To gain 10% on selling sample milk at the cost price of pure milk, the quantity of water to be mixed with 50 kg. of pure milk in

Options

A. 2.5 kg
B. 5Kg.
C. 7.5Kg.
D. 10Kg.

Answer

To solve the problem of how much water needs to be mixed with 50 kg of pure milk to achieve a 10% profit when selling the mixture at the cost price of pure milk, we can follow these steps:

### Step-by-Step Solution:

1. **Determine the Cost Price of Pure Milk:**
- Let's assume the rate of pure milk is ₹1 per kg.
- Therefore, the cost of 50 kg of pure milk = 50 kg × ₹1/kg = ₹50.

**Hint:** Always start by calculating the total cost of the items you are working with.

2. **Calculate the Desired Profit:**
- We want to gain a profit of 10% on the selling price.
- Profit = 10% of the cost price of 50 kg of pure milk.
- Profit = (10/100) × ₹50 = ₹5.

**Hint:** Remember that profit is calculated as a percentage of the cost price.

3. **Determine the Selling Price:**
- Since the mixture is to be sold at the cost price of pure milk, the selling price will also be ₹50.

**Hint:** The selling price in this case is equal to the cost price of the pure milk.

4. **Calculate the Total Amount to be Sold to Achieve the Profit:**
- To achieve a profit of ₹5, the total amount of the mixture sold must be the selling price plus the desired profit.
- Total amount to be sold = Selling Price + Profit = ₹50 + ₹5 = ₹55.

**Hint:** The total amount sold includes both the cost and the profit.

5. **Find the Total Quantity of the Mixture:**
- Let the quantity of water to be mixed with 50 kg of pure milk be \( x \) kg.
- The total quantity of the mixture = 50 kg (pure milk) + \( x \) kg (water).

**Hint:** When mixing substances, always consider the total quantity of the mixture.

6. **Set Up the Equation for Selling Price:**
- The selling price per kg of the mixture = Total Selling Price / Total Quantity of Mixture.
- Selling Price per kg = ₹55 / (50 kg + \( x \) kg).

**Hint:** The selling price per kg is derived from the total selling price divided by the total quantity.

7. **Equate the Selling Price to the Cost Price of Pure Milk:**
- Since we are selling the mixture at the cost price of pure milk (₹1/kg), we can set up the equation:
\[ \frac{55}{50 + x} = 1 \]

**Hint:** This equation reflects the condition that the selling price of the mixture equals the cost price of pure milk.

8. **Solve for \( x \):**
- Cross-multiplying gives us:
\[ 55 = 50 + x \]
- Rearranging the equation gives:
\[ x = 55 - 50 = 5 \]

**Hint:** Isolate the variable to find the quantity of water needed.

### Final Answer:
The quantity of water to be mixed with 50 kg of pure milk is **5 kg**.