Nate and roger are painting a fence . nate can paint the entire fence in 4 hours and roger can paint it it in 5 hours. how long would it take them to paint the entire fence if they were working together?

Remark
These questions come up frequently -- usually about 1 a week and that's an estimate based on what I see, which is very small compared to the number of questions asked in a day, let alone a week. They look like they are test material.

Givens
Let Nate's time be n
Let Roger's time be r
Let the new time when both work together = t

General Formula
1 job/nate's time + 1 job / roger's time = 1 job / both

Substitute and solve
1/4 + 1/5 = 1/t

Change fractions so they have a common denominator

5/20 + 4/20 = 1/t Add the fractions.

9/20 = 1/t  cross multiply.
9t = 20      Divide by 9
t = 20/9
t = 2 2/9
t = 2.222 hours <<<< Answer

t = 2 hours and 0.2222*60 minutes
t = 2 hours and 13 minutes (approximately). <<<< answer