Question

The minute-hand and second-hand of a clock cross each other \underline{\hspace{1cm}} times between 09:15:00 AM and 09:45:00 AM on a day.

• A. 30
• B. 15
• C. 29
• D. 31

Hint:

Minute and second hand cross each other 59 times in one hour. In 30 minutes, they cross about half, i.e. ~29 times.

Answer 1

The Correct Option is C

Step 1: Speeds of hands.
- The second hand makes one revolution per 60 seconds. Hence, speed = \(360^\circ/60 = 6^\circ/s\).
- The minute hand makes one revolution per 3600 seconds. Hence, speed = \(360^\circ/3600 = 0.1^\circ/s\).

Step 2: Relative angular speed.
Relative speed = \(6 - 0.1 = 5.9^\circ/s\).

Step 3: Time for successive coincidences.
Whenever the hands coincide, the relative angular displacement = \(360^\circ\). \[ T = \frac{360}{5.9} \approx 61.02 \, s \]

Step 4: Duration given.
From 9:15 to 9:45, total time = 30 minutes = \(1800 \, s\).

Step 5: Number of coincidences.
\[ N = \frac{1800}{61.02} \approx 29.49 \] Hence, the hands cross **29 times** in that interval. \[ \boxed{29} \]