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

This problem can be solved easily using the method of alligation.
$$\begin{array}{ccc}\text{Profit of 1st part (8\%)}& & \text{Profit of 2nd part (18\%)}\\ & \text{Mean Profit (14\%)}& \\ (18-14)=4& & (14-8)=6\end{array}$$

The ratio of the quantities sold at $8\%$ profit and $18\%$ profit is:

$$\text{Ratio}=4:6=2:3$$

The total quantity of sugar is $1000\text{kg}$.
The quantity sold at $18\%$ profit corresponds to $3$ parts out of $(2+3)=5$ parts:

$$\text{Quantity}=\frac{3}{5}\times 1000=3\times 200=600\text{kg}$$