There are 5 bells which toll together at 10:00 A.M. They toll after 8,9,10,11,12 seconds respectively. How many times will they toll together again in the next 4 hours?3

To determine how many times the 5 bells will toll together again in the next 4 hours, we need to find the least common multiple (LCM) of their tolling intervals (8, 9, 10, 11, and 12 seconds).

Step-by-step solution:

1. Prime Factorization:

   • 8 = 2^3
   • 9 = 3^2
   • 10 = 2 × 5
   • 11 = 11 (prime number)
   • 12 = 2^2 × 3

2. LCM Calculation:

   • The LCM is found by taking the highest power of each prime number that appears in the factorizations.
   • LCM = 2^3 × 3^2 × 5 × 11

3. Calculate the LCM:

   • 2^3 = 8
   • 3^2 = 9
   • 5 = 5
   • 11 = 11
   • LCM = 8 × 9 × 5 × 11
   • LCM = 72 × 5 × 11
   • LCM = 360 × 11
   • LCM = 3960 seconds

4. Convert 4 hours to seconds:

   • 4 hours = 4 × 60 minutes = 240 minutes
   • 240 minutes = 240 × 60 seconds = 14400 seconds

5. Determine the number of times they toll together:

   • The bells toll together every 3960 seconds.
   • Number of times in 14400 seconds = 14400 / 3960
   • Number of times = 3.636 (approximately)

Since they toll together at the start (10:00 A.M.), we count the initial toll and the subsequent tolls within the 4-hour period.

6. Count the tolls:
   • Initial toll at 10:00 A.M.
   • Subsequent tolls: 1 (3960 seconds), 2 (7920 seconds), 3 (11880 seconds)

Therefore, the bells will toll together 4 times in the next 4 hours.