Bill, bob, and barry are hired to paint signs. in 8 hours bill can paint 1 sign, bob can paint 2 signs, and barry can paint 1 and 1 / 3 signs. they all come to work the first day, but barry doesn't like the job and quits after 3 hours. bob works half an hour longer than barry and quits. how long will it take bill to finish 2 signs they were supposed to paint?

Define rates at which the individuals paint.
Bill:  x = 1/8  signs/h
Bob: y = (2 signs)/(8 h) = 1/4  signs/h
Barry: z = (4/3 signs)/(8 h) = 1/6 signs/h

The job is to paint 2 signs, working together.

In the first 3 hours, the amount painted is
(x+y+z signs/h)*(3 h) = (1/8 + 1/4 + 1/6)*3 = 13/8 signs
Remaining: 2 - 13/8 = 3/8 signs

Barry quits.
In the next 1/2 hour (30 minutes), the amount painted is
(x + y signs/h)*(0.5 h) = (1/8 + 1/4)*0.5 = 3/16 signs
Remaining: 3/8 - 3/16 = 3/16 signs

Now Bob quits.
The time for Bill to finish the job is
(3/16  signs)/(1/8 signs/h) = 3/2 h

Bill finishes the job in 1.5 hours (or 1 hour, 30 minutes)