Mr. X sells two properties P1 and P2 for Rs. 1,00,000 each. He sells property P1 at a loss of 20 percent on its cost price. What percentage profit must he earn on property P2 so that overall there is neither gain nor loss on the sale of both properties together?

Introduction / Context:
This question illustrates how a loss on one item affects the required profit on another item when the seller wants to break even on the entire transaction. It tests understanding of cost price, selling price, and net gain or loss when two sales at the same selling price are involved.

Given Data / Assumptions:

• Selling price of property P1 = Rs. 1,00,000.
• Selling price of property P2 = Rs. 1,00,000.
• Property P1 is sold at a loss of 20 percent.
• There is no overall gain or loss on both properties combined, so total selling price equals total cost price.

Concept / Approach:
If the selling prices are equal for both properties, then we can compute the cost price of P1 using the given loss percentage. Then we know the total cost price of both properties must equal the total selling price (because net result is no profit, no loss). This allows us to find the cost price of P2. Finally we compute the percentage profit on P2 required to offset the loss on P1.

Step-by-Step Solution:
Let C1 be the cost price of property P1. Loss on P1 = 20 percent, so selling price of P1 = C1 * (1 - 20 / 100) = 0.80 * C1. Given selling price of P1 = Rs. 1,00,000, so 1,00,000 = 0.80 * C1. Thus C1 = 1,00,000 / 0.80 = Rs. 1,25,000. Let C2 be the cost price of property P2. Total selling price of both properties = 1,00,000 + 1,00,000 = Rs. 2,00,000. Since there is no overall gain or loss, total cost price must also be Rs. 2,00,000. Therefore C1 + C2 = 2,00,000. So C2 = 2,00,000 - 1,25,000 = Rs. 75,000. Profit on property P2 = 1,00,000 - 75,000 = Rs. 25,000. Profit percentage on P2 = 25,000 / 75,000 * 100 = 33.33 percent.

Verification / Alternative check:
Total cost price = 1,25,000 + 75,000 = 2,00,000. Total selling price = 1,00,000 + 1,00,000 = 2,00,000. Net gain or loss = 2,00,000 - 2,00,000 = 0. This confirms that a profit of 33.33 percent on P2 exactly cancels the 20 percent loss on P1, producing no overall profit or loss.

Why Other Options Are Wrong:
29.97% and 25% and 22.22%: These profit rates do not generate enough profit on P2 to counterbalance the 20 percent loss on P1, so the overall transaction would still result in a net loss. Only 33.33 percent gives perfect balance.

Common Pitfalls:
One common mistake is to try to average the percentage loss and percentage profit directly. Another error is to assume equal cost prices instead of equal selling prices. Remember that equal selling prices with unequal gain or loss percentages always require careful calculation of cost prices for each item.

Final Answer:
He must make a profit of 33.33% on property P2 to have no overall gain or loss.