Hands Apart: 3:30
Read the puzzle
A square clock face with the numbers 12, 3, 6 and 9 and tick marks for every hour shows half past three; a digital display underneath reads 3:30. The minute hand points straight down at the 6, and the hour hand sits between the 3 and the 4. What is the smaller angle between the two hands?
What is the SMALLER angle between the hands?
At 3:30 the hour hand is halfway between 3 and 4.
75°
Hour hand at 105°, minute hand at 180°: 75° apart.
How to solve it
Measure both hands clockwise from the 12. Each hour gap is 360 ÷ 12 = 30°, and the minute hand moves 6° per minute.
The minute hand at 30 minutes is at 30 × 6 = 180°, on the 6.
The hour hand moves 0.5° per minute. At 3:30 it has covered 3 × 30 = 90° for three full hours plus 30 × 0.5 = 15° for the half hour, so it is at 105°, halfway between the 3 and the 4.
Subtract: 180 − 105 = 75°. Going the other way round would be 360 − 75 = 285°, so the smaller angle is 75°.
The common wrong answer is 90°, which comes from counting three hour gaps from the 3 to the 6 as if the hour hand were still pointing exactly at the 3. By half past, it has crept half a gap forward, which removes 15° from the gap. Check with the formula |30 × H − 5.5 × M|: |90 − 165| = 75°.
