Five bells begin tolling together and then toll again at intervals of 3, 5, 7,…
2024
Five bells begin tolling together and then toll again at intervals of 3, 5, 7, 9, and 11 seconds respectively. What fraction of the next 60 minutes will have elapsed when all five bells next toll together simultaneously?
- A.
0.4
- B.
0.9
- C.
0.9625
- D.
0.7654
Show answer & explanation
Correct answer: C
When several periodic events all start together, they will next coincide again at every common multiple of their individual periods, the smallest such gap being the Least Common Multiple (LCM) of the periods. To express how far into a fixed total duration that simultaneous recurrence falls, divide the time of that recurrence by the total duration.
Factorize each interval: 3 = 3, 5 = 5, 7 = 7, 9 = 32, 11 = 11.
Take the LCM using the highest power of each prime: LCM = 32 × 5 × 7 × 11 = 3465 seconds.
Convert 3465 seconds to minutes: 3465 ÷ 60 = 57.75 minutes.
Express this as a fraction of the total 60-minute duration: 57.75 ÷ 60 = 0.9625.
Working backwards, 0.9625 × 60 = 57.75 minutes = 3465 seconds, which matches the LCM computed above, confirming the result.
So the five bells will next toll together simultaneously at the 0.9625 fraction of the 60-minute duration.