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?

General
1

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 ?

2

To find out how many times the six bells toll together in 30 minutes, we need to determine the least common multiple (LCM) of the intervals at which they toll. The LCM is the smallest number that is divisible by all the given intervals.

The given intervals are 2, 4, 6, 8, 10, and 12 seconds. Let’s first convert 30 minutes into seconds, as the intervals are provided in seconds.

1 minute = 60 seconds
30 minutes = 30 * 60 seconds = 1800 seconds

Now, let’s find the LCM of the given intervals.

Prime factorize each interval:
2 = 2
4 = 2^2
6 = 2 * 3
8 = 2^3
10 = 2 * 5
12 = 2^2 * 3

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

So, the LCM of the intervals is 120 seconds.

Now, we can calculate how many times the six bells will toll together in 30 minutes:

Number of times = Total time / Time interval
= 1800 seconds / 120 seconds
= 15

Therefore, the six bells will toll together 15 times in 30 minutes.