A shopkeeper buys $500$ m of electronic wire at $50$ paise per m. He sold $50$ % of the wire at a profit of $5$ %. At what percent should the shopkeeper sell the remaining wire so as to gain $10$% on the whole transaction?

The shopkeeper bought 500 m of wire at 50 paise per meter, which is ₹0.50 per meter.

Total Cost Price $= 500 \times 0.50$

Total Cost Price $= ₹250$

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For a 10% profit on the whole transaction:

Profit $= 10\% \text{ of } 250$

Profit $= ₹25$

Total Selling Price required $= 250 + 25$

Total Selling Price required $= ₹275$

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The shopkeeper sold 50% of the wire at 5% profit.

Wire sold $= 50\% \text{ of } 500$

Wire sold $= 250 \text{ m}$

Cost Price of 250 m $= 250 \times 0.50$

Cost Price of 250 m $= ₹125$

Profit at 5% $= 5\% \text{ of } 125$

Profit at 5% $= ₹6.25$

Selling Price $= 125 + 6.25$

Selling Price $= ₹131.25$

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Amount still needed from remaining wire:

Required amount $= 275 - 131.25$

Required amount $= ₹143.75$

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The remaining wire is 250 m with Cost Price of ₹125.

This must be sold for ₹143.75.

Profit on remaining wire $= 143.75 - 125$

Profit on remaining wire $= ₹18.75$

Profit percentage $= \dfrac{18.75}{125} \times 100$

Profit percentage $= 15\%$

Therefore, the shopkeeper should sell the remaining wire at 15% profit.