Suppose that it takes John 9 hours to paint a barn if he works alone and it takes Carl 6 hours to paint the same barn if he works alone. How long will it take them to paint the barn, in hours, if they work together?

If John takes 9 hours to paint a barn, then he does 1/9th of a barn per hour. Similarly, Carl does 1/6th of a barn per hour. Together, they can do 1/9 + 1/6, or 5/18ths of a barn per hour. Then 1 barn takes 1 / (5/18), or 18/5ths hours with the two of them working together