Question

Three bells ring at an interval of 9 minutes, 15 minutes and 21 minutes. The bells will next ring together at 11.00pm. Find the time the bells had last rang together?

​

Answer 1

The lowest coMmon denominator of 9, 15, and 21 is 315 (3•3•5•7). These 3 bells ring together every 315 minutes after 5:45 pm.

45 min • 1/60 hr/min = 0.75 hr

5:45 = 5 + 0.75 hr = 5.75 hr

5:45 pm to 10:00 am next day in minutes is:

available time = (12 - 5.75 + 10) hr • 60 min/hr

available time = 16.25 • 60 = 975 min

So as much as 975 min are available for the synchronized or periodic ringings.

Number of simultaneous ringings in 975 min:

975/315 = 3.1

3•315 = 945 min

975 - 945 = 30 mins after 3rd ringing after 5:45 pm.

315 min = 315•1/60 hr/min = 5.25 hr. 5.25 hr between simultaneous rings.

The next 3 ringings after 5:45 pm (5.75 hr) are at 5.75 + 5.25 hr = 11:00 pm, and then 11.0 pm + 1 = 12 am + 4.25 = 4.25 am (4:15 am), and then 4.25 + 5:25 hr = 9:30 am. There are 30 minutes left to 10:00 am (975 mins after 5:45 pm), so a 4th ringing does not have enough time.

ANSWER: Including 5:45 pm, the bells ring together at 11:00 pm, 4:25 am, and 9:30 am — a total of 4 times