E/Z stereoisomers : assigning priorities with a cycloalkane group
The reasoning you have used is clearly faulty as there is no meaning in assigning more number of carbon to one end in a cyclic system.

Just look at the both the branches you have marked using orange and blue. The logic followed is in accordance to Cahn-Ingold-Prelog(CIP) rules.


The first atom in branches, marked as blue and as orange , have two hydrogen and one carbon each.So, there is nothing to clearly distinguish between the two branches.
So, we look at the only distinguishing feature between the two branches , which is the substituents on the carbon atom(marked as 2) attached to the carbon atom(marked as 1) on both sides. 
We can see that the carbon atom marked as 2 on orange marked side has two carbon atoms and one hydrogen atom attached to it. This is of higher priority than the carbon on the blue marked side which has one carbon atom and two hydrogen atom attached to it.


Hence the Z-nomenclature as answered in the book.