Question

Five bells begin to ring together and ring respectively at intervals of 6, 5, 7, 10 and 12 seconds. How many times will they ring together in one hour excluding the one at the start?

• A. 7 times
• B. 8 times
• C. 9 times
• D. 11 times

Hint:

When multiple periodic events start together, their next simultaneous occurrence time is the LCM of their periods. In a time window, count the multiples of that LCM.

Answer 1

The Correct Option is B

Step 1: Combine the intervals two at a time: \(\mathrm{LCM}(6,5)=30\), then \(\mathrm{LCM}(30,7)=210\).

Step 2: Bring in the remaining intervals: \(\mathrm{LCM}(210,10)=210\), then \(\mathrm{LCM}(210,12)=420\) seconds between coincidences.

Step 3: Divide the one-hour duration by this interval: \(3600\div420=8.57\ldots\), so the bells coincide \(\lfloor8.57\rfloor=8\) times after the start.
\[ \boxed{8\ \text{times}} \]