Solution:

Total effort required by 6 painters for 27 days = 6 × 27 = 162 man-days.

Initial effort by 6 painters for 7 days = 6 × 7 = 42 man-days.

Job still requires 162 − 42 = 120 man-days, and there are 12 painters available to do it.

Additional effort required by 12 painters for 120 man-days = 120 / 12 = 10 days.

Thus, 10 more days are required to complete the painting job.

Alternative Shorter Solution:

6 painters require 27 − 7 = 20 more days to complete the job.
Therefore, 6 + 6 = 12 painters will require 20 × 6 / 12 = 10 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?