A merchant has $1000$ kg of sugar, part of which he sells at $8\%$ profit and the rest at $18\%$ profit. He gains $14\%$ on the whole. The quantity sold at $18\%$ profit is

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Concept & Rule of Alligation: The rule of alligation is a quick way to find the ratio in which two ingredients or components are mixed to produce a mixture of a desired average value. \frac{Quantity of Cheaper}{Quantity of Dearer} = \frac{Dearer Rate - Mean Rate}{Mean Rate - Cheaper Rate} Step-by-Step Solution: • Given: Total quantity = 1000 kg. Profit on first part = 8%, Profit on second part = 18%, Mean profit = 14%. • By the rule of alligation: • Ratio of quantity at 8% to quantity at 18% = (18 - 14) : (14 - 8) = 4 : 6 = 2 : 3. • Total ratio parts = 2 + 3 = 5. • The quantity sold at 18% profit corresponds to 3 parts out of 5. • Quantity sold at 18% profit = (3/5) × 1000 = 600 kg. Exam Strategy & Shortcut: Use the alligation cross method visually: 8% (Part 1) ---- 18% (Part 2) \ / 14% (Mean) / \ (18 - 14) ---- (14 - 8) 4 : 6 Ratio = 2:3. Quick mental math: (3/5) of 1000 is 600. Common Pitfall: Mixing up the ratios! A common mistake is assigning the calculated ratio 4 to the 18% profit instead of the 8% profit. Remember, the difference crosses over diagonally. Final Answer: Therefore, the correct answer is 600 kg.

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