Ben started his walk at 9:00 AM and walked at a constant speed of 2 and 1/2 miles per hour. He walked for a total of 4 hours before taking a break to rest and have lunch. After enjoying his meal, Ben continued his journey. However, this time he decided to increase his walking speed by 1/4 mile per hour. Let’s calculate how long it took him to cover the next part of his journey. After walking for some more time, Ben realized that he had been walking for a total of 9 hours since he started his journey.
Calculate the distance he covered.
Calculate how long it took him to cover the next part of his journey.
Calculate how much distance he had covered at this point.
In the first 4 hours, Ben walked at a speed of 2.5 miles per hour for a total distance of 2.5 * 4 = <<2.5*4=10>>10 miles.
After the break, Ben increased his speed to 2.5 + 1/4 = 2.75 miles per hour.
To calculate the time it took him to cover the next part of his journey, we subtract the time he had already walked (4 hours) from the total time (9 hours): 9 - 4 = <<9-4=5>>5 hours.
Therefore, Ben took 5 hours to cover the next part of his journey.
The distance covered during the next part of his journey can be calculated by multiplying his speed (2.75 miles per hour) by the time it took him (5 hours): 2.75 * 5 = <<2.75*5=13.75>>13.75 miles.
At this point, Ben had covered a total distance of 10 + 13.75 = <<10+13.75=23.75>>23.75 miles. Answer: \boxed{23.75}.