It would have been clearer had you written,
"How would I solve 'z divided by 4/9 equals 3/8' ?
You meant
z
-------- = 3/8, or 9z/4 = 3/8.
4
--
9
The LCD here is 8. Mult. both sides by 8: 18z = 3.
Solve this last result for z: z = 3/18, or z = 1/6 (answer)