I tend to think it is a process rather like finding a common denominator. Usually, you can look at the few fractions and see it.
The most rigorous way is to write every equation:
(a)NaNO3 + (b)NaOH +(c) Zn -> (d)NH3 + (e)Na2ZnO2 + (f)H2O
Then do a balance for each element, ie:
Sodium:
a + b = 2e
Nitrogen:
a = d
Oxygen:
3a = 2e +f
etc ...
As a practical matter, zero in on the fact that all the hydrogen is in (b). And Zinc tells you (c) = (e). Since there are 5 hydrogens on the left, (b) HAS to be a multiple of 5.
As a "trick", just look for those elements that are most simply connected right and left, then try just try them. If you were to try the 5,10,15 series for (b) it would go quickly.
Like in simplifying a fraction after finding a common denominator, you want the final equation in the smallest denominator. If you add 1/2 + 1/3 + 1/4 you can either spot they all could have 12, or use 24 as the common denominator ... 12/24 + 8/24 + 6/24 =15/24 = 26/24 =13/12
But the main trick is to spot the elements that are simply connected, and then plug those in to get the other elements.
[Mentor Note: explicit solution removed]
Generally, I am looking for the easiest integer relationships, then trying integer multiples of those ratios until I spot one. I find that trial and error approach fastest.