Exact angle between clock hands at 4:20 — Compute the smaller angle between the hour and minute hands at 4:20.

Introduction / Context:
Angles on a clock are computed from the hands’ absolute angles. The hour hand moves 0.5° per minute plus 30° per hour; the minute hand moves 6° per minute. The smaller angle is the absolute difference of these two absolute angles, clipped to ≤ 180° if needed.

Given Data / Assumptions:

• Time: 4:20.
• Hour-hand angle = 30 × H + 0.5 × M.
• Minute-hand angle = 6 × M.

Concept / Approach:
Compute each hand’s angular position from 12 O’clock and then take the absolute difference. If the result exceeds 180°, subtract from 360° to get the smaller angle. Here it will already be acute.

Step-by-Step Solution:
Hour-hand angle: 30 × 4 + 0.5 × 20 = 120 + 10 = 130°.Minute-hand angle: 6 × 20 = 120°.Difference: |130 − 120| = 10°.

Verification / Alternative check:
At 4:00 the angle is 120°; in 20 minutes, the hour hand moves 10° while the minute hand moves 120° from 0°, placing them 10° apart.

Why Other Options Are Wrong:
12.5° and 15° are common guesses but do not match the computed positions; “None of these” is unnecessary.

Common Pitfalls:
Forgetting the hour hand’s additional 0.5° per minute, or assuming it stays fixed at the hour mark.

Final Answer:
10°