The ratio of quantities of sugar in which sugar costing Rs. $40/\text{kg}$ and Rs. $30/\text{kg}$. In what ratio both sugar should be mixed so that there is no gain or loss when the mixture is sold at Rs. $32/\text{kg}$?

We can apply the rule of alligation here.
Cost price of cheaper sugar ($C$) $=\text{Rs. }30$
Cost price of dearer sugar ($D$) $=\text{Rs. }40$
Mean price required ($M$) $=\text{Rs. }32$ (since there is no profit or loss, the selling price equals the mean cost price).

By alligation rule formula:

$$\frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}}=\frac{D-M}{M-C}$$

$$\frac{\text{Quantity of Cheaper}}{\text{Quantity of Dearer}}=\frac{40-32}{32-30}=\frac{8}{2}=\frac{4}{1}$$

So, the ratio of cheaper to dearer sugar is $4:1$.
The question lists the items in the order of Rs. $40/\text{kg}$ (dearer) and Rs. $30/\text{kg}$ (cheaper).
Therefore, the required ratio of (Dearer : Cheaper) $=1:4$.