Solution:

Total effort required by 7 painters for 55 days = 7 × 55 = 385 man-days.

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

Job still requires 385 − 70 = 315 man-days, and there are 35 painters available to do it.

Additional effort required by 35 painters for 315 man-days = 315 / 35 = 9 days.

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

Alternative Shorter Solution:

7 painters require 55 − 10 = 45 more days to complete the job.
Therefore, 7 + 28 = 35 painters will require 45 × 7 / 35 = 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?