Step 1: Understanding the Question:
The question requires us to find the actual cost price of a book based on two different scenarios involving profit and loss percentages.
By representing the cost prices as variables, we can form a system of linear equations in two variables and solve them simultaneously to find the required value.
Step 2: Key Formula or Approach:
Selling Price = Cost Price + Profit (or - Loss).
Total Gain = Total Profit - Total Loss.
Step 3: Detailed Explanation:
Let the actual cost price of the pen be denoted by \( P \).
Let the actual cost price of the book be denoted by \( B \).
In the first scenario, Karim sells the pen at a 5% loss, which is mathematically represented as \( -0.05P \).
He sells the book at a 15% gain, which is represented as \( +0.15B \).
The total net gain in this first scenario is given as Rs. 7.
Therefore, we can set up our first equation: \( -0.05P + 0.15B = 7 \).
To eliminate the decimals and make calculations easier, multiply the entire equation by 100: \( -5P + 15B = 700 \).
Dividing this equation by 5 yields a simplified Equation 1: \( -P + 3B = 140 \).
In the second scenario, Karim sells the pen at a 5% gain (\( +0.05P \)) and the book at a 10% gain (\( +0.10B \)).
The total net gain in this second scenario is Rs. 13.
This gives us our second equation: \( 0.05P + 0.10B = 13 \).
Again, multiply by 100 to remove decimals: \( 5P + 10B = 1300 \).
Dividing by 5 yields a simplified Equation 2: \( P + 2B = 260 \).
Now, we can solve Equation 1 and Equation 2 simultaneously by adding them together.
Adding the two equations eliminates \( P \): \( (-P + 3B) + (P + 2B) = 140 + 260 \).
This simplifies to \( 5B = 400 \).
Solving for \( B \), we divide 400 by 5, which gives \( B = 80 \).
Hence, the actual cost price of the book is Rs. 80.
Step 4: Final Answer:
The actual price of the book is Rs. 80.