Two items A and B have equal cost prices. Item A is sold at a profit of 40%, and item B is sold at a price that is 20% less than the selling price of item A. If the total profit on both items together is Rs. 156, what is the cost price of item A?

Introduction / Context:
This question deals with two items that share the same cost price but are sold under different conditions. The first item is sold at a 40% profit and the second item at a reduced selling price, 20% less than the first item's selling price. The combined profit in rupees is given, and from that we must deduce the common cost price. This type of question is common in competitive exams as it involves combining profit percentage, absolute profit, and relationships between different selling prices in a multi-step calculation.

Given Data / Assumptions:
- Cost price of item A = cost price of item B = C rupees. - Item A is sold at a profit of 40% on its cost price. - Selling price of item B is 20% less than the selling price of item A. - Total profit on both items together is Rs. 156. - We must find the value of C, the cost price of item A.

Concept / Approach:
First, express the selling price of item A in terms of C using the 40% profit: SP_A = 1.40C. Then, because item B is sold at 20% less than this selling price, SP_B = SP_A * (1 - 0.20) = 0.80 * SP_A. Substituting SP_A, this gives SP_B in terms of C. Once both selling prices are expressed in terms of C, we can compute total selling price and total cost price, and use the given absolute profit of Rs. 156 to form an equation. Solving this equation for C gives the common cost price of both items.

Step-by-Step Solution:
Let the cost price of each item (A and B) be C rupees. Item A is sold at 40% profit, so SP_A = C * (1 + 0.40) = 1.40C. The selling price of item B is 20% less than SP_A. So, SP_B = SP_A * (1 - 0.20) = 1.40C * 0.80 = 1.12C. Total cost price of both items = C + C = 2C. Total selling price = SP_A + SP_B = 1.40C + 1.12C = 2.52C. Total profit = Total SP - Total CP = 2.52C - 2C = 0.52C. We are given that total profit = Rs. 156, so 0.52C = 156. Thus, C = 156 / 0.52. Compute C: 156 / 0.52 = 156 * (100 / 52) / 100 = 300. Therefore, the cost price of item A is Rs. 300.

Verification / Alternative check:
If C = 300, then SP_A = 1.40 * 300 = Rs. 420. SP_B = 0.80 * 420 = Rs. 336. Total SP = 420 + 336 = Rs. 756. Total CP = 300 + 300 = Rs. 600. Profit = 756 - 600 = Rs. 156, which matches the given total profit. Hence the value C = 300 is fully consistent with the problem data.

Why Other Options Are Wrong:
- If C were 350, total profit would be 0.52 * 350 = 182, not 156. - If C were 400, total profit would be 0.52 * 400 = 208. - If C were 450, total profit would be 0.52 * 450 = 234. - Rs. 250 similarly produces a total profit different from 156.

Common Pitfalls:
A common mistake is to apply the 20% reduction directly to the cost price of item B instead of applying it to the selling price of item A. Some students also incorrectly average the percentages or treat them as being applied on the combined cost price without separating the items. Another pitfall is to try to work with approximate values rather than expressing everything cleanly in terms of C, which makes solving the equation straightforward.

Final Answer:
The cost price of item A is Rs. 300.