Clock Strikes Thirteen

Solution

Answer: 82.5

A clock has 60 minute marks, and each mark represents 6 degrees on a circle.

At 4:00, the minute hand is on the 12 and the hour hand is on the 4. The smallest angle is \( 4 * 5 * 6 = 120 \) degrees.

At 4:15, the minute hand is on the 3. The hour hand has moved \( \frac{15}{60} \) of the way from 4 to 5, so it is 1.25 minute marks past the 4. The separation is \( 6.25 \) minute marks, giving an angle of \( 6 * 6.25 = 37.5 \) degrees.

Therefore, the difference in angles is \( 120 - 37.5 = 82.5 \) degrees.