A retailer receives a 40% discount on the printed price (marked price) of an article and then sells the article at the full printed price. What is the retailer's gain percentage on this transaction?

Introduction / Context:
This question deals with the concept of trade discount and profit. The retailer buys goods from a wholesaler at a discount on the marked price and then sells them to customers at the full marked price, thereby earning a profit. The task is to determine this profit percentage based on the discount percentage.

Given Data / Assumptions:
- Printed (marked) price of the article is P. - Retailer gets 40% discount on P. - Retailer sells at the full printed price P. - No other expenses like tax or transport are considered.

Concept / Approach:
We treat the marked price as a base to find cost price and selling price. If the discount is 40%, the retailer pays only 60% of the marked price. Then he sells at 100% of the marked price. Profit percentage is calculated on cost price. The formulas used are:
cost price = marked price * (100 - discount percent) / 100 profit = selling price - cost price profit percent = (profit / cost price) * 100

Step-by-Step Solution:
Step 1: Let marked price P = 100 units for easy calculation. Step 2: Discount is 40%, so cost price for retailer C = 100 - 40 = 60. Step 3: Selling price S = full marked price = 100. Step 4: Profit = S - C = 100 - 60 = 40. Step 5: Profit percent = (40 / 60) * 100 = 66.666..., which is 66 2/3%.

Verification / Alternative check:
If we work with an actual marked price, say Rs 300, cost to retailer would be 60% of 300 = Rs 180. Selling price remains Rs 300. Profit = 120. Profit percent = 120 / 180 * 100 = 66.666..., again confirming 66 2/3%. Since percentage calculations scale linearly, any chosen marked price gives the same profit percent.

Why Other Options Are Wrong:
Option 40 would be correct only if cost price were 100 and selling price were 140, which is not the situation here. Option 55 and 75 do not satisfy the relation of a 40% discount on marked price followed by sale at marked price when checked algebraically. Only 66 2/3% satisfies the conditions given in the problem.

Common Pitfalls:
A frequent error is to subtract the discount from selling price and think that the remaining percentage is profit. Another pitfall is to compute profit percentage on the marked price instead of on cost price. Always calculate profit percentage as profit divided by cost price, multiplied by 100.

Final Answer:
The retailer's gain percentage is 66 2/3%.