To find the correct oxidation state of Zn in ZnO (Zinc oxide), and each element in the molecule, we use a few rules and some simple math.

First, since the ZnO molecule doesn’t have an overall charge (like NO3- or H3O+) we could say that the total of the oxidation numbers for ZnO will be zero since it is a neutral molecule.

We write the oxidation number (O.N.) for elements that we know and use these to figure out oxidation number for Zn.

----------
GENERAL RULES
Free elements have an oxidation state of zero (e.g. Na, Fe, H2, O2, S8).
In an ion the all Oxidation numbers must add up to the charge on the ion.
In a neutral compound all Oxidation Numbers must add up to zero.
Group 1 = +1
Group 2 = +2
Hydrogen with Non-Metals = +1
Hydrogen with Metals (or Boron) = -1
Fluorine = -1
Oxygen = -2 (except in H2O2 or with Fluorine)
Group 17(7A) = -1 except with Oxygen and other halogens lower in the group
----------