A shopkeeper gives a 20% discount on the marked price of a pair of books. In addition, for every 9 pairs of books that a customer pays for, the shopkeeper gives 1 more pair of books free. If on this combined offer he still earns an overall profit of 26% on his cost price, by what percentage is the marked price greater than the cost price?

Introduction / Context:
This question combines two common commercial practices: percentage discount on marked price and giving free units as part of a promotional offer. Despite these apparent reductions, the shopkeeper still manages to earn a profit. The problem asks us to determine how much the marked price is increased above the cost price to sustain a 26% overall profit under these conditions. Such questions are designed to test multi-step reasoning with discount, free goods, and profit percentage all in one scenario, which is typical of competitive exams on quantitative aptitude.

Given Data / Assumptions:
- Discount on marked price = 20%. - For every 9 pairs of books purchased, 1 extra pair is given free, so the customer receives 10 pairs but pays for only 9. - Overall profit on the entire transaction = 26% on cost price. - All pairs of books are identical and have the same cost price and marked price. - We must find the percentage by which marked price exceeds cost price.

Concept / Approach:
Let the cost price per pair of books be C and the marked price per pair be M. Because of the 20% discount, the effective selling price per paid pair is 0.8M. When a customer buys 9 pairs, he pays for 9 pairs but receives 10 pairs (9 paid + 1 free). The shopkeeper's total revenue for those 10 pairs is 9 * 0.8M = 7.2M, while his total cost is 10C. The overall profit percentage is (Revenue - Cost) / Cost * 100, which is given as 26%. Using this relationship we can form an equation involving M and C and solve for M in terms of C to obtain the markup percentage on cost price.

Step-by-Step Solution:
Let cost price of one pair of books = C rupees. Let marked price of one pair of books = M rupees. Discount on marked price = 20%, so selling price per paid pair = 0.80M. In the offer, customer pays for 9 pairs and gets 10 pairs (1 pair free). Total revenue for 10 pairs = 9 * 0.80M = 7.2M. Total cost for 10 pairs = 10C. Profit = Revenue - Cost = 7.2M - 10C. Given overall profit% = 26%, so (7.2M - 10C) / (10C) * 100 = 26. So, (7.2M - 10C) / 10C = 0.26. Therefore, 7.2M - 10C = 2.6C, because 0.26 * 10C = 2.6C. So, 7.2M = 12.6C. Thus, M = 12.6C / 7.2 = 1.75C. Marked price is therefore 1.75 times the cost price, i.e., a 75% markup.

Verification / Alternative check:
If C = Rs. 100, then M = 1.75 * 100 = Rs. 175. Selling price per paid pair after 20% discount = 0.80 * 175 = Rs. 140. For 9 paid pairs, revenue = 9 * 140 = Rs. 1,260. Cost of 10 pairs = 10 * 100 = Rs. 1,000. Profit = 1,260 - 1,000 = Rs. 260. Profit% = 260 / 1,000 * 100 = 26%, which matches the given data.

Why Other Options Are Wrong:
- 35%, 65%, and 50% are lower markups and would not generate enough profit to reach 26% after discount and free goods. - 26% is the overall profit percentage, not the markup on the marked price; equating them would be a conceptual mistake. - Only a 75% markup on cost price is consistent with a 20% discount plus the 9 + 1 free scheme and 26% overall profit.

Common Pitfalls:
A common misunderstanding is to treat the 26% profit directly as the difference between cost price and marked price without accounting for the discount and free pair. Some students ignore the free pair entirely, effectively assuming revenue is for 9 pairs but cost is also for 9 pairs, which is incorrect. Another pitfall is computing profit percentage on marked price rather than on cost price, which changes the base and leads to wrong answers.

Final Answer:
The marked price is 75% higher than the cost price.