In a mixture of a 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, let's denote the total quantity of the mixture as \( x \) liters.

### Step 1: Determine the initial quantities of milk and water in the mixture.
Given that the mixture contains 26% water, we can calculate the quantities of water and milk:
- Water = 26% of \( x \) = \( 0.26x \)
- Milk = 74% of \( x \) = \( 0.74x \)

**Hint:** Remember that the percentages of milk and water must add up to 100%.

### Step 2: Understand the effect of adding 7 liters of pure milk.
When we replace 7 liters of the mixture with 7 liters of pure milk, the new quantity of milk in the mixture becomes:
- New Milk = Old Milk - Milk removed + Pure Milk added
- Milk removed = 7 liters of the mixture which contains 74% milk.
- Milk removed = \( 0.74 \times 7 = 5.18 \) liters.

So, the new quantity of milk becomes:
- New Milk = \( 0.74x - 5.18 + 7 \)

**Hint:** Calculate how much milk is removed and then add the pure milk to find the new total.

### Step 3: Set up the equation for the new percentage of milk.
After adding the 7 liters of pure milk, the total quantity of the mixture remains \( x \) liters. The new percentage of milk is given as 76%. Therefore, we can set up the equation:
\[ \frac{0.74x - 5.18 + 7}{x} = 0.76 \]

**Hint:** Remember that the percentage can be expressed as a fraction of the total quantity.

### Step 4: Simplify the equation.
First, simplify the numerator:
\[ 0.74x + 1.82 = 0.76x \]

Now, rearranging the equation gives:
\[ 0.76x - 0.74x = 1.82 \]
\[ 0.02x = 1.82 \]

**Hint:** Isolate \( x \) to find its value.

### Step 5: Solve for \( x \).
Now, divide both sides by 0.02:
\[ x = \frac{1.82}{0.02} = 91 \]

**Hint:** Ensure you perform the division correctly to find the total quantity.

### Conclusion:
The total quantity of the mixture is \( 91 \) liters.

**Final Answer:** The quantity of the mixture is 91 liters.