Solution:

Total effort required by 6 painters for 35 days = 6 × 35 = 210 man-days.

Initial effort by 6 painters for 7 days = 6 × 7 = 42 man-days.

Job still requires 210 − 42 = 168 man-days, and there are 24 painters available to do it.

Additional effort required by 24 painters for 168 man-days = 168 / 24 = 7 days.

Thus, 7 more days are required to complete the painting job.

Alternative Shorter Solution:

6 painters require 35 − 7 = 28 more days to complete the job.
Therefore, 6 + 18 = 24 painters will require 28 × 6 / 24 = 7 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?