The moving walkway issue with the rest 120 feet

[misrapreeti1]

The "˜moving walkway’ is a 300-foot long walkway consisting of a 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 relative to the ground. Once Bill reaches the group of people, he stops walking and stands with them until the walkway ends. What is Bill’s average rate of movement for his trip along the moving walkway?

A. 2 feet per second
B. 2.5 feet per second
C. 3 feet per second
D. 4 feet per second
E. 5 feet per second

The answer seems to be 'E'. And the solution is ->

The group was 120 feet ahead on the walkway when Bill started. Then Bill took 40 sec. to catch up with them, during which time the group continued travelling at the rate of the walkway. Thus, the group moved another (3 ft./sec.)(40 sec.) = 120 feet before Bill caught up. Now the group (with Bill) is 120 feet + 120 feet = 240 feet into the walkway, leaving 300-240 = 60 feet remaining on the conveyor belt.

But my QUESTION is that what happens to the time taken by Bill when he moves an additional 120 feet (that his friends have moved within the 40 secs).

Should not the total time be - 40 + 40 + 20 instead of 40 + 20??

Thanks!!!

[sunny.jain]

I solved the Question like this:-

PPl are moving at the speed of conveyor belt - 3 feet/second
Bill is moving at speed of - 6 feet/second

Relative speed with which Bill is approaching toward ppl = 6 - 3 = 3 feet/second

now distance is = 120 feet
so Bill need 40 seconds to cover this distance with his relative speed --> 120/3 = 40 seconds.

But actual distance bill will travel in these 40 seconds is = 40* 6 = 240
this include the distance the group will travel when bill is approaching toward it.
otherwise bill must have catched them at a distance of 120 feet.

so total time bill will take to reach other end of belt :- 300 - 240 = 60/3 = 20 + 40 = 60 seconds

hence, average speed = 5 feet/second.

*********************************************************
Another approach :-
Bill will cover the distance of 120 feet = 20 seconds <-- 120/6
group will move by addition distance of = 60 feet <--- 3*20

Bill will cover the distance of this 60 feet in = 10 seconds
group will again move by addition distance of = 30 feet

Bill will cover the distance of this 30 feet in = 5 seconds
group will again move by addition distance of = 15 feet

and so on...
so time has a GP sequence -

20 , 10, 5, 2.5 ....
Total time to catch the group =

a(1 - r^n)
-----------
(1 - r)

20( 1 - (0.5)^n)
===> ---------------
(1 - 0.5)

===> (0.5)^n --> we dont know n, but we knows this term will be very small so neglect it.

===> 20 / 0.5 ==> 40 seconds

so bill will catch them in 40 seconds ==> 240 feet

rest is same as first approach.

[rajeev.bajpai]

40 + 20 is right.

Remember that Bill moved at a combined speed of 6 ft/sec and covered 6*40 = 240 ft. in first 40 seconds. He covered remaining 60 ft in 20 sec at 3 ft/second standing still on the conveyer belt. The reason you are confused is because you are missing the point - He CAUGHT up with his friends at 3ft/second RELATIVE to his friends but he actually travelled at 3+3 = 6 ft/second.

[RonPurewal]

The "˜moving walkway’ is a 300-foot long walkway consisting of a 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 relative to the ground. Once Bill reaches the group of people, he stops walking and stands with them until the walkway ends. What is Bill’s average rate of movement for his trip along the moving walkway?

A. 2 feet per second
B. 2.5 feet per second
C. 3 feet per second
D. 4 feet per second
E. 5 feet per second

The answer seems to be 'E'. And the solution is ->

The group was 120 feet ahead on the walkway when Bill started. Then Bill took 40 sec. to catch up with them, during which time the group continued travelling at the rate of the walkway. Thus, the group moved another (3 ft./sec.)(40 sec.) = 120 feet before Bill caught up. Now the group (with Bill) is 120 feet + 120 feet = 240 feet into the walkway, leaving 300-240 = 60 feet remaining on the conveyor belt.

But my QUESTION is that what happens to the time taken by Bill when he moves an additional 120 feet (that his friends have moved within the 40 secs).

Should not the total time be - 40 + 40 + 20 instead of 40 + 20??

Thanks!!!

post33059.html#p33059

sunny's infinite series approach also works, although it is not for the mathematically timid at heart. nicely done.