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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