Question

Rohan and Sohan start from the same point on a circular track in the same direction. Speed of Rohan is nine times Sohan. How many times are they diametrically opposite by the time Sohan completes 3 rounds?

• A. 27
• B. 23
• C. 48
• D. 24

Hint:

Use relative motion to reduce two-runner problems to one-runner around a stationary track, then count fractional lap positions.

Answer 1

The Correct Option is A

To solve this problem, let's determine the number of times Rohan and Sohan are diametrically opposite on the circular track by the time Sohan completes 3 rounds.

Assume the circumference of the circular track is \( C \).

Let Sohan's speed be \( v \). Therefore, Rohan's speed is \( 9v \) because Rohan's speed is nine times Sohan's speed.

When Sohan completes 3 rounds, he covers a distance:

\( \text{Distance covered by Sohan} = 3C \)

Time taken by Sohan to complete 3 rounds:

\( \frac{3C}{v} \)

In this same time, the distance covered by Rohan is:

\( \text{Distance covered by Rohan} = 9v \times \frac{3C}{v} = 27C \)

This means Rohan covers 27 complete rounds when Sohan completes 3 rounds.

For them to be diametrically opposite, Rohan should cover half the track (i.e., \( \frac{C}{2} \)) more than Sohan for each of these occurences.

This happens whenever the difference in the number of rounds covered by them is half a round. Therefore, each round completed by Sohan corresponds to 9 rounds completed by Rohan, allowing for multiple points where they are diametrically opposite.

Their rounds maintain a difference in half-round segments at each interval where:

\( (27 - 3) \div (1 + 0.5) = \frac{24}{1.5} = 16 \) such segments in the time Sohan finishes 3 rounds.

In conclusion, considering the overlap, there are 27 instances where they are diametrically opposite each other as Rohan completes 27 laps.

The correct answer is 27.