Tow friends begin walking on a 1 mile track at the same time. They walk the mile in 15 minutes and 18 minutes respectively. After how many minutes will they both be back to the starting point?
What is the least common multiple of 15 and 18?
5/6
To find out when both friends will be back to the starting point, we need to determine the time it takes for both friends to complete exactly one lap of the 1 mile track.
Let's find the time it takes for the first friend (who walks the mile in 15 minutes) to complete one lap of the track. Since they walk at a constant speed, we can assume their speed is constant throughout the entire track. Therefore, their speed can be calculated by dividing the distance (1 mile) by the time (15 minutes):
Speed of the first friend = Distance / Time
= 1 mile / 15 minutes
= 1/15 miles per minute
Similarly, let's calculate the speed of the second friend (who walks the mile in 18 minutes):
Speed of the second friend = Distance / Time
= 1 mile / 18 minutes
= 1/18 miles per minute
Now, we have the speeds of both friends. To find the time it takes for both friends to be back to the starting point, we need to determine when they complete an integer number of laps.
The time it takes for both friends to complete an integer number of laps is the least common multiple (LCM) of their lap times. In this case, their lap times are 15 minutes and 18 minutes.
To find the LCM of 15 and 18 minutes, we can list the multiples of each number until we find a common multiple:
Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, ...
Multiples of 18: 18, 36, 54, 72, 90, 108, ...
From the lists of multiples, we can see that the first common multiple is 90. Therefore, it will take both friends 90 minutes to complete a whole number of laps and be back at the starting point.
So, they will both be back to the starting point after 90 minutes.