The cost price of 12 pens is equal to the selling price of 8 pens of the same type. What is the percentage gain made on each pen?

Introduction / Context:
This is a standard profit percentage question where a relationship between the cost price and selling price for different quantities is given. The problem checks whether you can convert that quantity relationship into a ratio of selling price to cost price for a single unit and then compute the gain percentage for each item.

Given Data / Assumptions:

• Cost price of 12 pens equals selling price of 8 pens.
• All pens are identical in type and quality.
• The trader buys and sells at uniform prices per pen.
• We must determine the percentage gain on each pen.

Concept / Approach:
Let cost price per pen be C and selling price per pen be S. Then total cost for 12 pens is 12C and total selling price for 8 pens is 8S. The statement that these are equal gives an equation 12C = 8S. From this equation we find the ratio S / C, which directly tells us by what factor the selling price exceeds the cost price. Profit percentage is then (S - C) / C * 100.

Step-by-Step Solution:
Step 1: Let cost price per pen be C and selling price per pen be S.Step 2: Given that 12C = 8S.Step 3: Simplify the equation: divide both sides by 4 to get 3C = 2S.Step 4: Rearranging, S = 3C / 2 = 1.5C.Step 5: Profit per pen = S - C = 1.5C - C = 0.5C.Step 6: Profit percentage = (Profit / Cost price) * 100 = (0.5C / C) * 100 = 50%.

Verification / Alternative check:
Assume C = Rs. 2 per pen. Then S = 1.5 * 2 = Rs. 3 per pen.Total cost of 12 pens = 12 * 2 = Rs. 24. Total selling price of 8 pens = 8 * 3 = Rs. 24, which matches the given condition.Profit per pen = 3 - 2 = Rs. 1, so profit percentage = 1 / 2 * 100 = 50%.

Why Other Options Are Wrong:
Gains of 12% or 30% imply that S is only slightly greater than C, which contradicts the ratio 12C = 8S. A gain of 60% would require S = 1.6C, not 1.5C, and 25% gain would mean S = 1.25C, again inconsistent with the condition. Only a 50% gain matches the derived ratio S = 1.5C.

Common Pitfalls:
Some learners misinterpret the equality of total cost and total selling price as applying to a single pen instead of to different quantities. Others incorrectly take ratios like 12:8 as cost to selling price directly without introducing unit cost symbols. Setting up the equation carefully with C and S is the most reliable method to avoid these mistakes.

Final Answer:
The gain made on each pen is 50% of the cost price.