In a mixture of milk and water, there is only 26% water. After replacing the mixture with 7 litres of pure milk, the percentage of milk in the mixture become 76%. The quantity of mixture is :

Options

A. 65 litre
B. 91 litre
C. 38 litre
D. none of these

Answer

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

### Step 1: Understand the initial mixture
In the initial mixture of milk and water, we know that 26% of the mixture is water. Therefore, the percentage of milk in the mixture is:
\[ \text{Percentage of milk} = 100\% - 26\% = 74\% \]

**Hint:** Calculate the percentage of milk by subtracting the percentage of water from 100%.

### Step 2: Set up the scenario after replacement
After replacing 7 liters of the mixture with 7 liters of pure milk, the new percentage of milk in the mixture becomes 76%. This means the percentage of water in the new mixture is:
\[ \text{Percentage of water} = 100\% - 76\% = 24\% \]

**Hint:** Remember that pure milk contains 0% water, which affects the overall percentage of water in the mixture.

### Step 3: Use the allegation method
We will apply the allegation method to find the ratio of the quantities of the original mixture and the pure milk added. We set up the allegation as follows:

- The percentage of water in the original mixture = 26%
- The percentage of water in the pure milk = 0%
- The percentage of water in the new mixture = 24%

Using the allegation formula, we find the ratio:
\[ \text{Ratio} = \frac{(24 - 0)}{(26 - 24)} = \frac{24}{2} = 12:1 \]

**Hint:** The allegation method helps us find the ratio of the quantities of two mixtures based on their concentrations.

### Step 4: Calculate the total units
From the ratio 12:1, we can say that for every 12 parts of the original mixture, there is 1 part of pure milk added. Therefore, the total number of parts (units) in the mixture after replacement is:
\[ \text{Total units} = 12 + 1 = 13 \text{ units} \]

**Hint:** Always add the parts of both mixtures to get the total units.

### Step 5: Relate the units to the volume of pure milk
Since we know that the 1 unit corresponds to the 7 liters of pure milk added, we can find the volume of the entire mixture:
\[ \text{Volume of mixture} = 13 \text{ units} \times 7 \text{ liters/unit} = 91 \text{ liters} \]

**Hint:** Use the known volume of one part to find the total volume by multiplying it with the total number of parts.

### Conclusion
Thus, the quantity of the mixture is:
\[ \boxed{91 \text{ liters}} \]