A vessel of 80 litre is filled with milk and water. 70% of milk and 30% of water is taken out of the vessel. It is found that the vessel is vacated by 55%. Find the initial quantity of milk and water.

Options

A. 20,60
B. 30,50
C. 50,30
D. 60,20

Answer

To solve the problem step by step, let's denote the initial quantities of milk and water in the vessel.

### Step 1: Define Variables
Let:
- \( x \) = initial quantity of milk (in liters)
- \( y \) = initial quantity of water (in liters)

From the problem, we know that:
- The total volume of the vessel is 80 liters.
- Therefore, we can write the equation:
\[ x + y = 80 \]

### Step 2: Determine the Amount Taken Out
According to the problem:
- 70% of the milk is taken out, which is \( 0.7x \).
- 30% of the water is taken out, which is \( 0.3y \).

### Step 3: Calculate the Total Volume Taken Out
The total volume taken out from the vessel is:
\[ 0.7x + 0.3y \]

### Step 4: Determine the Volume Vacated
The problem states that the vessel is vacated by 55%. Since the total volume of the vessel is 80 liters, the volume vacated is:
\[ 0.55 \times 80 = 44 \text{ liters} \]

### Step 5: Set Up the Equation
Now, we can set up the equation based on the volume taken out:
\[ 0.7x + 0.3y = 44 \]

### Step 6: Solve the System of Equations
Now we have a system of two equations:
1. \( x + y = 80 \)
2. \( 0.7x + 0.3y = 44 \)

We can solve these equations simultaneously. From the first equation, we can express \( y \) in terms of \( x \):
\[ y = 80 - x \]

Substituting this into the second equation:
\[ 0.7x + 0.3(80 - x) = 44 \]
Expanding this:
\[ 0.7x + 24 - 0.3x = 44 \]
Combining like terms:
\[ 0.4x + 24 = 44 \]
Subtracting 24 from both sides:
\[ 0.4x = 20 \]
Dividing by 0.4:
\[ x = \frac{20}{0.4} = 50 \]

### Step 7: Find the Quantity of Water
Now, substitute \( x \) back into the equation for \( y \):
\[ y = 80 - 50 = 30 \]

### Final Answer
Thus, the initial quantities are:
- Milk: 50 liters
- Water: 30 liters