Factors and Multiples - 3 bells tolling

Three bells toll at different intervals. Bell A tolls every 24 seconds, Bell B tolls every 60 seconds and Bell C tolls every 90 seconds. The three bells toll together at 1100 hrs.

a) find the number of times all three bells toll together between 1100 and 1200 hours.

b) find the number of times only two bells toll together between 1100 and 1200 hours.

I am able to solve (b) by finding the Lowest Common Multiple. But I can't solve (b) short of making of list of the times each bell tolls. My answer is 30. But, there must be a shorter way to solve this. Can somebody please help?

Re: Factors and Multiples - 3 bells tolling

A and B toll together every LCM(24, 60)= 120 seconds. There are 3600 seconds in one hour and 120 divides into 3600 30 times. A and B toll together (perhaps with C) 30 times between 1100 and 1200 hours. A and C toll together every LCM(24, 90)= 360 seconds and 360 divides into 36000 100 times. A and C toll together (perhaps with B) 100 times between 1100 and 1200. B and C toll together every LCM(60, 90)= 180 seconds and 180 divides into 3600 20 times. B and C toll together (perhaps with A) 20 times between 1100 and 1200.

So **at least** two bells toll together 30+ 100+ 20= 150 times. You say you have already found the answer to (a), the number of times all three bells toll together. Since those times will be counted in each of the three computations above, subtract **3** times that number from 150 to find the number of times exactly two bells toll together.

Re: Factors and Multiples - 3 bells tolling

Quote:

Originally Posted by

**HallsofIvy** ...and 360 divides into 36000 100 times. A and C toll together (perhaps with B) 100 times between 1100 and 1200.

I think this should be: 360 divides into **3600 10 times**. A and C toll together ... **10** times between 1100 and 1200."

Also - you need to be careful about how to handle the beginning and ending seconds - don't forget to include the fact that all three chime at 11:00. Bells A and B actually toll together 31 times between 1100 and 1200 inclusive, A and C 11 times, and B and C 21 times. So when subtracting three times the number of occurences of all three chiming together, be sure to include the fact that all three chime at t = 11:00.