Correct Option (A)
The problem requires determining the number of times two individuals, A and B, cross each other while moving in opposite directions around a circular park within a specified time frame.
- A and B commence their walk at 8:00 a.m. from the same point, moving in opposite directions.
- Speed of A = 2 rounds per hour.
- Speed of B = 3 rounds per hour.
- When two entities move in opposite directions, their relative speed is the sum of their individual speeds.
- Relative speed = Speed of A + Speed of B = 2 rounds/hour + 3 rounds/hour = 5 rounds/hour.
- This relative speed indicates that they will cross each other 5 times in one hour.
- The total time interval specified is from 8:00 a.m. to 9:30 a.m., which equates to 1.5 hours (1 hour and 30 minutes).
- In the first full hour (from 8:00 a.m. to 9:00 a.m.), they will complete 5 crossings.
- For the remaining 0.5 hours (from 9:00 a.m. to 9:30 a.m.), the number of crossings is calculated as Relative speed × Time = 5 rounds/hour × 0.5 hours = 2.5 times.
- Since the question asks for the number of times they "shall cross each other" before 9:30 a.m., it refers to the completed crossings within this period. Therefore, in the 0.5-hour interval, 2 full crossings are completed. The 0.5 indicates that they are halfway to completing a third crossing, which would occur after 9:30 a.m.
- Total number of crossings = 5 (in the first hour) + 2 (completed in the subsequent 0.5 hours) = 7 crossings.
Incorrect Options:
Options B (6), C (5), and D (8) are incorrect as they do not accurately represent the total number of crossings based on the calculated relative speed and the precise time interval. Potential reasons for arriving at these incorrect options include:
- Miscalculation of the relative speed, leading to an incorrect rate of crossings per hour.
- Incorrectly determining the total duration of the time interval.
- Failure to account for crossings in the partial hour (0.5 hours) or misinterpreting the fractional crossing (e.g., rounding 2.5 crossings up or down without considering only completed crossings within the specified time).
- Calculating crossings only for a full hour, which would yield 5 crossings (Option C).