Q.Show that minute hand of a clock gains $5°30'$ on the hour hand in one minute.

The hour and minute hands of a clock both make one full $360°$ rotation, but at different fixed rates: the minute hand turns $360°$ in $60$ minutes, i.e. $6°$ per minute; the hour hand turns $360°$ in $12$ hours, i.e. $30°$ per hour $=0.5°$ per minute.

At $H$ o'clock and $M$ minutes, the minute hand has turned $6M$ degrees from the $12$-mark and the hour hand has turned $30H + 0.5M$ degrees from the $12$-mark, so the angle between them is

$\left|\,30H - 5.5M\,\right|°$ …