Question

In a 4000 meter race around a circular stadium having a circumference of 1000 meters, the fastest runner and the slowest runner reach the same point at the end of the 5th minute, for the first time after the start of the race. All the runners have the same starting point and each runner maintains a uniform speed throughout the race. If the fastest runner runs at twice the speed of the slowest runner, then what shall be the time taken by the fastest runner to finish the said race?

• A. 15 minutes
• B. 20 minutes
• C. 10 minutes
• D. 13 minutes
• E. 5 minutes

Hint:

On a circular track, the time for two runners starting together to meet again is the lap length divided by their relative speed.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In a circular race, two runners starting from the same point meet for the first time when the distance gap between them is exactly equal to one full circumference of the track.
Step 2: Key Formula or Approach:
1. Relative Speed \(\times\) Time = Circumference.
2. Time to finish = Total race distance / Speed.
Step 3: Detailed Explanation:
1. Let the speed of the slowest runner be \(v\) m/min.
2. Since the fastest runner is twice as fast, their speed is \(2v\) m/min.
3. The relative speed between them is \(2v - v = v\) m/min.
4. They meet for the first time at the end of the 5th minute. The track circumference is 1000 meters.
\[ \text{Relative Speed} \times \text{Time} = \text{Circumference} \]
\[ v \times 5 = 1000 \implies v = 200 \text{ m/min} \]
5. The speed of the fastest runner is \(2v = 2 \times 200 = 400 \text{ m/min} \).
6. The total length of the race is 4000 meters.
7. Time taken by the fastest runner to finish the race:
\[ \text{Time} = \frac{\text{Total Distance}}{\text{Speed of Fastest Runner}} \]
\[ \text{Time} = \frac{4000}{400} = 10 \text{ minutes} \].
Step 4: Final Answer:
The fastest runner will take 10 minutes to finish the race.