Keith has to grind the valves on his Mercedes. He estimates it will take him 8 hours. His friends Joe and Russ have done the same job in 10 and 12 hours respectively. Keith had worked alone for 2 hours when Joe came along and helped him for 2 hours, then left. Joe worked another half an hour alone and then Russ came along and together they finished the job. How long did it take them to finish?

k = Keith rate = 1/8 per hour

j = Joe rate = 1/10 per hour

r = Russ rate = 1/12 per hour

Given:

(k*2) + (k+j)*2 + j*(0.5) + (j+r)*T = 1.0

Substitute known values and solve for "T":

(1/8*2) + (1/8+1/10)*2 + (1/10)*(0.5) + ((1/10)+(1/12))*T = 1.0

(1/4) + (1/4+1/5) + (1/20) + ((6/60)+(5/60))*T = 1.0

(1/4) + (9/20) + (1/20) + (11/60)*T = 1.0

(15/60) + (27/60) + (3/60) + (11/60)*T = 1.0

45 + 11T = 60

11T = 15

T = 15/11 = 1.3636 hours