A person sells his pen for Rs. 24 and the profit percentage is numerically equal to the cost price of the pen (in rupees). What is the cost price of the pen?

Introduction / Context:
This is a slightly tricky algebraic profit problem where the profit percentage is numerically equal to the cost price of the pen in rupees. The selling price is given, and we must use this relationship to set up and solve a quadratic equation. Such questions are commonly tested to see whether you can translate a verbal condition into a correct mathematical equation.

Given Data / Assumptions:
- Selling price (SP) of the pen = Rs. 24. - Cost price (CP) of the pen is unknown; call it x. - Profit percentage is numerically equal to x (the cost price in rupees). - Profit percentage is defined as Profit / CP * 100.

Concept / Approach:
Let CP = x. Then profit = SP - CP = 24 - x. Profit percentage = (Profit / CP) * 100 = (24 - x) / x * 100. According to the question, this percentage equals x. Therefore, we set up the equation (24 - x) / x * 100 = x and solve for x. This will result in a quadratic equation, from which we select the positive solution that makes sense in the context of cost price.

Step-by-Step Solution:
Step 1: Let CP = x. Step 2: Selling price SP = 24. Step 3: Profit = SP - CP = 24 - x. Step 4: Profit percentage = (Profit / CP) * 100 = (24 - x) / x * 100. Step 5: Given that profit percentage equals x. Step 6: So, (24 - x) / x * 100 = x. Step 7: Multiply both sides by x to get 100(24 - x) = x^2. Step 8: Expand: 2400 - 100x = x^2. Step 9: Rearrange to standard quadratic form: x^2 + 100x - 2400 = 0. Step 10: Factor or use the quadratic formula. The solutions are x = 20 or x = -120. Step 11: A cost price cannot be negative, so CP = Rs. 20.

Verification / Alternative check:
If CP = Rs. 20 and SP = Rs. 24, then profit = 4. Profit percentage = (4 / 20) * 100 = 20%. The profit percentage, 20, is indeed numerically equal to the cost price in rupees, which is also 20. The other solution x = -120 is discarded as it is not realistic for a cost price. Therefore, Rs. 20 is the only valid solution.

Why Other Options Are Wrong:
If CP were Rs. 12, profit would be 12 and profit percentage would be 100, not 12. If CP were Rs. 14, profit percentage would be approximately 71.43, not 14. For CP = Rs. 16, profit percentage would be 50, not 16. None of these satisfy the special condition that the profit percentage equals the cost price. Only CP = Rs. 20 fulfills the given relationship.

Common Pitfalls:
Students may forget to multiply by 100 when forming the profit percentage expression or may incorrectly interpret the condition and set 24 - x equal to x directly. Another mistake is ignoring the negative root without understanding why it is invalid. Remember that cost price must be positive and that percentage is always calculated as Profit divided by Cost Price times 100 unless stated otherwise.

Final Answer:
The cost price of the pen is Rs. 20.