A vendor buys bananas at the rate of 12 for Rs. 50 and sells them at the rate of 5 for Rs. 40. What is his profit percentage on the transaction?

Introduction / Context:
This question is another example of buying and selling goods at different quantity-based price rates, common in profit and loss problems. The vendor buys bananas at a certain rate (12 for Rs. 50) and sells them at a much higher rate (5 for Rs. 40). The task is to calculate the percentage profit on the cost price by converting these rates into per-banana cost price and per-banana selling price.

Given Data / Assumptions:

• Buying rate: 12 bananas for Rs. 50.
• Selling rate: 5 bananas for Rs. 40.
• We can assume a large number of bananas keeping the same rates.
• We must find the profit percentage on the cost price.

Concept / Approach:
The approach is to calculate the cost price per banana and the selling price per banana. Profit per banana is the difference between selling and cost prices. The profit percentage is then (profit per banana / cost price per banana) * 100. This method is independent of the actual number of bananas and relies only on the ratio of total cost to total quantity.

Step-by-Step Solution:
Cost for 12 bananas = Rs. 50. Cost price (CP) per banana = 50 / 12 rupees. Selling price for 5 bananas = Rs. 40. Selling price (SP) per banana = 40 / 5 = Rs. 8. Compute CP per banana: CP = 50 / 12 ≈ 4.1667 rupees. Profit per banana = SP − CP = 8 − (50 / 12). 8 = 96 / 12, so profit per banana = (96 / 12) − (50 / 12) = 46 / 12. Simplify 46 / 12 = 23 / 6 ≈ 3.8333 rupees. Profit percentage = (profit per banana / CP per banana) * 100. Profit percentage = ( (23 / 6) / (50 / 12) ) * 100. (23 / 6) / (50 / 12) = (23 / 6) * (12 / 50) = (23 * 12) / (6 * 50). (23 * 12) / (6 * 50) = (23 * 2) / 50 = 46 / 50 = 0.92. Therefore, profit percentage = 0.92 * 100 = 92%.

Verification / Alternative check:
Take a convenient total number, for example 60 bananas. At the buying rate, 12 bananas cost Rs. 50, so 60 bananas (5 groups of 12) cost 5 * 50 = Rs. 250. At the selling rate, 5 bananas sell for Rs. 40, so 60 bananas (12 groups of 5) sell for 12 * 40 = Rs. 480. Profit = 480 − 250 = Rs. 230. Profit percentage = (230 / 250) * 100 = 92%, which confirms our calculation.

Why Other Options Are Wrong:
82, 72, 62 and 68: These are all smaller than the actual profit percentage. They might result from partial simplification, using wrong denominators or incorrectly converting between the buy and sell rates. A quick check with 60 bananas, as shown above, clearly gives 92%, not any of these other values.

Common Pitfalls:
One frequent mistake is to compare 12 and 5 directly and assume a simple ratio-based profit without converting to monetary cost per unit. Another is to accidentally compute profit on the selling price instead of on the cost price. It is also easy to make algebraic errors when simplifying fractions; using a convenient total number of bananas is often the safest strategy.

Final Answer:
The vendor makes a profit of 92% on the bananas.