Solution:

Total effort required by 5 painters for 31 days = 5 × 31 = 155 man-days.

Initial effort by 5 painters for 10 days = 5 × 10 = 50 man-days.

Job still requires 155 − 50 = 105 man-days, and there are 15 painters available to do it.

Additional effort required by 15 painters for 105 man-days = 105 / 15 = 7 days.

Thus, 7 more days are required to complete the painting job.

Alternative Shorter Solution:

5 painters require 31 − 10 = 21 more days to complete the job.
Therefore, 5 + 10 = 15 painters will require 21 × 5 / 15 = 7 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?