A person buys pens at the rate of 25 pens for one rupee and later sells them at the rate of 15 pens for one rupee. What is his profit percentage on the transaction?

Introduction / Context:
This question deals with buying and selling items in different quantity-rate forms, a very common pattern in profit and loss problems. The person buys pens at a rate of 25 for Re. 1 and sells them at a rate of 15 for Re. 1. Since the cost and selling rates are expressed as “number of pens per rupee,” we must convert them into cost per pen and selling price per pen, then compute profit and profit percentage.

Given Data / Assumptions:

• Buying rate: 25 pens for Re. 1.
• Selling rate: 15 pens for Re. 1.
• We assume large enough quantity so that the rates hold exactly.
• We need the profit percentage on cost price.

Concept / Approach:
To find profit percentage, we need cost price and selling price per pen. From “25 pens for Re. 1” we get cost per pen. From “15 pens for Re. 1” we get selling price per pen. Profit per pen is the difference between selling price and cost price. Profit percentage is then (profit per pen / cost price per pen) * 100. The actual number of pens transacted does not matter because the ratio remains the same.

Step-by-Step Solution:
Cost for 25 pens = Re. 1. Therefore, cost price (CP) per pen = 1 / 25 rupee. Selling price for 15 pens = Re. 1. Therefore, selling price (SP) per pen = 1 / 15 rupee. Profit per pen = SP − CP = (1 / 15) − (1 / 25). Compute (1 / 15) − (1 / 25) = (25 − 15) / (15 * 25) = 10 / 375 = 2 / 75. So, profit per pen = 2 / 75 rupee. Cost price per pen = 1 / 25 = 3 / 75 rupee. Profit percentage = (profit per pen / cost price per pen) * 100. Profit percentage = ( (2 / 75) / (3 / 75) ) * 100. The 75 cancels out, giving (2 / 3) * 100. (2 / 3) * 100 = 66.67% (approximately), which is 66 2/3 percent.

Verification / Alternative check:
Take a convenient total number of pens, for example 75 pens. At the buying rate of 25 for Re. 1, cost for 75 pens = 3 rupees. At the selling rate of 15 for Re. 1, 75 pens will be sold in 5 groups of 15, so selling price = 5 rupees. Profit = 5 − 3 = 2 rupees. Profit percentage = (2 / 3) * 100 ≈ 66.67%, matching our earlier calculation.

Why Other Options Are Wrong:
16 2/3 and 33 1/3: These are much lower profit percentages and would arise if one incorrectly averaged rates or miscomputed per-pen amounts.
40 and 50: These are plausible but do not match the exact ratio 2 / 3 of cost. A quick numerical check with 75 pens shows that these percentages are not correct.

Common Pitfalls:
A typical mistake is to compare only the numbers 25 and 15 and conclude that the profit is (25 − 15) / 25 = 40%, which is wrong because the base should be cost per pen, not number of pens. Another error is to use rupees per pen instead of pens per rupee interchangeably without converting properly. Always convert both buying and selling rates to per-unit cost and then proceed to find profit percentage.

Final Answer:
The person makes a profit of approximately 66.67% on the pens.