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

### 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. $$ \frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}} = \frac{\text{Dearer Rate} - \text{Mean Rate}}{\text{Mean Rate} - \text{Cheaper Rate}} $$ ### 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 = $\frac{3}{5} \times 1000 = 600$ 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: $\frac{3}{5}$ 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**.