A vessel contains a mixture of 160L of milk and 20L of water. Then 'x' L of mixture is taken out and 20L of milk and 25L of water are added to the remaining mixture. If the difference between the quantity of milk and water is 100 ltr, then find 'x'?

Options

A. 27
B. 36
C. 45
D. 54

Answer

To solve the problem step by step, we will follow the logic outlined in the video transcript and break it down into simpler parts.

### Step 1: Understand the Initial Mixture
The vessel initially contains:
- Milk = 160 L
- Water = 20 L
- Total mixture = 160 L + 20 L = 180 L

### Step 2: Determine the Composition of the Mixture Taken Out
Let 'x' L of the mixture be taken out. The proportion of milk and water in the mixture is:
- Milk proportion = 160/180 = 8/9
- Water proportion = 20/180 = 2/9

Thus, the quantities taken out are:
- Milk taken out = (8/9)x L
- Water taken out = (2/9)x L

### Step 3: Calculate Remaining Quantities After Taking Out the Mixture
After removing 'x' L of the mixture, the remaining quantities are:
- Remaining milk = 160 - (8/9)x
- Remaining water = 20 - (2/9)x

### Step 4: Add New Quantities to the Remaining Mixture
Next, we add:
- 20 L of milk
- 25 L of water

So, the new quantities become:
- New milk = (160 - (8/9)x) + 20 = 180 - (8/9)x
- New water = (20 - (2/9)x) + 25 = 45 - (2/9)x

### Step 5: Set Up the Equation for the Difference
We know that the difference between the quantity of milk and water is 100 L:
\[ \text{Milk} - \text{Water} = 100 \]
Substituting the new quantities:
\[ (180 - (8/9)x) - (45 - (2/9)x) = 100 \]

### Step 6: Simplify the Equation
Now, simplify the equation:
\[ 180 - (8/9)x - 45 + (2/9)x = 100 \]
Combine like terms:
\[ 135 - (8/9)x + (2/9)x = 100 \]
This simplifies to:
\[ 135 - (6/9)x = 100 \]
\[ 135 - \frac{2}{3}x = 100 \]

### Step 7: Isolate 'x'
Rearranging gives:
\[ 135 - 100 = \frac{2}{3}x \]
\[ 35 = \frac{2}{3}x \]
Now, multiply both sides by 3:
\[ 105 = 2x \]
Finally, divide by 2:
\[ x = \frac{105}{2} = 52.5 \]

### Step 8: Conclusion
The value of 'x' is 52.5 L.