On selling a pen at 5% loss and a book at 15% gain, Karim gains Rs 7. If he sells the pen at 5% gain and the book at 10% gain, then he gains Rs 13. The actual price of the book is

Options

A. 100
B. 80
C. 10
D. 400

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information given.

### Step 1: Define Variables
Let:
- The cost price of the pen = \( x \) rupees
- The cost price of the book = \( y \) rupees

### Step 2: Selling Price Calculations
1. **Selling Price of the Pen at 5% Loss**:
\[ \text{Selling Price of Pen} = x - \left(5\% \text{ of } x\right) = x \times \frac{95}{100} = \frac{95x}{100} \]

2. **Selling Price of the Book at 15% Gain**:
\[ \text{Selling Price of Book} = y + \left(15\% \text{ of } y\right) = y \times \frac{115}{100} = \frac{115y}{100} \]

### Step 3: Set Up the First Equation
According to the problem, when Karim sells the pen at a 5% loss and the book at a 15% gain, he gains Rs 7. Therefore, we can write:
\[ \left(\frac{95x}{100} + \frac{115y}{100}\right) - (x + y) = 7 \]
Simplifying this gives:
\[ \frac{95x + 115y - 100x - 100y}{100} = 7 \]
\[ \frac{-5x + 15y}{100} = 7 \]
Multiplying through by 100:
\[ -5x + 15y = 700 \quad \text{(Equation 1)} \]

### Step 4: Selling Price Calculations for the Second Scenario
1. **Selling Price of the Pen at 5% Gain**:
\[ \text{Selling Price of Pen} = x + \left(5\% \text{ of } x\right) = x \times \frac{105}{100} = \frac{105x}{100} \]

2. **Selling Price of the Book at 10% Gain**:
\[ \text{Selling Price of Book} = y + \left(10\% \text{ of } y\right) = y \times \frac{110}{100} = \frac{110y}{100} \]

### Step 5: Set Up the Second Equation
When Karim sells the pen at a 5% gain and the book at a 10% gain, he gains Rs 13. Therefore:
\[ \left(\frac{105x}{100} + \frac{110y}{100}\right) - (x + y) = 13 \]
Simplifying this gives:
\[ \frac{105x + 110y - 100x - 100y}{100} = 13 \]
\[ \frac{5x + 10y}{100} = 13 \]
Multiplying through by 100:
\[ 5x + 10y = 1300 \quad \text{(Equation 2)} \]

### Step 6: Solve the System of Equations
Now we have two equations:
1. \( -5x + 15y = 700 \)
2. \( 5x + 10y = 1300 \)

Adding both equations:
\[ (-5x + 15y) + (5x + 10y) = 700 + 1300 \]
This simplifies to:
\[ 25y = 2000 \]
Dividing by 25:
\[ y = 80 \]

### Step 7: Conclusion
The actual price of the book is \( \text{Rs } 80 \).