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Escalator

by goyalsau » Thu Nov 18, 2010 8:51 pm
A and B walk up an escalator (moving straightway).The escalator moves at const speed ,A takes 3 steps for 2 steps of B,A gets to the top of the escalator after having taken 25 steps,while B takes only 20 steps to reach the top. If the escalator were turned off,how many steps would they have to take to walk up?

1)40 2)50 3}60 4)80 5) Can not be Determined.
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by limestone » Thu Nov 18, 2010 11:11 pm
Regardless the speed of the escalator, when A walks 25 steps, B must walk about :
25/3 * 2 = 16.7 steps

So when A finished, B was 25 - 16.7 = 8.3 steps away from A. Thus B needs to walk another 8.3 steps.
However, B walked more only (20 - 16.7)= 3.3 steps. Why? Because the escalator did the missing steps for B.
Thus the escalator moved 8.3 -3.3 = 5 steps.

Now we have the ratios of two pairs of speed:

Speed of A/ that of B: 3:2
Speed of B/ that of the escalator: 3.3:5 or 10:15 or 2:3

Reconcile all the ratio: speed of A/ that of B/ that of the escalator: 3:2:3

As the speed of A = the speed of the escalator, then if the escalator is turned off, A must have moved a doubled number of steps.
Thus the distance A must move is: 25*2 = 50.

Pick B.

Recheck:

A moved 25 steps; the escalator moved 25 steps; total : 50 ( speed of A = speed of the escalator)
B moved 20 steps; the escalator moved 30 steps; total : 50 (speed of B: speed of the escalator = 2:3)
Confirmed.
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by goyalsau » Fri Nov 19, 2010 1:46 am
Great Work,
But i don't i will be able to do a new problem based on this,
limestone wrote:Regardless the speed of the escalator, when A walks 25 steps, B must walk about :
25/3 * 2 = 16.7 steps

So when A finished, B was 25 - 16.7 = 8.3 steps away from A. Thus B needs to walk another 8.3 steps.
However, B walked more only (20 - 16.7)= 3.3 steps. Why? Because the escalator did the missing steps for B.
Thus the escalator moved 8.3 -3.3 = 5 steps.

Now we have the ratios of two pairs of speed:

Speed of A/ that of B: 3:2
Speed of B/ that of the escalator: 3.3:5 or 10:15 or 2:3

Reconcile all the ratio: speed of A/ that of B/ that of the escalator: 3:2:3

As the speed of A = the speed of the escalator, then if the escalator is turned off, A must have moved a doubled number of steps.
Thus the distance A must move is: 25*2 = 50.

Pick B.

Recheck:

A moved 25 steps; the escalator moved 25 steps; total : 50 ( speed of A = speed of the escalator)
B moved 20 steps; the escalator moved 30 steps; total : 50 (speed of B: speed of the escalator = 2:3)
Confirmed.
:roll:
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by limestone » Fri Nov 19, 2010 3:03 am
It seems to be a complex question. I wonder if GMAT will test something like this.

The problem here is majorly related to ratio.

Speed of A : 3 (steps per second for example)
Speed of B: 2
Speed of E: x ( E is the escalator)
We know that A moved 25 steps, B 20 steps. What we need to find here is the ratio between the speed of A or B and that of E. With such a ratio, we can calculate how many steps the escalator had help lifting A or B. For example:
Speed of E is 5. Then E helped A : 5/3*25 = 41.7 steps. E helped B: 5/2 * 20 = 50 steps.

Now apply this to the problem:
When the escalator was on:

Speed of A: 3 + x
Speed of B: 2 + x

A finished after 25 steps, which means the time is : 25/3 = 8.3 seconds. So A had moved:
8.3*(3+x) = 25 + 8.3x steps (this is also the length of the escalator)
At the moment A finished, B had moved:
8.3 * (2+x) = 16.7 * 8.3x steps

At the time A finish, B needed to move more : (25 + 8.3x) - (16.7 + 8.3x) = 8.3 steps
Total steps that B moved is 20, at the time A finished B had moved 16.7 steps. So actually B moved more 20 -16.7 = 3.3
Thus the distance 8.3 - 3.3 = 5 must be the distance that B is moved by the escalator.
Ratio between the speed of B and that of E : 3.3 : 5 = 10 : 15 = 2:3

Now we have the reconciled ratio among three of them :
A:B:E = 3:2:3
The distance E moved A : 25* 3/3 = 25
Total distance A moved ( by itself and by the escalator) = 25 + 25 = 50
The distance E moved B : 20* 3/2 = 30
Total distance B moved: 20 + 30 = 50
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by goyalsau » Fri Nov 19, 2010 4:02 am
Once again thanks a lot for the detailed explanation ,
I would like to know some more things, , :?: :?:
limestone wrote:It seems to be a complex question. I wonder if GMAT will test something like this.

The problem here is majorly related to ratio.

Speed of A : 3 (steps per second for example)
Speed of B: 2
Speed of E: x ( E is the escalator)
We know that A moved 25 steps, B 20 steps. What we need to find here is the ratio between the speed of A or B and that of E. With such a ratio, we can calculate how many steps the escalator had help lifting A or B. For example:
Speed of E is 5.
Then E helped A : 5/3*25 = 41.7 steps. E helped B: 5/2 * 20 = 50 steps.
Please correct me over here, { Speed of A is 3 steps per second , In all he climb 25 climb, Means 25/3 is the the time for climbing 25 steps,
Now if the speed of E is 5 steps per second, then D = S . T
25/3 ( 5 ) = 41.7
I know this is same as your 5/3*25 = 41.7 But i am not to able to understand How u made 5/3 .
If its just the way you right , and the logic is the same then its fine If its not, then please share your views.
[/quote]

limestone wrote: Now apply this to the problem:
When the escalator was on:

Speed of A: 3 + x
Speed of B: 2 + x

A finished after 25 steps, which means the time is : 25/3 = 8.3 seconds. So A had moved:
8.3*(3+x) = 25 + 8.3x steps (this is also the length of the escalator)
At the moment A finished, B had moved:
8.3 * (2+x) = 16.7 * 8.3x steps

At the time A finish, B needed to move more : (25 + 8.3x) - (16.7 + 8.3x) = 8.3 steps
Total steps that B moved is 20, at the time A finished B had moved 16.7 steps. So actually B moved more 20 -16.7 = 3.3
Thus the distance 8.3 - 3.3 = 5 must be the distance that B is moved by the escalator.
Ratio between the speed of B and that of E : 3.3 : 5 = 10 : 15 = 2:3

Now we have the reconciled ratio among three of them :
A:B:E = 3:2:3
Till here things are clear,
Now i know speed of A and speed of E is the same then in all 50 steps are there,
But

The distance E moved A : 25* 3/3 = 25 { i am not able to understand this ratio 3/3 }
What is reasoning I know it may be stupid question, But i am pretty sure when you will be ask me different problem there are some places where i will stuck
limestone wrote: Total distance A moved ( by itself and by the escalator) = 25 + 25 = 50
The distance E moved B : 20* 3/2 = 30
Total distance B moved: 20 + 30 = 50
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by limestone » Fri Nov 19, 2010 5:28 am
Please correct me over here, { Speed of A is 3 steps per second , In all he climb 25 climb, Means 25/3 is the the time for climbing 25 steps,
Now if the speed of E is 5 steps per second, then D = S . T
25/3 ( 5 ) = 41.7
I know this is same as your 5/3*25 = 41.7 But i am not to able to understand How u made 5/3 .
If its just the way you right , and the logic is the same then its fine If its not, then please share your views.
Yeah, it's the same approach. I just made a short cut.
If the speed of A to B is: x:y, then in a same period of time, the distance A will move to that of B is: x:y too.
In the above example:
Speed of A:3
Speed of E: 5
A moved 25 steps, hence E moved: 5/3 * 25 = 41.7
Till here things are clear,
Now i know speed of A and speed of E is the same then in all 50 steps are there,
But

The distance E moved A : 25* 3/3 = 25 { i am not able to understand this ratio 3/3 }
What is reasoning I know it may be stupid question, But i am pretty sure when you will be ask me different problem there are some places where i will stuck
As I said, ratio among A,B,E is : 3:2:3
Thus 3/3 in 25* 3/3 is from such a ratio. It means E shifted A ahead a distance of 25* 3/3 = 25 steps.[/quote]
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by Rahul@gurome » Fri Nov 19, 2010 5:37 am
There is a easier way to solve this problem. Easy to understand at least.
Treat the problem as a speed-distance problem where time is not same for the two cases, but distance is.

Say, the total number of steps = n and the speed of the escalator = x steps/min.
Speed of A = 3 steps/min and speed of B = 2 steps/min

Now A has taken 25 steps. Time taken by A = 25/3 min
Thus, n = (Combined speed of A and escalator)*(25/3) = (3 + x)*(25/3)

Now B has taken 20 steps. Time taken by B = 20/2 min = 10 min
Thus, n = (Combined speed of B and escalator)*(10) = (2 + x)*(10)

So, (3 + x)*(25/3) = (2 + x)*(10)
=> (3 + x)*(25) = (2 + x)*(30)
=> (75 + 25x) = (60 + 30x)
=> 5x = 15
=> x = 3

Replacing x = 3 in any of the individual equation results n = 50.

The correct answer is B.
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by goyalsau » Fri Nov 19, 2010 7:33 am
Rahul i don't want to take the credit from Limestone for all the hard work, But i must say

But you have once again Proved that

! THERE IS ALWAYS A BETTER WAY !
Rahul@gurome wrote:There is a easier way to solve this problem. Easy to understand at least.
Treat the problem as a speed-distance problem where time is not same for the two cases, but distance is.

Say, the total number of steps = n and the speed of the escalator = x steps/min.
Speed of A = 3 steps/min and speed of B = 2 steps/min

Now A has taken 25 steps. Time taken by A = 25/3 min
Thus, n = (Combined speed of A and escalator)*(25/3) = (3 + x)*(25/3)

Now B has taken 20 steps. Time taken by B = 20/2 min = 10 min
Thus, n = (Combined speed of B and escalator)*(10) = (2 + x)*(10)

So, (3 + x)*(25/3) = (2 + x)*(10)
=> (3 + x)*(25) = (2 + x)*(30)
=> (75 + 25x) = (60 + 30x)
=> 5x = 15
=> x = 3

Replacing x = 3 in any of the individual equation results n = 50.

The correct answer is B.
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by fskilnik@GMATH » Mon Nov 22, 2010 11:27 am
Hi there!

I believe limestone´s final arguments (from a very beautiful solution) quoted below
limestone wrote:As the speed of A = the speed of the escalator, then if the escalator is turned off, A must have moved a doubled number of steps.
Thus the distance A must move is: 25*2 = 50.
gave me the insight for a quick, clear and really simple solution... let´s have a look!

Considering n the number of steps (as Rahul defined), Va and Vb the velocities of A and B (when escalator is turned-off), and Ve the escalator´s velocity (all of them in "steps per minute", for sure) limestone told us that:

If Va : Ve = 1:1 , then 25 = n* 1/(1+1) because half the steps were walked by "pure A" and the other half by "escalator pushing A", correct ? In other words: 25 steps by A, another 25 steps by the escalator itself...

What if Va : Ve = 2:1 ? You may think 30 seconds to agree that we will have 25 = n * 2/(2+1) ...

So... let´s begin our solution!!

------------------------------------------------------------------------------------------------------------------------------

If Va : Ve = k:1 (please note that you can always put 1 there, there is no loss of generality... think 30 seconds about it) ? We will surely have

Statement (1): 25 = n * k/(k+1) (This is justified by the "motivation" given before we started!)

We are done! From the fact that Va:Vb = 3:2 (**) then:

Vb / Ve = (Vb/Va)*(Va/Ve) = (2/3)*(k/1) = (2k/3) : 1 :) (Did you think the 30 seconds I´ve asked you before?)

Therefore...

Statement (2): 20 = n * (2k/3) / (2k/3 + 1)

Hence (1) divided by (2) gives us 5/2 = (2k+3)/(k+1) (verify that) and therefore k = 1.

Now I´ll make limestone smile... have a look what happens to Va:Ve = k :1 and Vb:Ve = (2k/3) : 1 when k = 1 and compare to his first post final words:
limestone wrote:A moved 25 steps; the escalator moved 25 steps (Va:Ve = 1:1)
(speed of B: speed of the escalator = 2:3) , that is the same of Vb:Ve = (2/3) : 1
Finally from (say) statement (1) we have: 25 = n * (1/2) then n = 50 as expected... ;)

Regards,
Fabio.

(**) Here I guess the problem should be better stated, because I believe the "most natural" interpretation would be different, and NOT equivalent, by the way: (Va+Ve)/(Vb+Ve) = 3:2 ... think about it... we are IN the escalator, aren´t we?

I created a problem for my students based on this one, with the (**) suggestion implemented. It is below, I hope you like it!

----------------------------------------------------------------------------------------------------------------------------
M. Poirot and Miss Marple walk into a turned-off escalator to realize that he is able to walk 3 steps during the time she needs to walk only 2. When the escalator is turned-on (and is already working at constant speed), M. Poirot enters again the escalator to realize he needs to take 25 steps to go through it, while Miss Marple realizes she needs only 20 steps to reach its end (at the very same conditions). How many steps are there in the escalator ?

(A) 40
(B) 50
(C) 60
(D) 70
(E) It cannot be determined by the information given
----------------------------------------------------------------------------------------------------------------------------
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by limestone » Mon Nov 22, 2010 8:09 pm
Hi,

@Rahul: it's shorter to base on the equation of the escalator's distance to find the number of steps rather than find the ratios between A and the escalator. I didn't think about it and had made a long way. Very nice approach.

@fskilnik, your revised version is much easier to understand. Also your approach to find the ratios among Va, Vb, Ve is really beautiful. :D
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by fskilnik@GMATH » Tue Nov 23, 2010 3:09 am
limestone wrote:@fskilnik, your revised version is much easier to understand. Also your approach to find the ratios among Va, Vb, Ve is really beautiful. :D
I´m REALLY glad you liked it, limestone, because (as I said) I created my solution inspired by your approach/comments. The term "revised version" is really appropriate, therefore I am pleased "my partner" also enjoyed our 4-hands conclusions.

See you in other posts!

Cheers,
Fabio.
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