A person X is running around a circular track completing one round every 40 seconds. Another person Y running in the opposite direction meets X every 15 second. The time, expressed in seconds, taken by Y to complete one round is

Options

A. `12.5`
B. 24
C. 25
D. 55

Answer

To find the time taken by person Y to complete one round on the circular track, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the time taken by X to complete one round:**
Person X completes one round in 40 seconds.

2. **Determine the relative speed of X and Y:**
Since X and Y are running in opposite directions, their speeds will add up when they meet. Let the speed of X be \( v_X \) and the speed of Y be \( v_Y \).

3. **Calculate the speed of X:**
The speed of X can be calculated as:
\[ v_X = \frac{\text{Distance}}{\text{Time}} = \frac{2\pi r}{40} \]
where \( r \) is the radius of the circular track.

4. **Determine the time taken for them to meet:**
They meet every 15 seconds, which means the distance covered by both in that time is equal to one full round of the track.

5. **Calculate the distance covered by X in 15 seconds:**
The distance covered by X in 15 seconds is:
\[ d_X = v_X \times 15 = \left(\frac{2\pi r}{40}\right) \times 15 = \frac{30\pi r}{40} = \frac{3\pi r}{4} \]

6. **Calculate the distance covered by Y in 15 seconds:**
The distance covered by Y in 15 seconds is:
\[ d_Y = v_Y \times 15 \]

7. **Set up the equation for the total distance:**
The total distance covered by both X and Y in 15 seconds equals one full round of the track:
\[ d_X + d_Y = 2\pi r \]
Substituting the distances:
\[ \frac{3\pi r}{4} + v_Y \times 15 = 2\pi r \]

8. **Solve for \( v_Y \):**
Rearranging the equation gives:
\[ v_Y \times 15 = 2\pi r - \frac{3\pi r}{4} \]
Simplifying the right side:
\[ 2\pi r - \frac{3\pi r}{4} = \frac{8\pi r}{4} - \frac{3\pi r}{4} = \frac{5\pi r}{4} \]
Thus,
\[ v_Y \times 15 = \frac{5\pi r}{4} \]
Therefore,
\[ v_Y = \frac{5\pi r}{4 \times 15} = \frac{5\pi r}{60} = \frac{\pi r}{12} \]

9. **Calculate the time taken by Y to complete one round:**
The time taken by Y to complete one round \( T \) is given by:
\[ T = \frac{\text{Distance}}{\text{Speed}} = \frac{2\pi r}{v_Y} = \frac{2\pi r}{\frac{\pi r}{12}} = 2 \times 12 = 24 \text{ seconds} \]

### Final Answer:
Thus, the time taken by person Y to complete one round is **24 seconds**.