The profit obtained on selling an article for Rs. 540 is exactly equal to the loss incurred when the same article is sold for Rs. 370. If the article is instead sold for Rs. 910, what will be the profit percentage?

Introduction / Context:
This question uses a symmetric situation where the profit at a higher selling price equals the loss at a lower selling price. The same cost price is involved in both cases. From this symmetry we can determine the cost price and then compute the profit percentage when the article is sold at a new, higher selling price. It is a standard and elegant type of profit and loss problem.

Given Data / Assumptions:

• Selling price one is Rs. 540 and results in a profit.
• Selling price two is Rs. 370 and results in a loss.
• Magnitude of profit at 540 rupees equals magnitude of loss at 370 rupees.
• The cost price is the same in all cases.
• Selling price three is Rs. 910 and we must find the profit percentage at this price.

Concept / Approach:
Let the cost price be C. At selling price 540, profit equals 540 minus C. At selling price 370, loss equals C minus 370. The problem states that these magnitudes are equal, so we set 540 minus C equal to C minus 370 and solve for C. Once we know C, we compute profit at 910 rupees as 910 minus C and then express this as a percentage of C by using the standard profit percentage formula.

Step-by-Step Solution:
Let cost price be C rupees.At selling price 540, profit P1 equals 540 minus C.At selling price 370, loss L1 equals C minus 370.Given that profit equals loss in magnitude, we have 540 minus C equals C minus 370.Rearranging gives 540 plus 370 equals 2C.So 910 equals 2C and therefore C equals 455 rupees.Now consider selling price 910 rupees.Profit at 910 equals 910 minus 455 which is 455 rupees.Profit percentage equals 455 divided by 455 multiplied by 100 which equals 100 percent.

Verification / Alternative check:
We can verify the equal profit and loss condition using cost price 455 rupees. At 540 rupees, profit equals 540 minus 455 equals 85 rupees. At 370 rupees, loss equals 455 minus 370 which is also 85 rupees. Thus, the given statement of equal magnitude is satisfied. At 910 rupees, selling price is exactly double the cost price, which naturally leads to a profit equal to the cost price and therefore a 100 percent profit. This consistent reasoning confirms the result.

Why Other Options Are Wrong:
Profit percentages 25, 33, and 50 percent would correspond to much smaller profits relative to the cost price. For instance, a 50 percent profit would require a profit of only half of 455, that is about 227.5 rupees, which is far less than the actual profit of 455 rupees at a selling price of 910. Since the selling price is exactly twice the cost price, the only correct profit percentage is 100 percent.

Common Pitfalls:
Some learners incorrectly average the two selling prices and treat their difference from the cost price as profit. Others misinterpret the condition by equating profit percentages instead of profit amounts. The correct approach is to equate the rupee profit and rupee loss magnitudes, form a simple linear equation, and solve for the cost price. Once that is clear, the calculation of the new profit percentage is straightforward.

Final Answer:
When the article is sold for Rs. 910, the profit percentage is exactly 100 percent.