What is the smaller angle between the hands of a clock at 2:40 a.m.?

The correct answer is Option (4) → 160°

Measure both hands clockwise from 12: the minute hand at 6° a minute, the hour hand at 30° an hour plus 0.5° a minute.

• Hour hand at 2:40: \(2\times30^\circ+40\times0.5^\circ=60^\circ+20^\circ=80^\circ\)

• Minute hand at 40 minutes: \(40\times6^\circ=240^\circ\)

• Angle between them \(=240^\circ-80^\circ=160^\circ\), less than 180°, so it is the smaller angle

Why the other options are incorrect

• Option (1) 140° — Incorrect. The two angles the hands make at 2:40 are 160° and 200°, so 140° is neither of them.

• Option (2) 200° — Incorrect. This is the larger angle \(360^\circ-160^\circ\), not the smaller one.

• Option (3) 180° — Incorrect. This takes the hour hand still at the 2 mark (60°), forgetting its 20° move in 40 minutes.