Answer
$$t=\frac{99}{2}\ hours$$
Work Step by Step
We know that at the end of the time, the bog needs to be full. Thus, letting $t$ be the time it takes to fill, we find:
$$ 1=\frac{1}{9}t-\frac{1}{11}t\\ \frac{1}{9}t\cdot \:99-\frac{1}{11}t\cdot \:99=1\cdot \:99 \\ 2t=99\\ t=\frac{99}{2}\ hours$$
