A circular path is 800 m long. Beena and Arun run in opposite directions from the same point A on the path at the same time. They continue to run after they meet for the first time. Beena runs another $$53\frac{1}{3}$$​ seconds to arrive at A, while Arun runs another 2 minutes to get to point A.

Which of the following statements is/are correct?

Statement I: The speed of Beena is 6 m/sec.

Statement II: The ratio of the speeds of Arun and Beena is 2 : 3.

Correct option is C

Given:

• Circular track = 800 m
• Beena and Arun run in opposite directions starting from point A at the same time.
• After meeting,
• Beena takes 53⅓ sec = 160/3 sec to return to point A.
• Arun takes 2 min = 120 sec to return to point A.

We need to evaluate:

• Statement I: Beena’s speed = 6 m/s
• Statement II: Ratio of Arun’s speed to Beena’s = 2 : 3

Solution:

If two people start from the same point and run in opposite directions, they meet once their combined distance = full circle (800 m).

Let Beena's speed = B
Let Arun's speed = A

$$\text{They meet after time } t = \frac{800}{A + B} \\[1.5ex]\textbf{After meeting:} \\[1ex]\bullet\ \text{Beena takes } \frac{160}{3} \text{ sec to cover the remaining distance back to A.} \\[0.5ex]\bullet\ \text{Arun takes 120 sec to return to A.} \\[1.5ex]\textbf{So:} \\[1ex]\bullet\ \text{Distance Beena runs after meeting } = B \cdot \frac{160}{3} \\[0.5ex]\bullet\ \text{Distance Arun runs after meeting } = A \cdot 120 \\[1.5ex]\text{These distances sum up to the total circle:} \\[1.5ex]B \cdot \frac{160}{3} + A \cdot 120 = 800 \quad \text{(Equation 1)}$$​​

Let’s try Beena’s speed = 6 m/s (to check Statement I)

Then:

$$\bullet\ \text{Distance Beena runs after meeting} = 6 \cdot \frac{160}{3} = 320 \text{ meters} \\[1ex]\bullet\ \text{So Arun's remaining distance} = 800 - 320 = 480 \text{ meters} \\[1ex]\bullet\ \text{Arun's speed} = \frac{480}{120} = 4 \text{ m/s}$$​​

So, speeds are:

• Beena = 6 m/s
• Arun = 4 m/s

Then the ratio:

• Arun : Beena = 4 : 6 = 2 : 3

This matches Statement II

Final Checks:

Statement I:

Beena’s speed = 6 m/s → True

Statement II:

Arun : Beena = 2 : 3 → True

Final Answer: (C) Both I and II are correct.