To solve this problem, we need to determine the next time when both Sarah’s and Tasmin’s alarms will sound together after they initially sound at 7:45 a.m. Here's a detailed step-by-step solution:
1. Understanding the Alarm Intervals:
- Sarah’s alarm sounds every 9 minutes.
- Tasmin’s alarm sounds every 12 minutes.
2. Finding the Least Common Multiple (LCM):
To determine when both alarms will sound together, we need to find the least common multiple of the two intervals (9 and 12 minutes).
3. Next Time Alarms Sound Together:
The least common multiple of 9 and 12 is the smallest number of minutes both these intervals divide without leaving a remainder. This value is 36. So, both alarms will sound together 36 minutes after the initial time of 7:45 a.m.
4. Converting Minutes into Actual Time:
- Start by converting 7:45 a.m. to minutes:
- 7 hours = 7 * 60 = 420 minutes
- 45 minutes are already given
- Therefore, 7:45 a.m. is 420 + 45 = 465 minutes after midnight
- Add the 36-minute interval we calculated:
- 465 minutes (initial time) + 36 minutes (interval) = 501 minutes after midnight
5. Converting Total Minutes into Hours and Minutes:
We need to convert 501 minutes back into hours and minutes:
- 501 minutes ÷ 60 minutes/hour = 8 hours and 21 minutes (501 % 60 = 21)
So, both alarms will next sound together at 8:21 a.m.
Thus, the next time both alarms will sound together is at 8:21 a.m., 36 minutes after they initially sound at 7:45 a.m.
