Between 4 and 5 o'clock, at what exact time will the hands of a clock point in opposite directions (180°)?

Problem restatement
Find the moment between 4:00 and 5:00 when the minute and hour hands are exactly opposite (angle = 180°).

Given data

• At t minutes after 4, hour-hand angle = 120° + 0.5t°.
• Minute-hand angle = 6t°.
• Opposite direction → absolute difference = 180°.

Concept/Approach
Use relative speed of hands (5.5° per minute) or set up the angle-difference equation and solve for t.

Step-by-Step calculation
|6t − (120 + 0.5t)| = 1806t − 120 − 0.5t = 180 → 5.5t = 300t = 300 ÷ 5.5 = 54 6⁄11 minutesTime = 4 : 54 6⁄11

Verification/Alternative
Relative angular speed = 6 − 0.5 = 5.5°/min. From 4:00, the hands are 120° apart; to be opposite they need a 300° separation change: 300 ÷ 5.5 = 54 6⁄11 min.

Common pitfalls
Using 60/11 formula (overlap) instead of the 180° condition; forgetting the hour-hand motion (0.5°/min).

Final Answer
4:54 6⁄11