Final answer:
The LCM of the bells' tolling intervals is 360 seconds, and since there are 3600 seconds in an hour, they toll together 9 times within an hour (excluding the initial toll).
Step-by-step explanation:
The problem requires finding how many times four bells toll together within an hour, after starting to toll together. To solve this, we need to find the Least Common Multiple (LCM) of their tolling intervals: 8, 12, 15, and 18 seconds. The LCM of these numbers represents the interval at which all bells will toll together. Once the LCM is determined, we can then calculate how many times this interval fits into one hour, keeping in mind to exclude the initial toll at the start.
To find the LCM, prime factorize each interval and take the highest power of each prime number from all the factorizations:
- 8 = 2³
- 12 = 2² * 3
- 15 = 3 * 5
- 18 = 2 * 3²
The highest powers are 2³, 3², and 5, giving us an LCM of 2³ * 3² * 5 = 360 seconds. There are 3600 seconds in one hour, so the bells will toll together 3600/360 times, which is 10 times. Since we exclude the initial toll, they toll together 9 times in one hour.