In what ratio should freely available water be mixed with the wine worth Rs. 60 per litre so that after selling the mixture at Rs. 50 per litre, the profit will be 25%?

Options

A. `1:2`
B. `2:3`
C. `3:4`
D. `4:5`

Answer

To solve the problem step by step, we need to determine the ratio in which freely available water should be mixed with wine worth Rs. 60 per liter, so that after selling the mixture at Rs. 50 per liter, the profit will be 25%.

### Step 1: Understand the given values
- Cost price (CP) of wine = Rs. 60 per liter
- Cost price (CP) of water = Rs. 0 per liter (since it is freely available)
- Selling price (SP) of the mixture = Rs. 50 per liter
- Profit percentage = 25%

### Step 2: Calculate the effective cost price (CP) of the mixture
To find the effective cost price of the mixture that leads to a 25% profit when sold for Rs. 50, we can use the formula:

\[ \text{SP} = \text{CP} + \text{Profit} \]

Where profit can be calculated as:

\[ \text{Profit} = \frac{\text{Profit Percentage}}{100} \times \text{CP} \]

Given that the selling price (SP) is Rs. 50, we can rearrange the formula to find the effective cost price (CP):

\[ \text{CP} = \frac{\text{SP}}{1 + \frac{\text{Profit Percentage}}{100}} = \frac{50}{1 + 0.25} = \frac{50}{1.25} = 40 \]

### Step 3: Set up the equation for the mixture
Let the quantity of water be \( w \) liters and the quantity of wine be \( m \) liters. The total cost price of the mixture can be expressed as:

\[ \text{Total CP of mixture} = \text{CP of water} + \text{CP of wine} = 0 \cdot w + 60 \cdot m = 60m \]

The total quantity of the mixture is:

\[ \text{Total quantity} = w + m \]

The effective cost price of the mixture per liter is:

\[ \text{CP of mixture} = \frac{\text{Total CP}}{\text{Total quantity}} = \frac{60m}{w + m} \]

### Step 4: Set the CP of the mixture equal to Rs. 40
Since we calculated the effective CP of the mixture to be Rs. 40, we can set up the equation:

\[ \frac{60m}{w + m} = 40 \]

### Step 5: Solve the equation
Cross-multiplying gives us:

\[ 60m = 40(w + m) \]

Expanding the right side:

\[ 60m = 40w + 40m \]

Rearranging terms:

\[ 60m - 40m = 40w \]

\[ 20m = 40w \]

Dividing both sides by 20:

\[ m = 2w \]

### Step 6: Find the ratio of water to wine
From the equation \( m = 2w \), we can express the ratio of water to wine:

\[ \frac{w}{m} = \frac{w}{2w} = \frac{1}{2} \]

Thus, the ratio of water to wine is:

\[ \text{Ratio of water to wine} = 1:2 \]

### Final Answer
The ratio in which freely available water should be mixed with wine is **1:2**.

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