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 (in kg) sold at $18\%$ profit is (L.I.C.A.A.O., 2007)

### Concept & Rule of Alligation The Rule of Alligation is used to find the ratio in which two quantities at different rates or percentages must be mixed to produce a mixture at a given mean rate. $$ \frac{\text{Quantity of Cheaper (or Part 1)}}{\text{Quantity of Dearer (or Part 2)}} = \frac{\text{Dearer Rate} - \text{Mean Rate}}{\text{Mean Rate} - \text{Cheaper Rate}} $$ ### Step-by-Step Solution * **Part 1 Profit ($P_1$):** $8\%$ * **Part 2 Profit ($P_2$):** $18\%$ * **Mean Profit ($M$):** $14\%$ * Apply Alligation: * Ratio of Quantity 1 to Quantity 2 = $(18 - 14) : (14 - 8)$ * Ratio = $4 : 6 = 2 : 3$ * The total quantity is divided into $2 + 3 = 5$ parts. * **Total Quantity:** $1000$ kg. * Quantity sold at $18\%$ profit corresponds to the 3 parts. $$\text{Quantity at } 18\% = \frac{3}{5} \times 1000 = 3 \times 200 = 600 \text{ kg}$$ ### Exam Strategy & Shortcut Set up the alligation cross mentally: $8 \quad \quad \quad 18$ $\quad \searrow \quad \swarrow$ $\quad \quad 14$ $\quad \swarrow \quad \searrow$ $(18-14) \quad (14-8)$ $4 \quad : \quad 6$ Simplifies to $2 : 3$. Total 5 units = $1000$ kg. So 1 unit = $200$ kg. The $18\%$ side has 3 units, yielding $3 \times 200 = 600$. ### Common Pitfall A frequent mistake is matching the ratio parts to the wrong percentage. Always remember the cross-subtraction maps the right-side difference to the left-side quantity, and vice versa. Finding the quantity for $8\%$ (which is $400$) and selecting option (a) is a common trap. ### Final Answer Therefore, the correct answer is **600**.