Question

Five bells begin to toll together and toll respectively at intervals of 6,7,8,9 and 12 seconds. How many times, will they toll together in one hour excluding the one at the start?

• A. 5
• B. 6
• C. 7
• D. 4
• E. None of these

Answer 1

The Correct Option is C

Step 1: Understand the problem.
Five bells toll together at the start, and then toll respectively at intervals of 6, 7, 8, 9, and 12 seconds. We need to find how many times they will toll together in one hour, excluding the initial toll.

Step 2: Find the least common multiple (LCM) of the intervals.
The bells toll together when the time elapsed is a common multiple of their intervals. Therefore, we need to find the least common multiple (LCM) of the intervals 6, 7, 8, 9, and 12 seconds.

Find the LCM of 6, 7, 8, 9, and 12:
- Prime factorization of 6: \( 6 = 2 \times 3 \)
- Prime factorization of 7: \( 7 = 7 \)
- Prime factorization of 8: \( 8 = 2^3 \)
- Prime factorization of 9: \( 9 = 3^2 \)
- Prime factorization of 12: \( 12 = 2^2 \times 3 \)

The LCM is found by taking the highest powers of all prime factors:
\[ \text{LCM} = 2^3 \times 3^2 \times 7 = 8 \times 9 \times 7 = 504 \, \text{seconds} \]

Step 3: Calculate how many times the bells toll together in one hour.
The bells toll together every 504 seconds. One hour is 3600 seconds. To find how many times they toll together, divide 3600 by 504:
\[ \frac{3600}{504} \approx 7.14 \]
This means they toll together 7 times, but we exclude the initial toll. Therefore, the number of times they toll together excluding the start is:
\[ 7 - 1 = 6 \text{ times} \]

Step 4: Conclusion.
The bells toll together 6 times in one hour, excluding the initial toll.

Final Answer:
The correct option is (C): 3.