A shopkeeper sells an article at an 8 percent loss. If he increases the selling price of the article by Rs. 164, he earns a profit of 2.25 percent on the cost price. If he instead sells the article for Rs. 1760, what percentage profit does he make?

Introduction / Context:
This question belongs to profit and loss with changes in selling price. The shopkeeper initially sells at a loss, but when he increases the selling price by a fixed amount, he gains a small profit. From these two conditions we can determine the cost price, and then compute the profit percent for another given selling price. This structure is common in competitive exams.

Given Data / Assumptions:

• Initial selling price leads to an 8 percent loss.
• When the selling price is increased by Rs. 164, the profit becomes 2.25 percent.
• We need the profit percent if the article is sold for Rs. 1760.
• Cost price is constant in all the scenarios.

Concept / Approach:
The profit or loss percent is always based on cost price, whereas the difference between two selling prices is just a fixed rupee amount. The change from loss to profit gives the total change in profit, which equals the difference of the two selling prices. That relationship allows us to solve for the cost price. Once cost price is known, profit percent for any new selling price is straightforward to calculate.

Step-by-Step Solution:
Let the cost price be C rupees. Initial selling price SP1 corresponds to 8 percent loss, so SP1 = C * (1 - 8 / 100) = 0.92 * C. New selling price SP2 is SP1 + 164 and gives 2.25 percent profit. Thus SP2 = C * (1 + 2.25 / 100) = 1.0225 * C. Therefore 1.0225 * C - 0.92 * C = 164. So 0.1025 * C = 164. C = 164 / 0.1025 = 1600 rupees. Now consider the selling price of Rs. 1760. Profit at SP = 1760 = 1760 - 1600 = 160 rupees. Profit percent = 160 / 1600 * 100 = 10 percent.

Verification / Alternative check:
Check SP1: SP1 = 0.92 * 1600 = Rs. 1472. Loss at SP1 = 1600 - 1472 = Rs. 128, which is 8 percent of 1600. SP2 should be SP1 + 164 = 1472 + 164 = Rs. 1636. Profit at SP2 = 1636 - 1600 = Rs. 36, which is 36 / 1600 * 100 = 2.25 percent. All conditions are satisfied, therefore cost price and profit percentage for Rs. 1760 are correct.

Why Other Options Are Wrong:
2.5%, 5%, and 7.5%: These profit percentages do not match the profit of Rs. 160 on cost price Rs. 1600. Only 10 percent matches the correct calculation using the derived cost price.

Common Pitfalls:
A frequent mistake is to assume that the difference between loss percent and profit percent directly translates into the rupee difference in selling prices without scaling by cost price. Another error is misinterpreting 2.25 percent as a fraction or using approximate values carelessly. Always write selling prices in terms of cost price and use their difference to solve for the unknown cost.

Final Answer:
When the article is sold for Rs. 1760, the shopkeeper makes a profit of 10%.