Understanding the Bells Ringing Problem

This problem is about finding out when multiple events, which repeat at different fixed intervals, will occur simultaneously again after they have occurred together at a specific starting time. In this case, the events are the ringing of three different bells, and their intervals are given in seconds.

Applying the Concept of Least Common Multiple (LCM)

When objects or events repeat at regular intervals and start at the same time, they will occur together again at a time that is a multiple of each individual interval. To find the first time they will occur together again after the start, we need to find the smallest common multiple of all the intervals. This is precisely the definition of the Least Common Multiple (LCM).

Therefore, to find out after how many seconds the three bells will ring together again, we need to calculate the LCM of their individual ringing intervals: 36 seconds, 40 seconds, and 48 seconds.

Calculating the LCM of 36, 40, and 48

To find the LCM, we can use the prime factorization method. We find the prime factors of each number:

• Prime factorization of 36: \(36 = 2 \times 18 = 2 \times 2 \times 9 = 2^2 \times 3^2\)
• Prime factorization of 40: \(40 = 2 \times 20 = 2 \times 2 \times 10 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1\)
• Prime factorization of 48: \(48 = 2 \times 24 = 2 \times 2 \times 12 = 2 \times 2 \times 2 \times 6 = 2 \times 2 \times 2 \times 2 \times 3 = 2^4 \times 3^1\)

Now, to find the LCM, we take the highest power of each prime factor that appears in any of the factorizations:

• Highest power of 2: \(2^4\) (from 48)
• Highest power of 3: \(3^2\) (from 36)
• Highest power of 5: \(5^1\) (from 40)

The LCM is the product of these highest powers:

\(\text{LCM}(36, 40, 48) = 2^4 \times 3^2 \times 5^1\)

\(\text{LCM}(36, 40, 48) = 16 \times 9 \times 5\)

\(\text{LCM}(36, 40, 48) = 144 \times 5\)

\(\text{LCM}(36, 40, 48) = 720\)

So, the LCM of 36, 40, and 48 is 720.

Converting Seconds to Minutes

The LCM we calculated is in seconds because the given intervals were in seconds. The bells will ring together again after 720 seconds. The options are given in minutes, so we need to convert 720 seconds into minutes.

There are 60 seconds in 1 minute. To convert seconds to minutes, we divide the number of seconds by 60.

\(\text{Time in minutes} = \frac{\text{Time in seconds}}{60}\)

\(\text{Time in minutes} = \frac{720}{60}\)

\(\text{Time in minutes} = 12\)

So, the bells will ring together again after 12 minutes.

Summary of the Solution

To find when events repeating at different intervals will next occur together, calculate the Least Common Multiple (LCM) of the intervals. The intervals are 36, 40, and 48 seconds. The LCM is 720 seconds. Converting 720 seconds to minutes gives 12 minutes.

Revision Table: Key Concepts Review

Additional Information: When to Use LCM vs. HCF

This problem is a typical application of LCM. Problems that involve finding when multiple events will occur together again (like bells ringing, lights blinking, runners meeting on a track) usually require finding the LCM of the time intervals.

On the other hand, problems that involve dividing quantities into the largest possible equal parts (like distributing items into boxes, cutting pieces of cloth) usually require finding the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD).

Understanding the difference between LCM and HCF applications is crucial for solving such quantitative aptitude problems.