Six bells start ringing simultaneously. If they ring at intervals of 4, 6, 8, 10, 12, and 16 seconds respectively, how many times will they ring together in 40 minutes?

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Points to Remember:

  • This is a Least Common Multiple (LCM) problem combined with a unit conversion problem.
  • We need to find the LCM of the given intervals (4, 6, 8, 10, 12, 16 seconds).
  • We need to convert 40 minutes into seconds.

Introduction:

This question involves determining the frequency of a simultaneous event based on individual event frequencies. The problem hinges on finding the least common multiple (LCM) of a set of numbers representing the time intervals between rings of six bells. The LCM represents the shortest time interval after which all bells will ring together. We will then calculate how many times this simultaneous ringing occurs within a specified time frame.

Body:

1. Finding the Least Common Multiple (LCM):

To find when all six bells ring together, we need to find the least common multiple (LCM) of 4, 6, 8, 10, 12, and 16 seconds. We can use prime factorization to find the LCM:

  • 4 = 2²
  • 6 = 2 × 3
  • 8 = 2³
  • 10 = 2 × 5
  • 12 = 2² × 3
  • 16 = 2⁴

The LCM is found by taking the highest power of each prime factor present: 2⁴ × 3 × 5 = 16 × 3 × 5 = 240 seconds.

2. Converting Time Units:

The given time is 40 minutes. We need to convert this to seconds:

40 minutes × 60 seconds/minute = 2400 seconds

3. Calculating the Number of Simultaneous Rings:

Now we divide the total time in seconds (2400) by the LCM (240 seconds):

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2400 seconds / 240 seconds/ring = 10 rings

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Therefore, the six bells will ring together 10 times in 40 minutes.

Conclusion:

In summary, by finding the least common multiple of the individual ringing intervals (240 seconds) and converting the total time to seconds (2400 seconds), we determined that the six bells will ring simultaneously 10 times in 40 minutes. This problem highlights the importance of understanding fundamental mathematical concepts like LCM and unit conversion in solving real-world problems. This approach can be applied to various scenarios involving synchronized events, optimizing schedules, or coordinating processes requiring simultaneous actions. The solution emphasizes the power of systematic problem-solving and the importance of precision in calculations.

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