In order to find the ratio, or in this case, the scale of the map, you must remember to keep the units in base power 1.

In this question, they gave the values of the areas, in which the units are in power 2 (cm^2 and km^2). Therefore, you would need to square root the values (together with the units as well) in order to bring it down to base power 1 and find the scale.

Suppose in another question where instead of a map, they gave you a 3-dimensional model and they gave you the values of the volumes. The units for the volumes should be in power 3, such as cm^3, m^3, etc (note that 1 ml = 1 cm^3). Then you would have to cube root the values, together with the units, to find the scale.

So the key is to always bring it down to base power 1 by taking the necessary root (square or cube). The units in the values should give you a hint most of the time.
Area = Square Root
Volume = Cube Root

Hope this helps!