Solution:

Total effort required by 5 painters for 13 days = 5 × 13 = 65 man-days.

Initial effort by 5 painters for 5 days = 5 × 5 = 25 man-days.

Job still requires 65 − 25 = 40 man-days, and there are 10 painters available to do it.

Additional effort required by 10 painters for 40 man-days = 40 / 10 = 4 days.

Thus, 4 more days are required to complete the painting job.

Alternative Shorter Solution:

5 painters require 13 − 5 = 8 more days to complete the job.
Therefore, 5 + 5 = 10 painters will require 8 × 5 / 10 = 4 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?