Miles and Walker begin a workout together at the starting line of a closed-loop track. They run in opposite directions, each at their own constant pace, and continue running until they are both at the starting line at the exact same time again. They pass each other 11 times before this happens (not counting the start or end). Walker is slower than Miles but still manages to complete more than one lap. How many laps does Walker complete?
Walker completes five laps and Miles completes seven. Because Walker and Miles begin at the same location, when they pass each other for the first time, Walker has covered some distance around the track, and Miles has covered the remaining distance around the track. This means that together they have run a total distance equal to one lap around the track. This rule remains true for the entire run: every time they meet, Walker will have covered some part of the track since their last meeting, while Miles will have covered the rest, so their combined distance increases by exactly one full lap.
We are told they pass each other 11 times on the track before their final meeting at the starting line. That makes 12 total meetings, which means they run a combined total of exactly 12 laps. Because they both begin and end at the starting line, they must have each run a whole number of laps. So we are looking for two whole numbers that add to 12. Knowing that Walker is slower than Miles and that Walker completes more than one lap gives the following possibilities:
Walker: 2, Miles: 10
Walker: 3, Miles: 9
Walker: 4, Miles: 8
Walker: 5, Miles: 7
We’ll show why only the last pair is possible. If Walker runs two laps in the exact same time it takes Miles to run 10 laps, it means that Miles runs exactly five times as fast as Walker. But in that case, they would have met at the starting line much earlier! When Walker finished his first lap, Miles would have finished his fifth. The workout would have ended right there, after a combined six laps, rather than the 12 we know they run.
This same logic eliminates all the other options except for the Walker: 5, Miles: 7 case. The key is that their lap numbers cannot share a common divisor, or else they would have finished earlier.
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