This problem requires finding the least common multiple (LCM) of 9 and 12.
First, let's list the multiples of 9 and 12:
Multiples of 9: 9, 18, 27, 36, 45, ...
Multiples of 12: 12, 24, 36, 48, ...
The least common multiple (LCM) is the smallest number that is a multiple of both 9 and 12.
From the list, we see that 36 is the smallest number that appears in both lists. Therefore, it will take 36 weeks for Jiani and Rosa to get paid at the same time.