The diluted wine contains only 8 litres of wine and the rest is water. A new mixture whose concentration is 30%, is to be formed by replacing wine. How many litres of mixture shall be replaced with pure wine if there was initially 32 litres of water in the mixture ?

Options

A. 4
B. 5
C. 8
D. none of these

Answer

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

### Step 1: Understand the Initial Mixture
The initial mixture consists of:
- Wine: 8 liters
- Water: 32 liters

Total mixture = Wine + Water = 8 liters + 32 liters = 40 liters

**Hint:** Always start by identifying the components of the mixture and their quantities.

### Step 2: Determine the Required Concentration
We want to create a new mixture with a concentration of wine at 30%. This means that the concentration of water will be:
- Water concentration = 100% - 30% = 70%

**Hint:** Remember that the total concentration must always equal 100%.

### Step 3: Set Up the Allegation Method
We will use the allegation method to find out how much of the mixture needs to be replaced with pure wine. We will compare the concentrations of water in the current mixture and the new mixture.

Current concentration of water in the mixture:
- Water concentration = 80% (since there are 32 liters of water in a total of 40 liters)

Concentration of water in the pure wine:
- Water concentration = 0%

Concentration of water in the new mixture:
- Water concentration = 70%

**Hint:** The allegation method helps to find the ratio of two components based on their concentrations.

### Step 4: Apply the Allegation Formula
Using the allegation method:
- Difference between the new mixture's water concentration and the current mixture's water concentration:
- 70% (new mixture) - 0% (pure wine) = 70
- Difference between the current mixture's water concentration and the new mixture's water concentration:
- 80% (current mixture) - 70% (new mixture) = 10

Now, we can set up the ratio:
- Ratio = 70 : 10 = 7 : 1

**Hint:** The differences calculated will give you the ratio of the parts of the mixtures.

### Step 5: Calculate the Total Parts
The total parts in the ratio is:
- Total parts = 7 + 1 = 8 parts

**Hint:** Always sum the parts of the ratio to understand the total mixture composition.

### Step 6: Find the Volume of Each Part
Since the total mixture remains 40 liters, we can find the volume represented by each part:
- Volume of each part = Total mixture / Total parts = 40 liters / 8 parts = 5 liters

**Hint:** This step helps to determine how much each part of the ratio represents in terms of volume.

### Step 7: Determine the Volume of Mixture to be Replaced
From the ratio, the part representing the volume of the mixture to be replaced with pure wine is 1 part:
- Volume to be replaced = 1 part = 5 liters

**Hint:** The part that represents the replacement will give you the answer to how much needs to be replaced.

### Conclusion
Thus, the amount of the mixture that should be replaced with pure wine is **5 liters**.

**Final Answer:** 5 liters