Seven bells ring at intervals of 2, 3, 4, 6, 8, 9 and 12 minutes, respectively. They started ringing simultaneously at 7.10 in the morning. What will be the next time when they all ring simultaneously?

Given:

Seven bells ring at intervals of 2, 3, 4, 6, 8, 9, and 12 minutes, respectively.

They started ringing simultaneously at 7:10 a.m.

Formula Used:

The time when all bells ring simultaneously is given by the Least Common Multiple (LCM) of their intervals.

Calculation:

Find the LCM of the intervals: 2, 3, 4, 6, 8, 9, and 12.

Prime factorization:

2 = 2

3 = 3

4 = 2²

6 = 2 × 3

8 = 2³

9 = 3²

12 = 2² × 3

LCM = 2³ × 3² = 8 × 9 = 72 minutes

Total time = 7:10 a.m. + 72 minutes

72 minutes = 1 hour 12 minutes

⇒ Next simultaneous ringing time = 7:10 a.m. + 1 hour 12 minutes

⇒ 8:22 a.m.

The next time when all the bells ring simultaneously is 8:22 a.m.