Understanding Clock Hand Movement

The minute hand moves $360^\circ$ in 60 minutes ($6^\circ$ per minute). The hour hand moves $360^\circ$ in 12 hours (720 minutes), which is $0.5^\circ$ per minute. The relative speed between the minute and hour hand is $6^\circ - 0.5^\circ = 5.5^\circ$ per minute.

Calculating Angle Formation Frequency

The angle between the clock hands is given by the formula $\theta = |30H - 5.5M|$, where H is the hour and M is the minute.

We need to find when the angle $\theta$ is exactly $40^\circ$. This occurs in two scenarios:

1. The minute hand is $40^\circ$ ahead of the hour hand.
2. The minute hand is $40^\circ$ behind the hour hand.

These two scenarios correspond to the equations:

• $30H - 5.5M = 40$
• $30H - 5.5M = -40$

Rearranging the formulas to solve for M:

• $5.5M = 30H - 40 \implies M = \frac{60H - 80}{11}$
• $5.5M = 30H + 40 \implies M = \frac{60H + 80}{11}$

Counting Occurrences in 6 Hours

The time frame is from 1:00 pm to 7:00 pm (a duration of 6 hours). We need to check the number of valid minute values (0 to 60) for each hour H from 1 to 6.

• Hour 1 (1:xx pm):
  • $M = \frac{60(1) - 80}{11} = \frac{-20}{11}$ (Invalid)
  • $M = \frac{60(1) + 80}{11} = \frac{140}{11} = 12 \frac{8}{11}$ (Valid: 1 instance)
• Hour 2 (2:xx pm):
  • $M = \frac{60(2) - 80}{11} = \frac{40}{11} = 3 \frac{7}{11}$ (Valid)
  • $M = \frac{60(2) + 80}{11} = \frac{200}{11} = 18 \frac{2}{11}$ (Valid: 2 instances)
• Hour 3 (3:xx pm):
  • $M = \frac{60(3) - 80}{11} = \frac{100}{11} = 9 \frac{1}{11}$ (Valid)
  • $M = \frac{60(3) + 80}{11} = \frac{260}{11} = 23 \frac{7}{11}$ (Valid: 2 instances)
• Hour 4 (4:xx pm):
  • $M = \frac{60(4) - 80}{11} = \frac{160}{11} = 14 \frac{6}{11}$ (Valid)
  • $M = \frac{60(4) + 80}{11} = \frac{320}{11} = 29 \frac{1}{11}$ (Valid: 2 instances)
• Hour 5 (5:xx pm):
  • $M = \frac{60(5) - 80}{11} = \frac{220}{11} = 20$ (Valid)
  • $M = \frac{60(5) + 80}{11} = \frac{380}{11} = 34 \frac{6}{11}$ (Valid: 2 instances)
• Hour 6 (6:xx pm):
  • $M = \frac{60(6) - 80}{11} = \frac{280}{11} = 25 \frac{5}{11}$ (Valid)
  • $M = \frac{60(6) + 80}{11} = \frac{440}{11} = 40$ (Valid: 2 instances)

The total number of times the hands form a $40^\circ$ angle is the sum of instances from each hour:

Total Instances = $1 + 2 + 2 + 2 + 2 + 2 = 11$.

Alternatively, the clock hands form a specific angle (other than $0^\circ$ or $180^\circ$) 22 times in 12 hours. Therefore, in 6 hours, the number of times is $\frac{22}{12} \times 6 = 11$.

The final count is 11.