Solution:

Total effort required by 5 painters for 51 days = 5 × 51 = 255 man-days.

Initial effort by 5 painters for 6 days = 5 × 6 = 30 man-days.

Job still requires 255 − 30 = 225 man-days, and there are 25 painters available to do it.

Additional effort required by 25 painters for 225 man-days = 225 / 25 = 9 days.

Thus, 9 more days are required to complete the painting job.

Alternative Shorter Solution:

5 painters require 51 − 6 = 45 more days to complete the job.
Therefore, 5 + 20 = 25 painters will require 45 × 5 / 25 = 9 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?