A man buys a table and a chair together for Rs. 500; he sells the table at a loss of 10% and the chair at a gain of 10% and still makes an overall gain of Rs. 10; what is the cost price of the chair in rupees?

Introduction / Context:
This question is a classic example of a mixed profit and loss problem involving two items whose combined cost is known. One item is sold at a loss and the other at a gain, and an overall profit is given. From this information, we must determine the individual cost price of one item. Problems like this are common in the profit and loss section of aptitude exams and help train algebraic thinking with percentages.

Given Data / Assumptions:

• Total cost price of a table and a chair together is Rs. 500.
• The table is sold at a 10% loss.
• The chair is sold at a 10% profit.
• The overall gain on the entire transaction is Rs. 10, so the total selling price is Rs. 510.
• We need to find the cost price of the chair.

Concept / Approach:
Let the cost price of the table be T and of the chair be C. Then T + C = 500. The selling price of the table is 0.9 * T because of the 10% loss, and the selling price of the chair is 1.1 * C because of the 10% gain. The sum of these two selling prices is 510. This gives a system of two equations in T and C which we solve using basic algebra, then identify the cost price of the chair. This method is systematic and avoids guesswork.

Step-by-Step Solution:
Let cost price of table be T and cost price of chair be C.We know T + C = 500.Table is sold at 10% loss, so selling price of table = 0.9 * T.Chair is sold at 10% profit, so selling price of chair = 1.1 * C.Overall gain is Rs. 10, so total selling price = 500 + 10 = 510.Therefore 0.9 * T + 1.1 * C = 510.From T + C = 500, express C as 500 - T and substitute.So 0.9 * T + 1.1 * (500 - T) = 510.This simplifies to 0.9T + 550 - 1.1T = 510, giving -0.2T + 550 = 510.Thus -0.2T = -40, so T = 200 and hence C = 500 - 200 = 300.

Verification / Alternative check:
Using T = 200 and C = 300, compute actual selling prices. Table at 10% loss sells for 200 * 0.9 = 180. Chair at 10% gain sells for 300 * 1.1 = 330. Total selling price is 180 + 330 = 510. Since total cost price is 500, the profit is 510 - 500 = 10, which matches the given data. This confirms that the cost price of the chair is indeed Rs. 300.

Why Other Options Are Wrong:
If the chair cost were Rs. 200, then table cost would be Rs. 300, and the resulting selling prices would not give an overall profit of exactly Rs. 10.

Similarly, chair cost of Rs. 250 or Rs. 400 would break the equation 0.9T + 1.1C = 510 when combined with T + C = 500, leading to either a different net profit or a net loss.

Common Pitfalls:
Students sometimes misread the phrase "10% loss" and "10% gain" and mistakenly subtract or add 10 rupees directly instead of 10% of the respective cost price. Others assume that equal and opposite percentages automatically cancel, forgetting that the cost prices of the two items are not equal. The safe approach is always to define variables T and C, set up equations for total cost and total selling price, and then solve algebraically.

Final Answer:
The cost price of the chair is Rs. 300.