Solution:

Total effort required by 8 painters for 19 days = 8 × 19 = 152 man-days.

Initial effort by 8 painters for 4 days = 8 × 4 = 32 man-days.

Job still requires 152 − 32 = 120 man-days, and there are 24 painters available to do it.

Additional effort required by 24 painters for 120 man-days = 120 / 24 = 5 days.

Thus, 5 more days are required to complete the painting job.

Alternative Shorter Solution:

8 painters require 19 − 4 = 15 more days to complete the job.
Therefore, 8 + 16 = 24 painters will require 15 × 8 / 24 = 5 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?