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.
