Explanation:

- Arpita needs to complete homework in five subjects, each taking 40 minutes. So, she needs a total of 5 subjects x 40 minutes = 200 minutes or 3 hours and 20 minutes for all her homework.

- She aims to finish by 10:00 PM. Counting back 3 hours and 20 minutes brings us to 6:40 PM.

- Option:1, 6:30 PM - If she starts at 6:30 PM, she will finish at 9:50 PM. Since this is before 10:00 PM, she will complete in time.

- Option:2, 6:40 PM - Starting at 6:40 PM means she will finish exactly at 10:00 PM, meeting the deadline just right.

- Option:3, 7:10 PM - This start time leads to a finish at 10:30 PM, making her late for the program.

- Option:4, 7:20 PM - Starting at 7:20 PM results in finishing at 10:40 PM, which is well past the deadline.