A woman sells 20 articles for Rs. 160 and incurs a loss of 40%; how many such articles must she sell for Rs. 240 in order to earn a profit of 20%?

Introduction / Context:
This question connects loss in one selling situation with profit in another, both involving the same type of article. First, the woman sells a certain number of articles at a loss, which lets us compute the cost price per article. Then, we find how many articles she must sell at a different total amount to achieve a desired profit percentage. This is a typical profit and loss problem that blends unit cost and total transaction analysis.

Given Data / Assumptions:

• She sells 20 articles for Rs. 160.
• On this sale she incurs a loss of 40%.
• She wants to earn a profit of 20% on future sales.
• We need to find how many articles must be sold for Rs. 240 to earn this 20% profit.
• Cost price per article remains constant.

Concept / Approach:
First, we find the selling price per article in the first case, then use the given loss percentage to find the cost price per article. After we know the cost price per article, we compute the selling price per article that corresponds to a 20% profit. Finally, we divide the total desired revenue of Rs. 240 by this per unit selling price to find the number of articles required. This sequence ensures that both loss and profit conditions are properly connected through the same cost price.

Step-by-Step Solution:
Step 1: In the first scenario, 20 articles sell for Rs. 160.Step 2: Selling price per article in the first scenario = 160 / 20 = Rs. 8.Step 3: She incurs a loss of 40% at this selling price.Step 4: When loss is 40%, selling price = 60% of cost price.Step 5: Therefore, 8 = 0.60 * Cost price per article.Step 6: Cost price per article = 8 / 0.60 = 8 / 0.6 = Rs. 13.33 approximately (or exactly 40 / 3).Step 7: Now she wants to earn a profit of 20%.Step 8: For 20% profit, selling price per article = Cost price * 1.20.Step 9: Selling price per article for 20% profit = (40 / 3) * 1.20 = (40 / 3) * (6 / 5) = 16.Step 10: So, required selling price per article to get 20% profit is Rs. 16.Step 11: She wants total revenue of Rs. 240 in this scenario.Step 12: Number of articles needed = 240 / 16 = 15.

Verification / Alternative check:
We can verify by checking the total cost and profit for 15 articles. Cost per article = Rs. 40 / 3. Total cost for 15 articles = 15 * (40 / 3) = 200. Total selling price = 15 * 16 = 240. Profit = 240 - 200 = Rs. 40. Profit percentage = (40 / 200) * 100 = 20%. Thus the requirement of 20% profit is satisfied when selling 15 articles for Rs. 240.

Why Other Options Are Wrong:

• 16, 17, 14: These values give per article selling prices of 240 / 16 = 15, 240 / 17 (approximately 14.12) and 240 / 14 (approximately 17.14) respectively. None of these match the required Rs. 16 per article that corresponds to exactly 20% profit based on the true cost price.

Common Pitfalls:

• Using average of loss and profit percentages directly, which is not valid.
• Forgetting to convert loss percentage into a relationship between cost price and selling price correctly.
• Rounding cost price too early instead of leaving it as an exact fraction, leading to slight errors in final division.

Final Answer:
She must sell 15 articles for Rs. 240 to earn a profit of 20% on her cost price.