Moving Walkway Question

[Guest]

I do not understand the shortcut way to solve this question in the last paragraph. How can we disregard the rate he went for the 1st 120 feet?

The "˜moving walkway’ is a 300-foot long conveyor belt that moves continuously at 3 feet per second. When Bill steps on the walkway, a group of people that are also on the walkway stands 120 feet in front of him. He walks toward the group at a combined rate (including both walkway and foot speed) of 6 feet per second, reaches the group of people, and then remains stationary until the walkway ends. What is Bill’s average rate of movement for his trip along the moving walkway?
2 feet per second
2.5 feet per second
3 feet per second
4 feet per second
5 feet per second

I do not understand the shortcut part of this question at the end below.

To determine Bill’s average rate of movement, first recall that Rate × Time = Distance. We are given that the moving walkway is 300 feet long, so we need only determine the time elapsed during Bill’s journey to determine his average rate.

There are two ways to find the time of Bill’s journey. First, we can break down Bill’s journey into two legs: walking and standing. While walking, Bill moves at 6 feet per second. Because the walkway moves at 3 feet per second, Bill’s foot speed along the walkway is 6 - 3 = 3 feet per second. Therefore, he covers the 120 feet between himself and the bottleneck in (120 feet)/(3 feet per second) = 40 seconds.

Now, how far along is Bill when he stops walking? While that 40 seconds elapsed, the crowd would have moved (40 seconds)(3 feet per second) = 120 feet. Because the crowd already had a 120 foot head start, Bill catches up to them at 120 + 120 = 240 feet. The final 60 feet are covered at the rate of the moving walkway, 3 feet per second, and therefore require (60 feet)/(3 feet per second) = 20 seconds. The total journey requires 40 + 20 = 60 seconds, and Bill’s rate of movement is (300 feet)/(60 seconds) = 5 feet per second.

This problem may also be solved with a shortcut. Consider that Bill’s journey will end when the crowd reaches the end of the walkway (as long as he catches up with the crowd before the walkway ends). When he steps on the walkway, the crowd is 180 feet from the end. The walkway travels this distance in (180 feet)/(3 feet per second) = 60 seconds, and Bill’s average rate of movement is (300 feet)/(60 seconds) = 5 feet per second.

The correct answer is E.

[StaceyKoprince]

First, get some paper so you can draw this out while you read this. Also, make two little markers from paper or two pieces of candy or something.

Draw a line to represent the walkway. Label it 300 feet. Put one of your markers about 180 feet from the end - this is where the crowd is when Bill steps on. Put the other at the beginning (to represent Bill).

Now, let's just do something conceptual - put a finger on each dot and move them along the walkway, but move "Bill" faster so that he catches up with the group before they finish. If we start counting at the moment that Bill steps on the walkway, then the amount of time that Bill spends to get to the end is the same amount of time that the group spends to get to the end, even though Bill travels farther than the group.

I need to calculate the distance that Bill travels and divide it by the time it takes him to travel. The distance is easy - 300 feet. The time I could calculate two ways: the amount of time it takes Bill or the amount of time it takes the group. The two times are equal.

As it happens, it's easier to calculate the group's time b/c the group stays at a steady speed. The group has to go 180 feet at 3 feet/sec, so it takes the group 60 sec to get to the end. Remember that we start to count this time at the point at which Bill steps on, so Bill also takes 60 sec to get to the end. All I need is total distance / total time, and now I have both of those numbers.