At 3:40, what is the smaller angle between the hour hand and the minute hand?

Problem restatementCompute the smaller angle between the clock hands at 3:40.

Given data

• Hour-hand position at H:M = 30H + 0.5M (degrees from 12).
• Minute-hand position at H:M = 6M (degrees from 12).

Concept/ApproachAngle formula: |30H − 5.5M| degrees gives the smaller angle for most times.

Step-by-Step calculation Angle = |30 × 3 − 5.5 × 40| = |90 − 220| = 130°

Verification/AlternativeExplicit positions: hour hand = 30×3 + 0.5×40 = 120 + 20 = 140°; minute hand = 6×40 = 240°; difference = |240 − 140| = 100°? Careful—hour hand at 3:40 is actually 30×3 + 0.5×40 = 90 + 20 = 110°. Minute hand = 240°. Difference = 130°. (Corrected arithmetic.)

Common pitfalls

• Forgetting the hour hand also moves during the minutes.

Final Answer130°