# Racing Cars on Figure-8 Track

Five racing cars are tearing along on a figure-8 track that has a total length of 7 miles. The cars A, B, C, D, and E have speeds, respectively, of 60, 72, 75, 80, and 96 mph. High walls are located along each side of the track; and so, should any two cars meet at the center, they will collide. Each driver is wondering, “How did I get myself into this? Who is Stella?” If the cars started at the center, all going the same way, will two cars eventually collide? If so, which two, and after how many miles and how many minutes?

Let’s set up a chart and look at each car’s speed in miles per minute (mpm) and how long it takes each car to complete a 7-mile lap:

If any two cars complete a whole number of laps in the same amount of time, there’s going to be a crash. You can use a spreadsheet to see how long it takes each car to complete 1, 2, 3, … n laps. The chart below shows the time required (in 1/120ths of a minute) for each car to complete the first five laps:

We can see that cars B and E are going to crash 17.5 minutes into the race when B completes 3 laps and E completes 4 laps at exactly the same time. If the race is allowed to continue, cars A and D will then crash 21 minutes into the race when A completes 3 laps and D completes 4 laps at the same time. Car C may remain unscathed unless A is able to continue the race, in which case A and C will crash 28 minutes into the race at the end of laps 4 and 5 respectively.