You run one lap around a mile track every 8 minutes. Your friend runs around the same track every 10 minutes. You both start at the starting line at the same time. a. How far have each of you run when you first meet again at the starting line? b. How far have each of you run the next time you meet at the starting line?
LCM(8,10) = 40
(a) 40/8 and 40/10 miles
(b) double that
To find the distance each of you have run when you first meet again at the starting line, we need to calculate the least common multiple (LCM) of 8 minutes and 10 minutes.
LCM(8, 10) = 40 minutes
a. So, when you first meet again at the starting line, 40 minutes have passed. In 40 minutes, you will have completed 40/8 = 5 laps. Therefore, you have run 5 miles when you first meet again at the starting line.
Your friend, who runs a lap every 10 minutes, would have completed 40/10 = 4 laps. Therefore, your friend has run 4 miles when you first meet again at the starting line.
b. To find the distance each of you have run the next time you meet at the starting line, we need to calculate the LCM of 8 minutes and 10 minutes again.
LCM(8, 10) = 40 minutes
So, the next time you both meet at the starting line is after 40 minutes. By then, you would have completed 40/8 = 5 laps, totaling 5 miles.
Similarly, your friend would have completed 40/10 = 4 laps, totaling 4 miles.
Therefore, next time you both meet at the starting line, you will have run 5 miles and your friend will have run 4 miles.
To answer these questions, we need to determine when you and your friend will meet at the starting line again.
Let's first calculate the time it takes for you and your friend to meet again.
To find the time, we need to find the least common multiple (LCM) of 8 minutes and 10 minutes. The LCM of two numbers is the smallest multiple that both numbers share evenly.
The prime factorization of 8 is 2 * 2 * 2, and the prime factorization of 10 is 2 * 5.
To find the LCM, we take the highest power of each prime factor that appears in either number: 2 * 2 * 2 * 5 = 40.
Therefore, the LCM of 8 minutes and 10 minutes is 40 minutes.
a. How far have each of you run when you first meet again at the starting line?
If you run one lap every 8 minutes and your friend runs one lap every 10 minutes, we can figure out how many laps each of you will have completed when you meet.
Since the LCM is 40 minutes, we can divide it by 8 minutes to find out how many laps you will have run. 40 ÷ 8 = 5 laps.
Similarly, we can divide the LCM by 10 minutes to find out how many laps your friend will have run. 40 ÷ 10 = 4 laps.
Therefore, when you both meet at the starting line for the first time, you will have run 5 laps (5 miles) and your friend will have run 4 laps (4 miles).
b. How far have each of you run the next time you meet at the starting line?
Since we've established that the LCM is 40 minutes, we can determine how many times you and your friend will have met at the starting line within that time frame.
If you both start at the same time and you complete one lap every 8 minutes while your friend completes one lap every 10 minutes, we need to find the multiples of 8 and 10 that fall within the 40-minute period.
Multiples of 8: 8, 16, 24, 32, 40
Multiples of 10: 10, 20, 30, 40
By examining the multiples, we can see that you meet your friend at the starting line at minutes 40, which is the LCM.
Therefore, when you both meet at the starting line for the second time, you will have run 5 laps (5 miles) again, and your friend will have run 4 laps (4 miles) again.
In conclusion, when you first meet at the starting line, you will have run 5 laps (5 miles) while your friend will have run 4 laps (4 miles). The next time you meet at the starting line, the distances covered will be the same as before, 5 laps (5 miles) for you and 4 laps (4 miles) for your friend.