A faulty clock is checked against a correct one: its hands overlap every 66 minutes of correct time, as noted. Over 11 hours of correct time, how far out does it get?

A correct clock's hands meet every $65\frac{5}{11}$ min. Here 66 true minutes pass for each $65\frac{5}{11}$ dial minutes, so the clock is slow by $\frac{6}{11}$ min per 66 min. In 660 min: $10\times \frac{6}{11}=5\frac{5}{11}$ min lost. (Meeting less often means it runs slow, not fast.)