How many times in a day are the hands of a clock straight (in the same line, either together or opposite)?

Problem restatementCount the number of times in 24 hours that the hour and minute hands are collinear: either overlapping (0°) or opposite (180°).

Concept/ApproachIn 12 hours, the hands overlap 11 times and are opposite 11 times. These are distinct events (no double-counting). Over 24 hours, the counts double.

Step-by-step reasoning Overlaps in 12 h = 11; oppositions in 12 h = 11 Total in 12 h = 11 + 11 = 22 Total in 24 h = 2 × 22 = 44

Verification/AlternativeBetween any two consecutive overlaps (~65 5/11 minutes apart), there is exactly one opposition (~32 8/11 minutes after an overlap). This yields paired counts of 11 each per 12 hours.

Common pitfalls

• Confusing “straight” with “overlap only”; “straight” includes both 0° and 180°.
• Assuming 24 or 48 based on hours rather than relative motion counts.

Final Answer44 times