Two runners are racing on a circular track that is 500 m long. They start from the same point on the track and run in the same direction. The ratio of their speeds is 2 : 3. The faster runner travelled a total distance of 8 km. How many times did the runners meet each other on the track? (1 km = 1000 m)
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
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Two runners are racing on a circular track that is 500 m long. They start from the same point on the track
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The runners meet whenever the faster runner completes ONE MORE LAP than the slower runner.Mikrislac wrote: ↑Mon Aug 10, 2020 11:20 pmTwo runners are racing on a circular track that is 500 m long. They start from the same point on the track and run in the same direction. The ratio of their speeds is 2 : 3. The faster runner travelled a total distance of 8 km. How many times did the runners meet each other on the track? (1 km = 1000 m)
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
The ratio of their speeds = 2 : 3.
Implication:
Every time the faster runner completes 3 laps, the slower runner completes only 2 laps, with the result that the faster runner completes one more lap than the slower runner and meets the slower runner.
Thus:
The total number of meetings = the total number of 3-lap trips completed by the faster runner.
Distance traveled by the faster runner = 8 km = 8000 meters
Since each lap = 500 meters, we get:
Number of laps completed by the faster runner = 8000/500 = 16 laps.
Since the faster runner travels a total of 16 laps, the number of 3-lap trips completed by the faster runner = 5.
The correct answer is E.
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Solution:Mikrislac wrote: ↑Mon Aug 10, 2020 11:20 pmTwo runners are racing on a circular track that is 500 m long. They start from the same point on the track and run in the same direction. The ratio of their speeds is 2 : 3. The faster runner travelled a total distance of 8 km. How many times did the runners meet each other on the track? (1 km = 1000 m)
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Since the faster runner traveled 8 km and since the ratio of the speed of the faster runner to that of the slower runner is 3/2, the slower runner traveled 8/(3/2) = 16/3 km. It follows that the faster runner traveled
8 - 16/3
24/3 - 16/3
8/3 km
more than the slower runner.
Notice that the two runners will meet every time the faster runner travels an integer multiple of 500 meters more than the slower runner. That is, when the faster runner travels 500 m, 1000 m, 1500 m etc. more than the slower runner, the two runners will be at the same position.
Thus, in order to determine the number of times the two runners meet, we need to determine the number of 500 meters in 8/3 kilometers. Since 500 meters = 0.5 kilometers and since (8/3)/(0.5) ≈ 5.33, the two runners will meet 5 times.
Answer: E
