A bell rings every 24 minutes and another every 18. They both ring together at 6pm. What time will they next ring together at the same time
?
What is the least common multiple?
18, 36, 54, 72, 90
24, 48, 72, 96
To find out when the two bells will ring together at the same time again, we need to determine the least common multiple (LCM) of their ringing intervals.
The first bell rings every 24 minutes, and the second bell rings every 18 minutes. We need to find the smallest number that is divisible by both 24 and 18.
To find the LCM, we can use prime factorization.
The prime factorization of 24 is: 2^3 * 3
The prime factorization of 18 is: 2 * 3^2
To find the LCM, we take the highest power of each prime factor that appears in either factorization:
- The highest power of 2 is 2^3 = 8
- The highest power of 3 is 3^2 = 9
Multiply these highest powers together to find the LCM: LCM(24, 18) = 8 * 9 = 72
So, the two bells will ring together at the same time every 72 minutes.
Since they rang together at 6pm, we need to calculate how many 72-minute intervals have passed since then to determine the next time they will ring together.
First, convert 6pm into minutes:
6pm = 6 * 60 = 360 minutes
Next, divide 360 by 72 to find out how many intervals have passed:
360 / 72 = 5 intervals
Finally, multiply the number of intervals by the interval length and add it to the starting time:
5 * 72 = 360
360 + 360 = 720
Therefore, the two bells will next ring together at the same time at 720 minutes after 6pm, which is equal to 6am the following day.