Answer
$\frac{11}{6}$ hours.
Work Step by Step
Let $2c$ be the speed of Cathie, then the speed of Abbie is $6c$ and the speed of Beth is $3c$. Thus the equation for them painting together: $(6c+3c+2c)1=1\\c=\frac{1}{11}$. Thus Abbie's speed is $\frac{11}{6}$ of the painting per hour, thus she would need \frac{11}{6}$ hours.
