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Agatha Christie´s Escalator

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Agatha Christie´s Escalator

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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

Answer: (B)

(This beautiful problem was posted by Saurabh Goyal in this forum in 2010. The wording was slightly modified.)

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Portuguese-speakers :: https://www.gmath.com.br

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Let Tp and Tm be time climbing moving escalator for Poirot and Marple and Rp and Rm their respective rates, and Re escalator speed. Based on given info, Poirot's rate is 3/2 or 50% greater than Marple

The distance they each travel is the same, so
Tp(Re + 1.5Rm) = Tm(Re + Rm)

So Tm/Tp = (Re+1.5Rm)/(Re+Rm)

Based on the further info that P and M take 25 and 20 steps in their respective times on the escalator, their rates can be expressed as

25/Tp and 20/ Tm

Since we already know P' s rate is 1.5 times M's rate, we can say

25/Tp = 1.5x20/Tm

So Tm/Tp = 6/5

So 6/5 = Tm/Tp = (Re+1.5Rm)/(Re+Rm)

Cross multiplying
6Re + 6Rm = 5Re + 7.5Rm

Solving for Re indicates = 1.5Rm

1.5Rm also= Rp from before, so Poirot walking and escalator speeds together are twice his natural speed

So Poirot only had to take half as many steps ,25, as he would have, implying there are 50 steps to the escalator

It implies this because if the escalator weren't moving, his total rate would be half, therefore in the same time his 25 steps would take him only half way up the escalator



Last edited by regor60 on Wed Aug 29, 2018 1:43 pm; edited 1 time in total

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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
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 some elements in common with the solution 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 for additional contributions before I post my way of looking into this "Agatha Christie´s mystery". Smile

Regards,
fskilnik.

_________________
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br



Last edited by fskilnik@GMATH on Fri Aug 31, 2018 8:10 am; edited 1 time in total

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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
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:
P/M = 3/2.

Ratios can be MULTIPLIED:
P/E * E/M = P/M
Since P/M = 3/2, we get:
P/E * E/M = 3/2.

We can PLUG IN THE ANSWERS, which represent the number of steps on the escalator.
When the correct answer is plugged in, P/E * E/M = 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 P/E = 25/35 = 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 E/M = 40/20 = 2.
P/E * E/M = 5/7 * 2 = 10/7.
In this case, the result in red 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 P/E = 25/15 = 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 E/M = 20/20 = 1.
P/E * E/M = 5/3 * 1 = 5/3.
In this case, the result in red 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.

The correct answer is B.

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 P/E = 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 E/M = 30/20 = 3/2.
P/E * E/M = 1 * 3/2 = 3/2.
Success!

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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
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:
P/M = 3/2.

Ratios can be MULTIPLIED:
P/E * E/M = P/M .
Since P/M = 3/2, we get:
P/E * E/M = 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:
P/E = 25/(x-25).
When Marple travels 20 steps, the number of steps traveled by the escalator = x-20.
Thus:
E/M = (x-20)/20.

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

25/(x-25) * (x-20)/20 = 3/2

25/20 * (x-25)/(x-20) = 3/2

5/4 * (x-20)/(x-25) = 3/2

(x-20)/(x-25) = 12/10

(x-20)/(x-25) = 6/5

5x - 100 = 6x - 150

50 = x.

The correct answer is B.

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Private Tutor for the GMAT and GRE
GMATGuruNY@gmail.com

If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.

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For more information, please email me at GMATGuruNY@gmail.com.
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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
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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English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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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
Let T and t be the respective times that Poirot and Marple need to travel up the escalator when its running. Let R and r be their respective rates. Let e = the rate of the escalator. Since the distance they travel up the escalator is the same, we can create the equation:

T(R + e) = t(r + e)

Since Poirot walks 3 steps for every 2 steps Marple walks, Poirot’s rate is 1.5 times Marple’s. That is, R = 1.5r. Replacing this into the equation, we have:

T(1.5r + e) = t(r + e)

T/t = (r + e)/(1.5r + e)

Furthermore, since Poirot takes 25 steps to go through the escalator versus Marple’s 20 steps, we have R = 25/T and r = 20/t. Since R = 1.5r, we have 1.5(20/t) = 25/T. This gives us:

30/t = 25/T

T/t = 25/30

T/t = 5/6

So we have:

(r + e)/(1.5r + e) = 5/6

6(r + e) = 5(1.5r + e)

6r + 6e = 7.5r + 5e

e = 1.5r

From this, we can see that e = R. Since Poirot takes 25 steps to travel up the escalator and his rate is same as the escalator’s rate, the escalator must help him with another 25 steps. That is, the escalator has 50 steps when it’s not moving.

Answer: B

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