A man buys 500 metres of electronic wire at 50 paise per metre. He sells 50% of the wire at a profit of 5%. At what profit percentage must he sell the remaining wire so that his overall profit on the entire transaction is 10%?

Introduction / Context:
This question deals with variable selling prices and a target overall profit. The man sells half of the quantity at a known profit rate and the other half at an unknown rate, but the total profit percentage for the entire quantity is given. We need to determine the required profit percentage on the remaining portion.

Given Data / Assumptions:
- Total quantity of wire = 500 metres. - Cost price per metre = Rs 0.50. - Half of the wire, that is 250 metres, is sold at a 5% profit. - Desired overall profit on the entire 500 metres = 10%. - Remaining 250 metres are sold at a uniform profit percentage to be determined.

Concept / Approach:
First compute the total cost of 500 metres. Then find the selling price for the first 250 metres at 5% profit. Next, compute the total selling price required to achieve a 10% overall profit. The difference between required total selling price and the selling price already obtained from the first half gives the revenue needed from the second half, from which we can find the required profit percentage.

Step-by-Step Solution:
Step 1: Total cost price C = 500 * 0.50 = Rs 250. Step 2: Cost price for first 250 metres = 250 * 0.50 = Rs 125. Step 3: Selling price for first 250 metres at 5% profit = 125 * 1.05 = Rs 131.25. Step 4: Required total selling price for 10% overall profit = C * 1.10 = 250 * 1.10 = Rs 275. Step 5: Required selling price from remaining 250 metres = 275 - 131.25 = Rs 143.75. Step 6: Cost price of remaining 250 metres = 125 (same as the first half). Step 7: Profit on remaining 250 metres = 143.75 - 125 = Rs 18.75. Step 8: Profit percent on remaining 250 metres = (18.75 / 125) * 100 = 15%.

Verification / Alternative check:
Check total selling price with the derived rate. If remaining 250 metres are sold at 15% profit, their selling price is 125 * 1.15 = Rs 143.75, matching Step 5. Then total selling price = 131.25 + 143.75 = Rs 275. Total profit = 275 - 250 = Rs 25, and overall profit percent = 25 / 250 * 100 = 10%, as required.

Why Other Options Are Wrong:
Profit rates of 13%, 12.5%, or 20% applied to the remaining wire would yield total selling prices that are either too low or too high to produce an overall profit of exactly 10%. Only 15% generates the precise additional revenue needed to hit the target overall profit.

Common Pitfalls:
Many learners mistakenly average 5% and 10% or try to apply 15% directly without checking the actual monetary values. Another error is to forget that both halves of the wire have equal cost price and that the total profit must be computed on the entire cost, not on each half separately.

Final Answer:
The remaining wire must be sold at a 15% profit.