Solution:

Total effort required by 8 painters for 14 days = 8 × 14 = 112 man-days.

Initial effort by 8 painters for 8 days = 8 × 8 = 64 man-days.

Job still requires 112 − 64 = 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:

8 painters require 14 − 8 = 6 more days to complete the job.
Therefore, 8 + 8 = 16 painters will require 6 × 8 / 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?