Four bells toll at intervals of 5, 10, 12 and 15 minutes respectively. If they…
2021
Four bells toll at intervals of 5, 10, 12 and 15 minutes respectively. If they start tolling together at 3, after what interval will they toll together and how many times they toll together in 18 hours?
- A.
2 hours, 9 times
- B.
3 hours, 6 times
- C.
1 hour, 18 times
- D.
30 minutes, 36 times
Attempted by 26 students.
Show answer & explanation
Correct answer: C
Concept
When several events repeat at fixed intervals and start together, they next coincide after a time equal to the Lowest Common Multiple (LCM) of their individual intervals. The number of coincidences in a given span is found by dividing that span by the LCM.
Application
Find the LCM of the four intervals 5, 10, 12 and 15 minutes. Prime factors: 5 = 5, 10 = 2·5, 12 = 2²·3, 15 = 3·5.
Take the highest power of each prime: 2² · 3 · 5 = 4 · 3 · 5 = 60. So the LCM = 60 minutes = 1 hour.
Hence the bells next toll together after 60 minutes, i.e. every 1 hour.
Total span = 18 hours = 1080 minutes. Number of times they toll together after the start = 1080 ÷ 60 = 18.
Therefore they toll together every 1 hour and do so 18 times in 18 hours.
Here the moment they all start together at 3 o'clock is the common starting point; the count above is the number of later coincidences within the 18-hour span.
Cross-check
Counting the coincidences as a multiplication: in 18 hours there is one coincidence per hour after the start, giving 18 × 1 = 18 coincidences, which matches the division above.