Hands Apart: 9:45
Read the puzzle
A round clock face with the numbers 12, 3, 6 and 9 and minute tick marks shows quarter to ten; a digital display underneath reads 9:45. The minute hand points straight left at the 9, and the hour hand sits most of the way from the 9 toward the 10. What is the smaller angle between the two hands?
What is the SMALLER angle between the hands?
The hour hand moves 30° each hour. How far has it gone after 45 minutes?
22.5°
The minute hand is on 9; the hour hand is 3/4 of the way to 10: 3/4 × 30° = 22.5°.
How to solve it
Measure from the 12 going clockwise. Each hour gap is 30°, the minute hand moves 6° per minute and the hour hand moves 0.5° per minute.
The minute hand at 45 minutes is at 45 × 6 = 270°, on the 9.
The hour hand is at 9 × 30 = 270° for nine full hours plus 45 × 0.5 = 22.5° for the extra 45 minutes, so it sits at 292.5°. Put another way, 45 minutes is three quarters of an hour, and 3/4 × 30° = 22.5°.
Subtract: 292.5 − 270 = 22.5°. The other way round is 360 − 22.5 = 337.5°, so the smaller angle is 22.5°.
The usual wrong answer is 30°, from picturing the hour hand already on the 10. At 9:45 it still has a quarter of a gap, 7.5°, to go. Others say 0°, thinking both hands sit on the 9. Check with |30 × H − 5.5 × M| = |270 − 247.5| = 22.5°.
