Three lighthouses on three nearby islands flash their lights every evening. On one island, the light flashes every 45 seconds. The lighthouse on the second island flashes every 2 minutes. The third lighthouse flahses a light every 1 and 1/4 minutes. Suppose the lights flash together at 8:35pm. At what time will they flahs together agian? Explain your answer.
First the times in seconds.
Then least common multiple
45 seconds, 120 seconds, 75 seconds
factor them all
3 *3 * 5
2 * 2 * 2 * 3 * 5
3 * 5 * 5
so we need
2 * 2 * 2 * 3 * 5 * 5
thet is
8*3*25
=600 seconds
= 10 minutes
8:45
It's actually 2*2*2*3*3*5*5 = 1800 seconds which is 30 minutes. It will be 9:05 when they all flash together again.
To determine when the lighthouses will flash together again, we need to find the least common multiple (LCM) of their flash times.
First, let's convert all the flash times to seconds for easier calculations:
Lighthouse 1 flashes every 45 seconds.
Lighthouse 2 flashes every 2 minutes = 2 x 60 = 120 seconds.
Lighthouse 3 flashes every 1 and 1/4 minutes = 1.25 x 60 = 75 seconds.
Now, let's find the LCM of these three numbers: 45, 120, and 75.
We can start by finding the LCM of 45 and 120:
The multiples of 45 are: 45, 90, 135, 180, 225, 270, 315, 360...
The multiples of 120 are: 120, 240, 360, 480, 600...
From the lists of multiples, we can see that 360 is the smallest number that appears in both lists. Therefore, the LCM of 45 and 120 is 360.
Now, let's find the LCM of 360 and 75:
The multiples of 360 are: 360, 720, 1080...
The multiples of 75 are: 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, 825, 900...
From the lists of multiples, we can see that 360 is the smallest number that appears in both lists. Therefore, the LCM of 360 and 75 is also 360.
So, the lighthouses will flash together again every 360 seconds.
Now, let's calculate the time in hours and minutes using the starting time of 8:35pm.
8:35pm = 8 hours and 35 minutes = 8 x 60 minutes + 35 minutes = 515 minutes.
Since the lighthouses flash together every 360 seconds, and there are 60 seconds in a minute, the time interval in minutes is 360/60 = 6 minutes.
To find out when they will flash together again, we divide the total minutes (515) by the time interval in minutes (6):
515 minutes / 6 minutes = 85 remainder 5.
This means that the lighthouses will flash together again after 85 intervals of 6 minutes, with an additional 5 minutes.
To convert this back into hours and minutes:
85 intervals of 6 minutes = 85 x 6 = 510 minutes.
Adding the additional 5 minutes: 510 minutes + 5 minutes = 515 minutes.
Now, we need to add the calculated minutes back to the starting time of 8:35pm.
515 minutes = 515 / 60 = 8 hours and 35 minutes.
Therefore, the lighthouses will flash together again at 8 hours and 35 minutes after 8:35pm, which is 4:10am the next day.