Picture a vibrant circular 1000-yard track viewed from an overhead perspective. It is surrounded by lush greenery and sitting in a wide expanse under a clear blue sky. On the starting line are three cyclists, ready to race. Cyclist A is on a vivid red bicycle, Cyclist B is on a sharp silver bike, and Cyclist C is on a bright blue bicycle. They are pedalling at different rates, creating a dynamic and engaging scene filled with movement. The energy and dynamism of the race are captured, showing the competitors eagerly racing around the track. Everything conveys a sense of motion and competition.

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?

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.

I still don't get this

its OBVIOUSLY 9....geez

Well, it looks like we have a race going on! Let's see if we can bring these cyclists together again.

To find the least number of minutes it will take for all three cyclists to be together again, we need to find the least common multiple (LCM) of their rates. In this case, their rates are 700, 800, and 900 yards per minute.

The LCM of 700, 800, and 900 is 3600. This means that every 3600 yards, all three cyclists will be together again.

Since the circular track is 1000 yards in circumference, we can divide 3600 by 1000 to find the number of laps it will take for the cyclists to be together again.

3600 / 1000 = 3.6

So, it will take approximately 3.6 laps for all three cyclists to be together again, but since we can't have a fraction of a lap, we'll round up.

Therefore, the least number of minutes it must take for all three cyclists to be together again is 4 minutes.

To find the least number of minutes it will take for all three cyclists to be together again, we need to find the least common multiple (LCM) of the distances covered by each cyclist in one minute.

The distances covered by each cyclist in one minute are as follows:
- Cyclist A covers 700 yards
- Cyclist B covers 800 yards
- Cyclist C covers 900 yards

To find the LCM, we need to find the smallest number that is divisible by all of these distances.

To do this, we can start by finding the prime factorization of each distance:

700 = 2 * 2 * 5 * 5 * 7
800 = 2 * 2 * 2 * 2 * 5 * 5
900 = 2 * 2 * 3 * 3 * 5 * 5

Next, we find the highest power of each prime factor that appears in any of the prime factorizations. In this case, the highest power of 2 is 2^2, the highest power of 3 is 3^2, and the highest power of 5 is 5^2. We don't need to consider the factor 7 because it only appears in the prime factorization of Cyclist A's distance.

Finally, we multiply these highest powers together to find the LCM:

LCM = 2^2 * 3^2 * 5^2 = 4 * 9 * 25 = 900

Therefore, the least number of minutes it will take for all three cyclists to be together again is 900 minutes.