Three runners Ram, Karan, and Mohan start from a fixed post around an oval at speeds 5 km/h, 5 1/7 km/h, and 5 1/3 km/h respectively. They all return to the fixed post exactly in 3 3/4 hours. What is the greatest possible track length (in km) consistent with this?
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A28 / 5 km
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B5 / 28 km
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C11 / 28 km
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D28 / 11 km
Answer
Correct Answer: 5 / 28 km
Explanation
Introduction / Context:When multiple runners all return to a common starting post at the same time, the track length must divide the distance each runner covers in that time, so that each completes an integer number of laps. The largest feasible track length equals the greatest common divisor (GCD) of the three distances covered.
Given Data / Assumptions:
- Time t = 3 3/4 h = 15/4 h.
- Speeds: v1 = 5, v2 = 5 1/7 = 36/7, v3 = 5 1/3 = 16/3 (km/h).
Concept / Approach:Distances: d1 = v1*t = 5*(15/4) = 75/4; d2 = (36/7)*(15/4) = 135/7; d3 = (16/3)*(15/4) = 20. The maximum track length L is the GCD of {d1, d2, d3}.
Step-by-Step Solution:
Express with a common denominator 28: d1 = 525/28, d2 = 540/28, d3 = 560/28.GCD(525, 540, 560) = 5; therefore L = 5/28 km.Verification / Alternative check:Check laps: (525/28)/(5/28) = 105; (540/28)/(5/28) = 108; (560/28)/(5/28) = 112. All integers.
Why Other Options Are Wrong:Other fractions do not divide each distance into an integer number of laps.
Common Pitfalls:Using LCM of times without converting to distances, or taking an average instead of the GCD.
Final Answer:5 / 28 km