A vessel 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% and has 160 L mixture.Find the quantity of milk and water in this mixture.

Options

A. Milk = 100 L, Water = 60 L
B. Milk = 50 L, Water = 110 L
C. Milk = 70 L, Water = 90 L
D. Milk = 60 L, Water = 100 L

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Understand the problem
We have a vessel containing a mixture of milk and water. We know that 70% of the milk and 30% of the water are taken out. After this removal, it is stated that the vessel is vacated by 55% and contains 160 liters of the mixture.

### Step 2: Set up the equations
Let the initial quantity of milk in the vessel be \( X \) liters. Since the total mixture is 160 liters, the initial quantity of water will be \( 160 - X \) liters.

### Step 3: Calculate the quantities taken out
From the vessel:
- The amount of milk taken out = \( 70\% \) of \( X \) = \( 0.7X \)
- The amount of water taken out = \( 30\% \) of \( (160 - X) \) = \( 0.3(160 - X) \)

### Step 4: Set up the equation based on the vacated volume
According to the problem, the vessel is vacated by 55%. This means that the remaining mixture in the vessel is \( 45\% \) of the original volume (which is 160 liters). Therefore, the remaining volume is:
\[ 0.45 \times 160 = 72 \text{ liters} \]

### Step 5: Set up the equation for the remaining mixture
The total amount of mixture taken out is equal to the initial mixture minus the remaining mixture:
\[ (0.7X + 0.3(160 - X)) = 160 - 72 \]
This simplifies to:
\[ 0.7X + 0.3(160 - X) = 88 \]

### Step 6: Expand and simplify the equation
Expanding the equation:
\[ 0.7X + 48 - 0.3X = 88 \]
Combining like terms:
\[ 0.4X + 48 = 88 \]

### Step 7: Solve for \( X \)
Subtract 48 from both sides:
\[ 0.4X = 88 - 48 \]
\[ 0.4X = 40 \]
Now divide both sides by 0.4:
\[ X = \frac{40}{0.4} = 100 \text{ liters} \]

### Step 8: Find the quantity of water
Now that we have \( X \) (the quantity of milk), we can find the quantity of water:
\[ \text{Water} = 160 - X = 160 - 100 = 60 \text{ liters} \]

### Final Answer
The quantity of milk is 100 liters, and the quantity of water is 60 liters.