You can use:

Painters times days (in one situation) = Painters times days (in the other situation).

10x = 6·13
10x = 90
 Divide both sides by 10
 = 
  x = 9

Or you can think it out this way:  
  
The least common multiple of 6 painters and 10 painters
is 30 painters.

Since 6 painters take 15 days to paint the school,
30 painters could paint the school in only 3 days. 
That's because 30 is 5 times as many painters and 
so they could finish in 1/5th of the time, and 
1/5th of 15 days is 3 days. 

10 painters is only 1/3 as many painters as 30 painters,
so it would take them 3 times as long as it would take 
30 painters, and 3×3 is 9, so it would take them 9 hours.

Edwin