Solution:

Total effort required by 4 painters for 40 days = 4 × 40 = 160 man-days.

Initial effort by 4 painters for 10 days = 4 × 10 = 40 man-days.

Job still requires 160 − 40 = 120 man-days, and there are 20 painters available to do it.

Additional effort required by 20 painters for 120 man-days = 120 / 20 = 6 days.

Thus, 6 more days are required to complete the painting job.

Alternative Shorter Solution:

4 painters require 40 − 10 = 30 more days to complete the job.
Therefore, 4 + 16 = 20 painters will require 30 × 4 / 20 = 6 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?