Solution:

Total effort required by 4 painters for 21 days = 4 × 21 = 84 man-days.

Initial effort by 4 painters for 9 days = 4 × 9 = 36 man-days.

Job still requires 84 − 36 = 48 man-days, and there are 16 painters available to do it.

Additional effort required by 16 painters for 48 man-days = 48 / 16 = 3 days.

Thus, 3 more days are required to complete the painting job.

Alternative Shorter Solution:

4 painters require 21 − 9 = 12 more days to complete the job.
Therefore, 4 + 12 = 16 painters will require 12 × 4 / 16 = 3 more days to complete the job.

Food for thought:

The number of days for the job is inversely proportional to the number of painters. So, is 'number of days × number of painters = constant'?

What assumptions were made to arrive at the solution?
Is something assumed about the output of the painters?