A vessel contain mixture of milk and water in the ratio of 6: 1 respectively. If 21 liters mixture taken out and replaced with 77 liters of milk, then the resultant mixture becomes twice of the initial mixture. Find the quantity of the initial mixture?

Options

A. 56 liters
B. 42 liters
C. 84 liters
D. 35 liters

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Understand the initial conditions
We have a mixture of milk and water in a vessel in the ratio of 6:1. This means for every 6 parts of milk, there is 1 part of water.

### Step 2: Define the initial quantities
Let the total quantity of the mixture be \( x \) liters. Therefore, the quantity of milk in the mixture is \( \frac{6}{7}x \) liters and the quantity of water is \( \frac{1}{7}x \) liters.

### Step 3: Calculate the quantity of mixture taken out
When 21 liters of the mixture is taken out, the quantities of milk and water removed can be calculated based on the initial ratio:
- Milk taken out = \( \frac{6}{7} \times 21 = 18 \) liters
- Water taken out = \( \frac{1}{7} \times 21 = 3 \) liters

### Step 4: Calculate the remaining quantities after removal
After removing 21 liters of the mixture, the remaining quantities of milk and water in the vessel will be:
- Remaining milk = \( \frac{6}{7}x - 18 \) liters
- Remaining water = \( \frac{1}{7}x - 3 \) liters

### Step 5: Add 77 liters of milk
Now, we replace the 21 liters of mixture with 77 liters of milk. The new quantity of milk becomes:
- New milk quantity = \( \left(\frac{6}{7}x - 18\right) + 77 \)

### Step 6: Write the new total quantities
The new total quantities in the vessel will be:
- New milk quantity = \( \frac{6}{7}x - 18 + 77 = \frac{6}{7}x + 59 \)
- Water quantity remains the same = \( \frac{1}{7}x - 3 \)

### Step 7: Set up the equation for the resultant mixture
According to the problem, the resultant mixture becomes twice the initial mixture. Therefore, the total quantity of the new mixture is:
- Total new mixture = \( \left(\frac{6}{7}x + 59\right) + \left(\frac{1}{7}x - 3\right) = x + 56 \)

### Step 8: Set up the equation for the ratio
Since the new mixture is said to be twice the initial mixture, we can set up the equation:
\[ x + 56 = 2x \]

### Step 9: Solve for \( x \)
Rearranging the equation gives:
\[ 56 = 2x - x \]
\[ 56 = x \]

### Step 10: Conclusion
The quantity of the initial mixture is \( 56 \) liters.