An article is sold for Rs 753, and the profit earned is equal to the loss incurred when the same article is sold for Rs 455. At what price should the article be sold to obtain a profit of 50%?

Introduction / Context:
This question uses a symmetry condition between profit and loss at two different selling prices. The profit when selling at Rs 753 equals the loss when selling at Rs 455. From this, we can deduce the cost price. Once the cost price is known, we can easily compute the selling price required for a desired profit of 50%. This style of problem is common in exams and tests your algebraic setup skills.

Given Data / Assumptions:

• First selling price SP₁ = Rs 753 (profit case).
• Second selling price SP₂ = Rs 455 (loss case).
• Profit at SP₁ is equal to loss at SP₂.
• We need selling price for a 50% profit.
• Cost price (CP) is the same in all cases.

Concept / Approach:
If CP is C, profit at SP₁ is (753 - C), and loss at SP₂ is (C - 455). These two are given equal, so 753 - C = C - 455. This equation allows us to solve for C. Then to get 50% profit, new selling price SP₃ must be 1.5 times CP, because 50% profit means adding half of cost price to cost. This two-step approach is reliable and direct.

Step-by-Step Solution:
Step 1: Let cost price = C.Step 2: Profit at SP₁ = 753 - C.Step 3: Loss at SP₂ = C - 455.Step 4: Given that profit equals loss, so 753 - C = C - 455.Step 5: Rearranging, 753 + 455 = 2C ⇒ 1,208 = 2C ⇒ C = 604.Step 6: For a 50% profit, SP₃ = C * 1.5 = 604 * 1.5 = 906.

Verification / Alternative check:
Check the equality condition using C = 604. Profit at Rs 753 is 753 - 604 = 149. Loss at Rs 455 is 604 - 455 = 149. They are equal, satisfying the main condition. For a 50% profit, profit amount should be 0.5 * 604 = 302, giving SP₃ = 604 + 302 = 906, which matches our computed selling price and confirms correctness.

Why Other Options Are Wrong:
Rs 855, Rs 955 and Rs 896 correspond to profit percentages that are not exactly 50% when applied to a cost price of Rs 604. For example, 855 gives profit of 855 - 604 = 251, which is about 41.56%, not 50%. Only Rs 906 yields precisely 50% profit relative to the cost price found from the equal profit and loss condition.

Common Pitfalls:
Sometimes students add or subtract the two selling prices directly without forming equations or mistakenly think that the cost price is the average of the two selling prices. While in this particular case C does turn out to be the average, that result follows from correctly setting 753 - C = C - 455, not from an unsupported shortcut. Always derive such relationships from first principles to avoid errors in less symmetric cases.

Final Answer:
To earn a 50% profit, the article should be sold for Rs. 906.