What approximate value will come in place of the question mark (?) in the following question?
In a mixture, ratio of milk and water is `6 : 1`. 21 liters of mixture is withdrawn, and 77 liters of water is added to the mixture. If the total quantity of resultant mixture is twice of the quantity of original mixture, what was the amount of milk in original mixture?

Options

A. A)24 liters
B. B)30 liters
C. C)48 liters
D. D)56 liters

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the initial mixture
The ratio of milk to water in the mixture is given as 6:1. This means for every 6 parts of milk, there is 1 part of water.

### Step 2: Calculate the total parts in the mixture
The total parts in the mixture = 6 (milk) + 1 (water) = 7 parts.

### Step 3: Define the total volume of the original mixture
Let the total volume of the original mixture be \( x \) liters. Since the ratio is 6:1, the amount of milk in the original mixture is \( \frac{6}{7}x \) and the amount of water is \( \frac{1}{7}x \).

### Step 4: Withdraw 21 liters of the mixture
When 21 liters of the mixture is withdrawn, the amount of milk and water removed can be calculated based on the ratio:
- Milk withdrawn = \( \frac{6}{7} \times 21 = 18 \) liters
- Water withdrawn = \( \frac{1}{7} \times 21 = 3 \) liters

### Step 5: Calculate the remaining quantities after withdrawal
After withdrawing 21 liters:
- Remaining milk = \( \frac{6}{7}x - 18 \)
- Remaining water = \( \frac{1}{7}x - 3 \)

### Step 6: Add 77 liters of water to the mixture
Now, we add 77 liters of water to the remaining water:
- New amount of water = \( \left(\frac{1}{7}x - 3\right) + 77 = \frac{1}{7}x + 74 \)

### Step 7: Calculate the total quantity of the resultant mixture
The total quantity of the resultant mixture is:
- Total mixture = Remaining milk + New amount of water
- Total mixture = \( \left(\frac{6}{7}x - 18\right) + \left(\frac{1}{7}x + 74\right) \)
- Total mixture = \( \frac{6}{7}x + \frac{1}{7}x + 74 - 18 \)
- Total mixture = \( x + 56 \)

### Step 8: Set up the equation based on the problem statement
According to the problem, the total quantity of the resultant mixture is twice the quantity of the original mixture:
- \( x + 56 = 2x \)

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

### Step 10: Find the amount of milk in the original mixture
Now, we can find the amount of milk in the original mixture:
- Amount of milk = \( \frac{6}{7}x = \frac{6}{7} \times 56 = 48 \) liters.

Thus, the amount of milk in the original mixture is **48 liters**.