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A particular clock-like device has a round face with three hands. The first hand completes one full rotation every hour, the second hand completes half a rotation per hour, and the third hand completes one-third of a rotation per hour. Initially, all hands are positioned at the 12 o'clock mark and start moving simultaneously. How many minutes will elapse before the hands align together again for the first time since starting?
A. 60
B. 120
C. 180
D. 240
E. 360
A. 60
B. 120
C. 180
D. 240
E. 360
22 Jan 2024, 10:18
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Official Solution:
A particular clock-like device has a round face with three hands. The first hand completes one full rotation every hour, the second hand completes half a rotation per hour, and the third hand completes one-third of a rotation per hour. Initially, all hands are positioned at the 12 o'clock mark and start moving simultaneously. How many minutes will elapse before the hands align together again for the first time since starting?
A. 60
B. 120
C. 180
D. 240
E. 360
The three hands of the clock have different rotation speeds:
• The first hand's period is one rotation in 1 hour.
• The second hand's period is one rotation in 2 hours.
• The third hand's period is one rotation in 3 hours.
To find when they align again, we need the least common multiple (LCM) of their rotation periods:
The LCM of 1, 2, and 3 is 6. Therefore, the hands will align again after 6 hours, which is 360 minutes.
Answer: E
A particular clock-like device has a round face with three hands. The first hand completes one full rotation every hour, the second hand completes half a rotation per hour, and the third hand completes one-third of a rotation per hour. Initially, all hands are positioned at the 12 o'clock mark and start moving simultaneously. How many minutes will elapse before the hands align together again for the first time since starting?
A. 60
B. 120
C. 180
D. 240
E. 360
The three hands of the clock have different rotation speeds:
• The first hand's period is one rotation in 1 hour.
• The second hand's period is one rotation in 2 hours.
• The third hand's period is one rotation in 3 hours.
To find when they align again, we need the least common multiple (LCM) of their rotation periods:
The LCM of 1, 2, and 3 is 6. Therefore, the hands will align again after 6 hours, which is 360 minutes.
Answer: E
07 Jun 2025, 06:46
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You can convert the rotation time to minutes and take the LCM
First hand will rotate in 60 mins
Second hand will rotate in 120 mins
Third hand will rotate in 180 mins
LCM of 60, 120, 180 is 360. all will align together first time in 360 minutes. i.e 6 hours
First hand will rotate in 60 mins
Second hand will rotate in 120 mins
Third hand will rotate in 180 mins
LCM of 60, 120, 180 is 360. all will align together first time in 360 minutes. i.e 6 hours
