Clock Problems

At 10:00 the minute hand is at the 12 mark ( $0^\circ$ or $360^\circ$ ) and the hour hand at the 10 mark ( $300^\circ$ ). The angle between them is $360^\circ -300^\circ =60^\circ$ .

Ten minutes later the minute hand has moved to the 2 mark or $60^\circ$ or $420^\circ$ . The hour hand has moved by $\dfrac {360^\circ} {12\times 60}\times 10=5^\circ$ and it is at $300 +5=305^\circ$ . The angle between the two hands is $420-305=\boxed {115^\circ}$ .

This solution is about 10:10 am.

We can take the whole clock face of this 12 hour clock as 360° (full turn).

Then:

1 hour represents a 360° ÷ 12 = 30° angle and

10 minutes (hour hand) represent a 30° ÷ 6 = 5° angle.

The hour hand is between 10 and 11, 5° (10 minutes) closer to 12, than the number 10 is.

Therefore, the angle between the hour hand and the number 12:

2 × 30°- 5° = 55°

The minute hand is pointing at the number 2.

Therefore, the angle between the number 12 and the minute hand:

2 × 30°= 60°

Hence, the angle between the hour hand and the minute hand:

$55° + 60° = \boxed {115°}$