2006 AMC 10A Problems/Problem 15

Problem

Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?

/begin{figure}
unitsize(1cm);
draw((5,0){up}..{left}(0,5),red);
draw((-5,0){up}..{right}(0,5),red);
draw((5,0){down}..{left}(0,-5),red);
draw((-5,0){down}..{right}(0,-5),red);
draw((6,0){up}..{left}(0,6),blue);
draw((-6,0){up}..{right}(0,6),blue);
draw((6,0){down}..{left}(0,-6),blue);
draw((-6,0){down}..{right}(0,-6),blue);
\end{figure}
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Solution

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Since , we note that Odell runs one lap in minutes, while Kershaw also runs one lap in minutes. They take the same amount of time to run a lap, and since they are running in opposite directions they will meet exactly twice per lap (once at the starting point, the other at the half-way point). Thus, there are laps run by both, or meeting points .

See Also