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## Circular Racing

On a circular cycle track, two cyclists are preparing for the next summer Olympics by traveling at constant speeds around a track. When they went in opposite directions, they met every 10 seconds; when they went in the same direction, one caught the other every 170 seconds. Find the speed of each cyclist if the circular track was 170 meters in circumference.

Hint: Draw a picture representing both situations. Does it make sense that one of the cyclists is traveling faster than the other? What does that mean when they are traveling in the opposite direction? Same direction?

Solution Commentary: It is possible to solve this problem by resorting to the "powerful" use of the d = rt formula. That is fine, but let's try to think through this problem without doing a great amount of complicated calculations.

Focus on the faster cyclist, who, when the cyclists are traveling in the same direction, must travel 170 meters more in order to lap the slower cyclist. But this takes 170 seconds...so how much faster is the one cyclist traveling?

Now, focus on the case when they are traveling in opposite directions. Could this not be viewed as one cyclist traveling 1 lap around the 170 meter course in 10 seconds?

Now, how can you combine the two pieces of information?