A dishonest milkman buy some milk at Rs. 10 /ltand mixed 5 ltwater to this milk and then sold it Rs. 10 /ltand gains 20 % profit. Find the quantity of milk that he bought.

Options

A. 20
B. 25
C. 30
D. 35

Answer

To solve the problem step by step, we will break down the information given and use it to find the quantity of milk that the dishonest milkman bought.

### Step 1: Understand the Selling Price and Profit
The milkman sells the mixture at Rs. 10 per liter and gains a profit of 20%.

**Hint:** Profit percentage can be used to find the cost price (CP) using the formula:
\[ \text{CP} = \frac{\text{SP} \times 100}{100 + \text{Gain \%}} \]

### Step 2: Calculate the Cost Price
Using the formula from Step 1:
- Selling Price (SP) = Rs. 10
- Gain % = 20%

Now, substituting the values:
\[ \text{CP} = \frac{10 \times 100}{100 + 20} = \frac{1000}{120} = \frac{50}{6} \text{ Rs. per liter} \]

**Hint:** This gives you the effective cost price of the mixture (milk + water).

### Step 3: Set Up the Allegation Method
Now, we will use the allegation method to find the ratio of milk to water in the mixture.

- Cost price of milk = Rs. 10 per liter
- Cost price of water = Rs. 0 per liter (since water is free)
- Cost price of the mixture = Rs. 50/6 per liter

**Hint:** The allegation method helps to find the ratio of two or more components in a mixture based on their cost prices.

### Step 4: Apply the Allegation Formula
Using the allegation method, we can set up the following:

\[ \text{Milk} \quad \text{Water} \]
\[ 10 \quad \quad 0 \]
\[ \frac{50}{6} \]

Now, calculate the differences:
- Difference between milk and mixture: \( 10 - \frac{50}{6} = \frac{60}{6} - \frac{50}{6} = \frac{10}{6} \)
- Difference between water and mixture: \( \frac{50}{6} - 0 = \frac{50}{6} \)

**Hint:** The ratios can be simplified to find the proportion of milk and water.

### Step 5: Calculate the Ratio
The ratio of milk to water is:
\[ \text{Ratio} = \frac{\frac{10}{6}}{\frac{50}{6}} = \frac{10}{50} = \frac{1}{5} \]

This means for every 1 part of milk, there are 5 parts of water.

**Hint:** This ratio will help us find the total quantity of milk and water in the mixture.

### Step 6: Determine the Total Quantity
Let the quantity of milk be \( x \) liters. Then, the quantity of water is \( 5x \) liters.

The total quantity of the mixture is:
\[ x + 5x = 6x \text{ liters} \]

**Hint:** Since the milkman sold the mixture at Rs. 10 per liter, we can use the total selling price to find the total quantity.

### Step 7: Calculate the Total Selling Price
The selling price of the entire mixture is:
\[ \text{Selling Price} = 10 \times 6x = 60x \]

### Step 8: Relate Selling Price to Cost Price
From the earlier calculation, we know the cost price of the mixture is:
\[ \text{Cost Price} = \frac{50}{6} \times 6x = 50x \]

Since we know he made a profit of 20%, we can set up the equation:
\[ 60x = 50x + 20\% \text{ of } 50x \]
\[ 60x = 50x + 10x \]
\[ 60x = 60x \]

This confirms our calculations are consistent.

### Step 9: Find the Quantity of Milk
From the ratio established earlier (1 part milk to 5 parts water), we can conclude:
- If the total quantity of the mixture is \( 6x \) liters, and the ratio of milk to the total mixture is \( \frac{1}{6} \), then:
\[ \text{Quantity of milk} = \frac{1}{6} \times 6x = x \]

Since we established that \( x = 5 \) liters of milk (from the ratio), we can conclude that the milkman bought:

**Final Answer: 5 liters of milk.**