From the time shown, after how many minutes will the hour and minute hands first be exactly 30° apart?

It is 3:40: hour hand 110°, minute hand 240°, so the minute hand is 130° ahead. That lead grows by 5.5° a minute; the smaller angle becomes 30° when the lead is 330°: (330 − 130) ÷ 5.5 = 200/5.5 = 36 4/11 min. (18 2/11 min = 100 ÷ 5.5 wrongly lets the lead shrink.)