At what time between 7 and 8 o'clock do the hands of a clock point in exactly opposite directions?

Explanation

1. 01

   At 7:00 the hour hand is $210^\circ$ ahead of the minute hand. For the hands to be opposite, the gap must shrink to $180^\circ$.

2. 02

   Gap to close $= 210 - 180 = 30^\circ$, at $5.5^\circ$ per minute: $30 \div 5.5 = \frac{60}{11} = 5\frac{5}{11}$ minutes.

3. 03

   So the hands are opposite at $5\frac{5}{11}$ minutes past 7.