A space research company wants to sell its two products A and B. If the product A is sold at 20% loss and the product B at 30% gain, the company will not lose anything. If the product A is sold at 15% loss and the product B at 15% gain, the company will lose Rs. 6 million in the deal. What is the cost of product B ?

Options

A. Rs. 140 million
B. Rs. 120 million
C. Rs. 100 million
D. Rs. 80 million

Answer

To solve the problem step by step, we will first define the costs of products A and B, and then set up equations based on the information provided.

### Step 1: Define Variables
Let:
- Cost of product A = \( X \) (in millions)
- Cost of product B = \( Y \) (in millions)

### Step 2: Set Up the First Equation
According to the problem, if product A is sold at a 20% loss and product B at a 30% gain, the company will not incur any loss.

- Selling price of product A at 20% loss = \( 80\% \) of \( X = 0.8X \)
- Selling price of product B at 30% gain = \( 130\% \) of \( Y = 1.3Y \)

Since the total selling price equals the total cost, we can write the equation:
\[ 0.8X + 1.3Y = X + Y \]

### Step 3: Rearranging the First Equation
Rearranging the equation gives:
\[ 0.8X + 1.3Y - X - Y = 0 \]
\[ -0.2X + 0.3Y = 0 \]
This simplifies to:
\[ 0.3Y = 0.2X \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
According to the problem, if product A is sold at a 15% loss and product B at a 15% gain, the company will incur a loss of 6 million.

- Selling price of product A at 15% loss = \( 85\% \) of \( X = 0.85X \)
- Selling price of product B at 15% gain = \( 115\% \) of \( Y = 1.15Y \)

The equation for this scenario, considering the loss of 6 million, is:
\[ 0.85X + 1.15Y = X + Y - 6 \]

### Step 5: Rearranging the Second Equation
Rearranging gives:
\[ 0.85X + 1.15Y - X - Y + 6 = 0 \]
\[ -0.15X + 0.15Y + 6 = 0 \]
This simplifies to:
\[ 0.15Y = 0.15X - 6 \quad \text{(Equation 2)} \]

### Step 6: Solve the Equations
From Equation 1, we have:
\[ Y = \frac{0.2}{0.3}X = \frac{2}{3}X \]

Substituting \( Y \) into Equation 2:
\[ 0.15 \left(\frac{2}{3}X\right) = 0.15X - 6 \]
\[ 0.1X = 0.15X - 6 \]
Rearranging gives:
\[ 0.15X - 0.1X = 6 \]
\[ 0.05X = 6 \]
\[ X = \frac{6}{0.05} = 120 \text{ million} \]

### Step 7: Find Cost of Product B
Substituting \( X \) back into the equation for \( Y \):
\[ Y = \frac{2}{3} \times 120 = 80 \text{ million} \]

### Final Answer
The cost of product B is \( 80 \) million.

---