Solution:

Total effort required by 7 painters for 26 days = 7 × 26 = 182 man-days.

Initial effort by 7 painters for 10 days = 7 × 10 = 70 man-days.

Job still requires 182 − 70 = 112 man-days, and there are 14 painters available to do it.

Additional effort required by 14 painters for 112 man-days = 112 / 14 = 8 days.

Thus, 8 more days are required to complete the painting job.

Alternative Shorter Solution:

7 painters require 26 − 10 = 16 more days to complete the job.
Therefore, 7 + 7 = 14 painters will require 16 × 7 / 14 = 8 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?