An item is sold at a loss of 10%. Had it been sold for Rs 9 more, there would have been a profit of 25/2% (that is, 12.5%). What is the cost price of the item?

Introduction / Context:
This is a standard profit and loss question where the item is sold at a loss in one scenario, and a slightly higher selling price would convert the loss into a profit. By comparing the two situations, we can set up an equation in terms of the cost price. Such questions test your ability to apply profit and loss formulas in two different cases for the same cost price.

Given Data / Assumptions:

• In the first case, the item is sold at a loss of 10%.
• If sold for Rs 9 more than the first selling price, there would be a profit of 25/2% = 12.5%.
• Cost price (CP) is the same in both cases.
• We must find the cost price of the item.

Concept / Approach:
If CP is C, then a 10% loss implies first selling price SP₁ = 0.9C. The second selling price is SP₂ = SP₁ + 9. We are also told that SP₂ corresponds to a 12.5% profit, so SP₂ = 1.125C. Equating these two expressions for SP₂ gives us a simple linear equation in C. Solving this equation yields the cost price.

Step-by-Step Solution:
Step 1: Let cost price = C.Step 2: At 10% loss, first selling price SP₁ = 0.9C.Step 3: Second selling price SP₂ = SP₁ + 9 = 0.9C + 9.Step 4: Second case has 12.5% profit, so SP₂ = 1.125C.Step 5: Equate them: 0.9C + 9 = 1.125C.Step 6: 9 = 1.125C - 0.9C = 0.225C ⇒ C = 9 / 0.225 = 40.

Verification / Alternative check:
Check with CP = Rs 40. At 10% loss, SP₁ = 0.9 * 40 = Rs 36. If sold for Rs 9 more, second selling price is SP₂ = 36 + 9 = Rs 45. Profit at SP₂ is 45 - 40 = Rs 5. Profit% = (5 / 40) * 100 = 12.5%, which equals 25/2%. This confirms that CP = Rs 40 fits both conditions exactly.

Why Other Options Are Wrong:
CP values like Rs 90, Rs 75 and Rs 55 do not yield a difference of exactly Rs 9 between the loss-making selling price and the profit-making selling price while also matching the specified percentages. For example, with CP = 90, 10% loss gives SP₁ = 81, and a 12.5% profit gives SP₂ = 101.25, whose difference is 20.25, not 9. Hence they are incorrect.

Common Pitfalls:
Some learners misread 25/2% as 25% or mistakenly consider the Rs 9 as a change in cost price instead of selling price. Another error is to attempt to find profit or loss percentages using arbitrary assumptions rather than forming a precise equation. Always translate both scenarios into algebraic expressions involving CP and then solve systematically.

Final Answer:
The cost price of the item is Rs. 40.