Question

A circular track is 1000 yards in circumference. Cyclists A, B, and C start at the same place and time, and race around the track at the following rates per minute: A at 700 yards, B at 800 yards, and C at 900 yards. What is the least number of minutes it mus take for all three to be together again?

Answer 1

The least number of minutes it takes for all three cyclists to be together again is 2800 minutes.

Step-by-step explanation:

To find the least number of minutes it takes for all three cyclists to be together again, we need to determine the least common multiple (LCM) of their rates. The rates are given as distances per minute, so we can calculate the LCM of 700 yards, 800 yards, and 900 yards. The LCM is the smallest number that is divisible by all three rates.

To calculate the LCM, we list their multiples:

700: 700, 1400, 2100, 2800, 3500, ...
800: 800, 1600, 2400, 3200, 4000, ...
900: 900, 1800, 2700, 3600, 4500, ...

The LCM of 700, 800, and 900 is 2800 yards. Therefore, it will take 2800 minutes for all three cyclists to be together again.

Answer 2

B gains 100 yards every minute. So, after 10 minutes, B has lapped A once.

C gains 100 yards on B every minute, so after 10 minutes C has lapped B.

Thus, after 10 minutes, A has gone 7 laps, B has gone 8 laps, C has gone 9 laps, and all are even again.