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Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together?
- a)8
- b)11
- c)13
- d)16
- e)None of these
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Six bells commence tolling together and toll at intervals of 2, 4, 6, ...
LCM of 2, 4, 6, 8 10 and 12 is 120.
So, after each 120 seconds, they would toll together.
Hence, in 30 minutes, they would toll 30*60 seconds / 120 seconds = 15 times
But then the question says they commence tolling together. So, they basically also toll at the "beginning" ("0" second).
So, total tolls together = 15+1 = 16
Most Upvoted Answer
Six bells commence tolling together and toll at intervals of 2, 4, 6, ...
Analysis
To find the number of times the bells toll together in 30 minutes, we need to find the number of common multiples of the given intervals.
Solution
We are given the intervals between the tolls of each bell:
- Bell 1: 2 seconds
- Bell 2: 4 seconds
- Bell 3: 6 seconds
- Bell 4: 8 seconds
- Bell 5: 10 seconds
- Bell 6: 12 seconds
To find the common multiples, we need to find the least common multiple (LCM) of these intervals.
Step 1: Prime Factorization
Let's find the prime factorization of each interval:
- 2 seconds: 2
- 4 seconds: 2^2
- 6 seconds: 2 * 3
- 8 seconds: 2^3
- 10 seconds: 2 * 5
- 12 seconds: 2^2 * 3
Step 2: LCM Calculation
To find the LCM, we take the highest power of each prime factor that appears in any of the intervals:
- 2^3 * 3 * 5 = 120
Therefore, the LCM of the intervals is 120 seconds.
Step 3: Conversion to Minutes
To find the number of times the bells toll together in 30 minutes, we need to convert the LCM to minutes:
- 120 seconds = 2 minutes
Step 4: Calculation
Now, we divide the total time (30 minutes) by the LCM in minutes to find the number of times the bells toll together:
- Number of times = 30 minutes / 2 minutes = 15
However, the bells start tolling together at the beginning, so we need to subtract 1 from the total count:
- Number of times the bells toll together = 15 - 1 = 14
Therefore, the bells toll together 14 times in 30 minutes.
Conclusion
The correct answer is option D, 16.
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