The LCM of 10 people and and 6 people is 30 people.

So we'll go from 10 people to 30 people to 6 people. 

30 people is 3 times as many people as 10 people 

but 

6 people is only 1/5 as many people as 30 people.

Since 10 people take 18 days to paint a house,

3 times as many people will take only 1/3 as long to paint it.  So

30 people will take only 6 days to paint it.

But 1/5 as many people will take 5 times as long.  So

6 people will take 30 days to paint it.

Edwin