let

x = the hours that Abbie needs to paint the living room alone

y = the hours that Beth needs to paint the living room alone

z = the hours that Cathie needs to paint the living room alone

t = 60 min = 1 hour the time they need to pain the room working all 3 together

Abbie paints twice as fast as Beth

x = 2*y

Abbie paints three times as fast as Cathie

x = 3*z

if they pain together they finish in 1 hour

1/x + 1/y + 1/z = 1

by solving the system of 3 equations

x = 2*y

x = 3*z

1/x + 1/y + 1/z = 1

we find that

x = 6 hours

Abbie needs 6 hours to finish the painting alone.