A bell rings every 2 hours, a second bell rings every 3 hours, and a third bell rings every 4 hours. If all 3 bells ring at 9:00 AM. , at what time will all 3 bells next ring?
If you have not learned about LCM before, then you can make a list:
9:00 11 13 15 17 19 21 23:00
9:00 12 15 18 21 0:00
9:00 13 17 21 01:00
21:00 = 9:00 pm is the next time all three bells will ring together.
If you are familiar with LCM, then the intervals are 2,3 and 4 hours.
The LCM of 2, 3 is 6 hours,
the LCM of 3,4 is 12,
the LCM of 2,4 is 4.
So take the longest period of 12 hours and add to 9:00am to get 9:00pm.
To find the time when all three bells will next ring, we need to find the least common multiple (LCM) of the three time intervals: 2 hours, 3 hours, and 4 hours.
One way to do this is to list out the multiples of each time interval until we find a common multiple.
Multiples of 2 hours: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...
Multiples of 3 hours: 3, 6, 9, 12, 15, 18, 21, ...
Multiples of 4 hours: 4, 8, 12, 16, 20, ...
From the lists above, we can see that the least common multiple of 2, 3, and 4 is 12. This means that all three bells will next ring after 12 hours.
Since the bells last rang at 9:00 AM, we can add 12 hours to determine the next time they will all ring:
9:00 AM + 12 hours = 9:00 PM
Therefore, all three bells will next ring at 9:00 PM.