Anil purchases two books for a total of Rs. 100. He sells the first book at a loss of 5% and the second book at a profit of 20%. If his overall profit on the two books together is 5%, what is the cost price of the first book?

Introduction / Context:
This question involves two items being sold at different profit and loss percentages, while an overall profit percentage on the combined transaction is specified. The task is to find the individual cost price of one item using total cost and total profit data. Such problems help candidates practise forming equations that balance total cost with total selling price and overall percentage profit.

Given Data / Assumptions:

• Total cost price of two books = Rs. 100.
• First book is sold at a 5% loss.
• Second book is sold at a 20% profit.
• Overall profit on both books together = 5%.
• We need to find the cost price of the first book.

Concept / Approach:
Let the cost prices of the first and second books be C1 and C2 respectively. Then C1 + C2 = 100. We express the selling prices in terms of C1 and C2 using the given percentages: first at 95% of its cost and second at 120% of its cost. Overall profit of 5% means that the combined selling price equals 105% of the total cost price. This gives another equation. Solving the two equations together yields C1 and C2, from which we pick C1 as the required answer.

Step-by-Step Solution:
Step 1: Let cost price of first book = C1 and cost price of second book = C2. Step 2: Given C1 + C2 = 100. (Equation 1) Step 3: First book is sold at 5% loss, so SP1 = 0.95 * C1. Step 4: Second book is sold at 20% profit, so SP2 = 1.20 * C2. Step 5: Overall profit = 5%, so total SP = 1.05 * (C1 + C2) = 1.05 * 100 = 105. Step 6: Therefore, 0.95C1 + 1.20C2 = 105. (Equation 2) Step 7: From Equation 1, C2 = 100 - C1. Step 8: Substitute in Equation 2: 0.95C1 + 1.20(100 - C1) = 105. Step 9: Expand: 0.95C1 + 120 - 1.20C1 = 105. Step 10: Combine like terms: (0.95 - 1.20)C1 + 120 = 105 → (-0.25)C1 = -15. Step 11: Hence C1 = (-15) / (-0.25) = 60.

Verification / Alternative check:
If C1 = 60, then C2 = 40. SP1 at 5% loss = 0.95 * 60 = 57. SP2 at 20% profit = 1.20 * 40 = 48. Total SP = 57 + 48 = 105. Total CP = 100. Overall profit = 105 - 100 = 5, which is 5% of 100. This matches the required overall profit, confirming that the cost of the first book is Rs. 60.

Why Other Options Are Wrong:
Choosing 58, 55, 45 or 52 as the cost of the first book leads to a different pair of cost prices and ultimately a different overall profit percentage. None of these values produce exactly 5% overall profit when the given 5% loss and 20% gain are applied, so they are inconsistent with the conditions of the problem.

Common Pitfalls:
Learners sometimes assume both books have equal cost or try to average the percentages 5% and 20% directly. Another error is to take 5% of Rs. 100 and try to distribute it incorrectly between the two books without setting up proper equations. Writing clear equations for total cost and total selling price is the most reliable approach.

Final Answer:
The cost price of the first book is 60 rupees.