z=3. If z varies inversely as w, and z=4 when w=9, then when w=12 the value of z is 3.
The key to solve this problem is using reverse proportionality, in which two magnitudes a and b are inversely proportional when there is a constant k such that
a⋅b=k, where constant k is called the proportionality constant.
Then if z varies inversely as w, and z=4 when w=9
z.w=k -------> 4.9=k -------> k=36
So, let's find z when w = 12. With k = 36
z.w=K, clear z from the equation
z=k/w -------> z=36/12 -----> z=3