Goods are purchased for Rs. 450. One third of the goods is sold at a loss of 10%. At what gain percentage should the remaining goods be sold so that the overall profit on the entire transaction is 20%?

Introduction / Context:
Profit and loss questions often involve situations where part of the stock is sold at one profit or loss percent and the rest at another rate. To achieve a target overall gain, we must correctly combine the partial selling prices. This question tests understanding of total cost price, partial selling prices and the concept of overall percentage profit on the whole transaction.

Given Data / Assumptions:

• Total cost price (CP) of goods = Rs. 450.
• One third of the goods is sold at a loss of 10%.
• Two thirds of the goods remain and are sold at some gain percentage.
• Required overall profit on the entire transaction = 20% of total cost price.
• All goods are identical and cost is proportional to quantity.

Concept / Approach:
The key idea is that overall profit depends on total selling price (SP) compared to total cost price. We first find the CP and SP of the part sold at a loss. Then we use the required overall SP (calculated from 20% gain on the total CP) to determine how much SP must come from the remaining goods. From that we can compute the gain percentage on the remaining part by comparing its SP with its CP.

Step-by-Step Solution:
Step 1: CP of one third of the goods = (1/3) * 450 = Rs. 150. Step 2: This part is sold at 10% loss, so SP on this part = 150 * (1 - 10/100) = 150 * 0.9 = Rs. 135. Step 3: CP of the remaining two thirds = 450 - 150 = Rs. 300. Step 4: Required overall profit = 20% of 450 = 0.20 * 450 = Rs. 90. Step 5: Therefore, required total SP for 20% gain = 450 + 90 = Rs. 540. Step 6: SP needed from the remaining goods = 540 - 135 = Rs. 405. Step 7: Gain on remaining goods = SP - CP = 405 - 300 = Rs. 105. Step 8: Gain percentage on remaining goods = (105 / 300) * 100 = 35%.

Verification / Alternative check:
We can verify by recomputing the entire transaction. One third sold for Rs. 135, two thirds sold for Rs. 405, giving total SP = 135 + 405 = Rs. 540. Comparing with total CP of Rs. 450, profit = 540 - 450 = Rs. 90. Profit percent = (90 / 450) * 100 = 20%, which matches the requirement. Hence 35% gain on the remaining goods is correct.

Why Other Options Are Wrong:
A gain of 28%, 22%, 19% or 30% on the remaining goods would produce a total SP different from Rs. 540 and therefore a different overall profit percent. Only 35% generates exactly 20% profit on the whole transaction, so the other options are numerically inconsistent with the given conditions.

Common Pitfalls:
Learners sometimes average percentages directly instead of working with actual money values. Another common mistake is to apply the 20% overall gain only to the remaining goods instead of the entire lot. Always compute CP and SP for each part separately, then combine them to check the overall profit or loss.

Final Answer:
The remaining goods must be sold at a gain of 35% to obtain 20% profit on the whole transaction.