Five bells start together and then ring at intervals of 6 s, 7 s, 8 s, 9 s, and 12 s respectively. In one hour, how many additional times (excluding the initial start) will they ring together?

Introduction / Context:
When events repeat at fixed intervals, the times they coincide are multiples of the least common multiple (LCM) of their intervals. Here we find the LCM of the five bell intervals and count how many multiples lie within one hour, excluding the initial time zero.

Given Data / Assumptions:

• Intervals: 6 s, 7 s, 8 s, 9 s, 12 s
• Total duration: 1 hour = 3600 s
• Exclude the start at t = 0

Concept / Approach:
They ring together every LCM(6, 7, 8, 9, 12) seconds. The count of together-rings after the start equals floor(3600 / LCM).

Step-by-Step Solution:
Prime factors: 6 = 2*3; 7 = 7; 8 = 2^3; 9 = 3^2; 12 = 2^2*3.LCM = 2^3 * 3^2 * 7 = 8 * 9 * 7 = 504 seconds.Number of coincidences after start = floor(3600 / 504) = 7.

Verification / Alternative check:
They meet at t = 504, 1008, 1512, 2016, 2520, 3024, and 3528 seconds (seven times) within 3600 s.

Why Other Options Are Wrong:
3, 5, 6, 9 result from incorrect LCM or duration division, or from including/excluding the start/end inconsistently.

Common Pitfalls:
Using gcd instead of LCM; adding one for the start when the question says to exclude it; rounding errors in the division.

Final Answer:
7