Pure milk costs ₹ 16 per litre. After adding water the milkman sells the mixture ₹15 per litre and thereby makes a profit of 25%. In what respective ratio does he mix milk with water?

Options

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

Explanation

To solve the problem step by step, we will follow the process of calculating the cost price of the mixture and then use the allegation method to find the ratio of milk to water.

### Step 1: Determine the Selling Price and Profit Percentage
- The selling price (SP) of the mixture is ₹15 per litre.
- The profit percentage is 25%.

### Step 2: Calculate the Cost Price (CP) of the Mixture
Using the formula for profit:
\[ SP = CP + \text{Profit} \]
Where:
\[ \text{Profit} = \frac{\text{Profit Percentage}}{100} \times CP \]
Thus, we can express SP as:
\[ SP = CP + \frac{25}{100} \times CP = CP \left(1 + \frac{25}{100}\right) = CP \left(\frac{125}{100}\right) \]
Now, substituting the SP:
\[ 15 = CP \left(\frac{125}{100}\right) \]
To find CP, we rearrange:
\[ CP = 15 \times \frac{100}{125} = 15 \times 0.8 = 12 \]
So, the cost price of the mixture is ₹12 per litre.

### Step 3: Identify the Cost Price of Milk and Water
- The cost price of pure milk is ₹16 per litre.
- The cost price of water is ₹0 per litre.

### Step 4: Use the Allegation Method to Find the Ratio
We will set up the allegation method:
- Higher value (milk) = ₹16
- Lower value (water) = ₹0
- Average value (mixture) = ₹12

Using the allegation formula:
- Difference between average value and lower value:
\[ 12 - 0 = 12 \]
- Difference between higher value and average value:
\[ 16 - 12 = 4 \]

### Step 5: Write the Ratio
The ratio of milk to water is given by the differences calculated:
\[ \text{Ratio of Milk : Water} = 12 : 4 \]
Simplifying this ratio:
\[ \text{Ratio of Milk : Water} = 3 : 1 \]

### Final Answer
The milkman mixes milk and water in the ratio of **3:1**.

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