How many times in a day, the hands of a clock are straight?

### Concept & Logic The hands of a clock are "straight" when they are coincident (angle = $0^\circ$) OR when they are pointing in opposite directions (angle = $180^\circ$). ### Step-by-Step Solution 1. **Coincident (0 degrees):** The hands coincide 11 times in 12 hours, which means 22 times in 24 hours. (They miss a coincidence between 11 and 1, meeting only at 12). 2. **Opposite (180 degrees):** The hands are in opposite directions 11 times in 12 hours, which means 22 times in 24 hours. (They miss being opposite between 5 and 7, being opposite only exactly at 6). 3. **Total Straight Line Occurrences:** Total = (Coinciding) + (Opposite) Total = 22 + 22 = 44 times. ### Exam Strategy & Shortcut Memorize the standard daily occurrences for clock hands: - Coincide: 22 times - Opposite: 22 times - Straight line (either of the above): $22 + 22 = 44$ times - Right angles: 44 times ### Common Pitfall A frequent mistake is interpreting "straight" as ONLY pointing in opposite directions ($180^\circ$), leading to an answer of 22. "Straight" implies forming a straight line, which includes the overlapping state. ### Final Answer Therefore, the correct answer is **44**.