - 3.4x = -61.2 + 34 \]
\[ -0.4x = -27.2 \]
\[ x = \frac{27.2}{0.4} = 68 \]

### Conclusion
The initial quantity of the mixture in the vessel before the replacement was **68 litres**.

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