Four bells begin to toll together and toll respectively at intervals of 6, 7, 8 and 9 seconds. In 1.54 hours, how many times do they toll together and in what interval (seconds)?

### Concept & Formula When multiple independent cyclical events (like bells tolling) begin at the same time, the interval at which they will next occur simultaneously is the Least Common Multiple (LCM) of their individual intervals. $$\text{Simultaneous Interval} = \text{LCM}(t_1, t_2, t_3, t_4)$$ To find the total number of simultaneous events within a given duration $T$: $$\text{Total Occurrences} = \left( \frac{T}{\text{Simultaneous Interval}} \right) + 1$$ *Note: The $+1$ accounts for the initial simultaneous toll at $t = 0$.* ### Step-by-Step Solution **Given:** * Tolling intervals: 6, 7, 8, and 9 seconds * Total time window = 1.54 hours **Calculation:** 1. Find the LCM of the intervals to get the common tolling interval: * $6 = 2 \times 3$ * $7 = 7$ * $8 = 2^3$ * $9 = 3^2$ * $\text{LCM} = 2^3 \times 3^2 \times 7 = 8 \times 9 \times 7 = 504 \text{ seconds}$ 2. Convert the total time window into seconds so units match: * $1.54 \text{ hours} = 1.54 \times 3600 \text{ seconds}$ * Total time = $5544 \text{ seconds}$ 3. Calculate the number of times they toll together: $$\text{Number of tolls} = \left( \frac{5544}{504} \right) + 1$$ $$\text{Number of tolls} = 11 + 1 = 12$$ ### Exam Strategy & Shortcut **Two-Part Verification:** The options contain two values (number of tolls, interval). Always solve for the easiest component first. The interval is clearly the LCM of 6, 7, 8, and 9. Since 8 and 9 are co-prime, their LCM is 72. Multiplying by 7 gives 504. This immediately eliminates options (b) and (d). You only need to divide 5544 by 504 to decide between 12 and 14. $504 \times 10 = 5040$, so it's slightly more than 10 times. Thus, $11 + 1 = 12$ is the logical answer. ### Common Pitfall The most frequent error is omitting the $+1$ for the initial toll. If a student divides 5544 by 504, they get 11. If 11 were an option, many would mistakenly select it. Always remember that "begin to toll together" means the clock starts exactly on a simultaneous toll. ### Final Answer **Therefore, the correct answer is 12, 504.**