Question

5 bells start commencing together at 9 AM. They ring at the intervals of 12 seconds, 18 seconds, 24 seconds, 36 seconds and 45 seconds. At what time do the bells ring together again?

• A. 9:10 AM
• B. 9:06 AM
• C. 9:08 AM
• D. 9:05 AM

Hint:

For any problem involving intervals of recurring events (such as traffic lights, bells, or running laps), finding the LCM of the intervals is always the standard approach.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
This problem requires us to determine when multiple events (ringing of bells) with different time periods will next occur simultaneously.

Step 2: Key Formula or Approach:
The time interval after which all bells ring together is the Least Common Multiple (LCM) of their individual ringing intervals.

Step 3: Detailed Explanation:
Let us calculate the LCM of the intervals 12, 18, 24, 36, and 45 seconds using prime factorization:

• Find prime factors for each number:
- $12 = 2^2 \times 3^1$
- $18 = 2^1 \times 3^2$
- $24 = 2^3 \times 3^1$
- $36 = 2^2 \times 3^2$
- $45 = 3^2 \times 5^1$

• Select the highest power of each prime factor present in the factorizations:
- Highest power of 2 is $2^3 = 8$.
- Highest power of 3 is $3^2 = 9$.
- Highest power of 5 is $5^1 = 5$.

• Calculate LCM:
\[ \text{LCM} = 8 \times 9 \times 5 = 360\text{ seconds} \]

• Convert the time duration from seconds into minutes:
\[ 360\text{ seconds} = \frac{360}{60} = 6\text{ minutes} \]

• Calculate the next ringing time:
Adding 6 minutes to the initial start time of 9:00 AM gives 9:06 AM.

Step 4: Final Answer:
The bells will ring together again at 9:06 AM, which corresponds to option (B).