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M. Poirot and Miss Marple walk into a turned-off escalator to realize

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M. Poirot and Miss Marple walk into a turned-off escalator to realize  [#permalink]

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New post Updated on: 29 Aug 2018, 20:10
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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 it 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 on the escalator?

(A) 40
(B) 50
(C) 60
(D) 70
(E) It cannot be determined by the information given

(I met this beautiful problem - wording slightly modified - in 2010. It was posted by Saurabh Goyal.)

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Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!


Originally posted by fskilnik on 29 Aug 2018, 13:23.
Last edited by Bunuel on 29 Aug 2018, 20:10, edited 1 time in total.
Renamed the topic.
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Re: M. Poirot and Miss Marple walk into a turned-off escalator to realize  [#permalink]

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New post 29 Aug 2018, 19:34
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Consider the steps taken to be speed and you will figure out the answer more easily.

Let Poirot (P) have a speed of 3 steps/min (think of steps in terms of distance such as meters)
Let Maple (M) have a speed of 2 steps/min

When the escalator is on, for them to walk on it their speed gets modified with the speed of the escalator which is say x steps/min

So new speed of P = 3+x
New speed of M = 2+x

Now total distance is total number of steps on the escalator or S steps

Now the total number of steps to reach the very end is kind of like saying that it took them the same time as it would take P to take 25 steps or for M to step 20 steps.

we get 2 equations like this (3+x) [speed of P] * (25/3) (time taken for P to cover 25 steps) = S (Total steps of the escalator i.e. distance)

Again for M we get (2+x) * (20/3) = S

Solving the 2 equations we get S = 50
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M. Poirot and Miss Marple walk into a turned-off escalator to realize  [#permalink]

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New post 29 Aug 2018, 23:33
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fskilnik wrote:
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 it 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 on the escalator?

(A) 40
(B) 50
(C) 60
(D) 70
(E) It cannot be determined by the information given

(I met this beautiful problem - wording slightly modified - in 2010. It was posted by Saurabh Goyal.)



Let speed of M.Poirot be p and that of Miss Marple be m
we are given p=3 and m=2
let speed of escalator be e
when p and m are on escalator their respective speeds are 3+e and 2+e

we are also given that time taken by p (tp) = 25/3
and time taken by m (tm) = 20/2 = 10

distance = number of steps = speed * time

distance = (e+3)*25/3 = (e+2)*10 , e= 3 = speed of escalator
distance =50 = number of steps
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M. Poirot and Miss Marple walk into a turned-off escalator to realize  [#permalink]

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New post Updated on: 31 Aug 2018, 09:07
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fskilnik wrote:
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 it 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 on the escalator?

(A) 40
(B) 50
(C) 60
(D) 70
(E) It cannot be determined by the information given

(I met this beautiful problem - wording slightly modified - in 2010. It was posted by Saurabh Goyal.)


Thank you for joining me at my FIRST "new post" here, after "leaving" this wonderful community in 2010/2011 to focus on my online classes (at that time only in Portuguese).

From now on, I hope I will be able to help and learn a lot from you all.

The first solution I would like to present (below) has many elements in common with both solutions posted above. It was created by Rahul (from GuroMe), someone who (at that time) posted some beautiful problems/solutions. We will see an example of his competence (and clearness) below!

(I hope he is well... if someone knows him, please send my best regards to him!)

--------------------------------------------------------------------------------------------------
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 (A = M.Poirot, B = Miss Marple)

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.
--------------------------------------------------------------------------------------------------

I will wait a bit longer to wait for additional contributions before I post my own way of looking into this "Agatha Christie´s mystery". :)

Regards,
fskilnik.
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Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT)
Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!


Originally posted by fskilnik on 30 Aug 2018, 09:33.
Last edited by fskilnik on 31 Aug 2018, 09:07, edited 1 time in total.
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Re: M. Poirot and Miss Marple walk into a turned-off escalator to realize  [#permalink]

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New post 30 Aug 2018, 12:36
fskilnik wrote:
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 it 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 on the escalator?

(A) 40
(B) 50
(C) 60
(D) 70
(E) It cannot be determined by the information given


Let P = Poirot's rate, M = Marple's rate, and E = the escalator's rate.
Since P travels 3 steps for every 2 steps that M travels:
\(\frac{P}{M} = \frac{3}{2}\)

Ratios can be MULTIPLIED:
\(\frac{P}{E} * \frac{E}{M} = \frac{P}{M}\)
Since \(\frac{P}{M} = \frac{3}{2}\), we get:
\(\frac{P}{E} * \frac{E}{M} = \frac{3}{2}\)

We can PLUG IN THE ANSWERS, which represent the number of steps on the escalator.
When the correct answer is plugged in, \(\frac{P}{E}* \frac{E}{M} = \frac{3}{2}\)
Start with an answer choice in the middle (B or C).


Answer choice C: 60 steps
Since P travels 25 steps to reach the top -- and the entire journey is 60 steps -- the escalator must travel 35 steps during P's journey, implying that \(\frac{P}{E} = \frac{25}{35} = \frac{5}{7}\)
Since M travels 20 steps to reach the top -- and the entire journey is 60 steps -- the escalator must travel 40 steps during M's journey, implying that \(\frac{E}{M} = \frac{40}{20} = 2\)
\(\frac{P}{E} * \frac{E}{M} = \frac{5}{7} * 2 = \frac{10}{7}\)
In this case, the resulting ratio is TOO SMALL.


Answer choice A: 40 steps
Since P travels 25 steps to reach the top -- and the entire journey is 40 steps -- the escalator must travel 15 steps during P's journey, implying that \(\frac{P}{E} = \frac{25}{15} = \frac{5}{3}\)
Since M travels 20 steps to reach the top -- and the entire journey is 40 steps -- the escalator must travel 20 steps during M's journey, implying that \(\frac{E}{M} = \frac{20}{20} = 1\)
\(\frac{P}{E} * \frac{E}{M} = \frac{5}{3} * 1 = \frac{5}{3}\)
In this case, the resulting ratio is TOO BIG.


Since C yields a result that is TOO SMALL, while A yields a result that is TOO BIG, the correct answer must be BETWEEN C AND A.



Answer choice B: 50 steps
Since P travels 25 steps to reach the top -- and the entire journey is 50 steps -- the escalator must travel 25 steps during P's journey, implying that \(\frac{P}{E} = \frac{25}{25} = 1\)
Since M travels 20 steps to reach the top -- and the entire journey is 50 steps -- the escalator must travel 30 steps during M's journey, implying that \(\frac{E}{M} = \frac{30}{20} = \frac{3}{2}\)
\(\frac{P}{E} * \frac{E}{M} = 1 * \frac{3}{2} = \frac{3}{2}\)
Success!
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M. Poirot and Miss Marple walk into a turned-off escalator to realize  [#permalink]

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New post 30 Aug 2018, 12:57
fskilnik wrote:
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 it 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 on the escalator?

(A) 40
(B) 50
(C) 60
(D) 70
(E) It cannot be determined by the information given


Algebraic approach:

Let P = Poirot's rate, M = Marple's rate, and E = the escalator's rate.
Since P travels 3 steps for every 2 steps that M travels:
\(\frac{P}{M} = \frac{3}{2}\)

Ratios can be MULTIPLIED:
\(\frac{P}{E} * \frac{E}{M} = \frac{P}{M}\)

Since \(\frac{P}{M} = \frac{3}{2}\), we get:
\(\frac{P}{E} * \frac{E}{M} = \frac{3}{2}\)

Let x = the number of steps on the escalator.

When Poirot travels 25 steps, the number of steps traveled by the escalator = x-25.
Thus:
\(\frac{P}{E} = \frac{25}{x-25}\)

When Marple travels 20 steps, the number of steps traveled by the escalator = x-20.
Thus:
\(\frac{E}{M}= \frac{x-20}{20}\)


Substituting \(\frac{P}{E} = \frac{25}{x-25}\) and \(\frac{E}{M}= \frac{x-20}{20}\) into \(\frac{P}{E} * \frac{E}{M} = \frac{3}{2}\), we get:

\(\frac{25}{x-25} * \frac{x-20}{20} = \frac{3}{2}\)

\(\frac{25}{20} * \frac{x-25}{x-20} = \frac{3}{2}\)

\(\frac{5}{4} * \frac{x-20}{x-25} = \frac{3}{2}\)

\(\frac{x-20}{x-25} = \frac{12}{10}\)

\(\frac{x-20}{x-25} = \frac{6}{5}\)

\(5x - 100 = 6x - 150\)

\(50 = x\)


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If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
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Re: M. Poirot and Miss Marple walk into a turned-off escalator to realize  [#permalink]

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New post 31 Aug 2018, 09:09
fskilnik wrote:
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 it 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 on the escalator?

(A) 40
(B) 50
(C) 60
(D) 70
(E) It cannot be determined by the information given


Hi, Mitch! Thank you for joining here, with two very nice solutions, by the way. (I remember your excellent posts from the "good old times", too. And I see you have not lost your power and competence... marvellous for all the GMAT community!)

Well, let me add still another approach... and start with some previous considerations:

M. Poirot (P) and Miss Marple (M) are "pushed" by the elevator (E) when it is turned-on. The fact time passes for all of them simultaneously (as Rahul wisely mentioned at first), allows us to use (for any given period of time) the direct proportionality of velocity (speed) V and distance (in steps) D... with that in mind, please note that:

If Vp : Ve = 2 : 1 (for instance) , then Dp : De = 2 : 1 and Dp = 2/(1+2) DpUe , I mean, the distance travelled by M.Poirot (Dp =number of steps he really walks) is 2/3 of the distance travelled by "Poirot + escalator pushing him" (in any given period of time).

If (in the general case) we have (say) Vp : Ve = m : n , we may write equivalently Vp : Ve = (m/n) : 1 , therefore, without loss of generality, we may assume we have Vp : Ve = k : 1 (where k is positive, not necessary an integer) and by the explanation given in the previous paragraph, from the fact that M. Poirot walked 25 steps to reach the top,

Statement (1) : 25 = n * k/(k+1) where n is our FOCUS, that is the number of steps in the escalator.

From the fact that Vp : Vm = 3 : 2 , then

Vm/Ve = (Vm/Vp)*(Vp/Ve) = (2/3)*(k/1) = (2k/3):1 , in other words, we can have the ratio Miss Marple´s velocity (speed) to the escalator´s one using Vp as a "bridge" (name used in our method to "connect ratios")...

Hence:

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

Finally, dividing (1) by (2) , we get (after simplifying a "4" with a "2", among other things) 5/2 = (2k+3)/(k+1) and it´s easy to obtain k = 1.

Therefore Vp : Ve = k : 1 = 1 : 1 , in other words, M. Poirot walks 25 steps, while the escalator "pushes him" other 25 steps, and the number of steps of the escalator is 50.

If you prefer, take k = 1 and substitute it in statement (1), for instance, to get 25 = n/2 , of course.

I hope you all enjoyed the problem and all these interesting solutions!

See you in other problems!

Regards,
fskilnik.
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Course release PROMO : finish our test drive till 30/Sep with (at least) 60 correct answers out of 92 (12-questions Mock included) to gain a 70% discount!

Re: M. Poirot and Miss Marple walk into a turned-off escalator to realize &nbs [#permalink] 31 Aug 2018, 09:09
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