Carlo Chan Solves The Escalator Puzzle

My previous post titled An Amusing Challenge has lured out some of the best minds from SLC. Engr. Carlo Chan and Engr. Ludy Aquino, both Board Placers in different years, and prides of SLC Engineering submitted their solutions minutes within each other. Two distinct approaches, one correct answer. I am publishing Carlo’s solution first after which I will append Ludy’s solution after he adds verbal explanations to his math equations so that us, lowly mortals could follow hehehe…

The Office Escalator

It’s 3 o’clock in the afternoon and everyone was in a hurry to get home. The Managing Director was generous enough to let everyone go early for the day, giving them time to prepare for New Year’s Eve. So I hurriedly shuffled the loose pile of paper on my desk and pushed each one into the trays, shut down my laptop and shoved it into its carrying case. “Happy New Year. See you next year, everyone”, I hollered to nobody in particular as I walked to the exits. Work could wait for next year which will start next Sunday! Everyone was filing out of the office into the parking lot. I needed to go, too! I could not wait for the slow descent of the escalator so I walked down even as it descended. It took me 10 steps with my normal gait to get down from the First Floor level to the Ground Floor level. I was about to dash away to the Parking Exits when Mar, looking down from the Second Floor level called my attention. “Hey, Sonny – you dropped your wallet.” In my eagerness, I dropped my wallet at the top of the descending flight that I just traversed. Wanting to get to my wallet as quickly as possible, I forgot to use the ascending flight and instead huff-and-puff my way up the descending flight at a rate which is 5 times my pace in going down. It took me 25 steps before I got to my wallet – thank God. Now Louisian Engineers, pray tell me how many steps has our office escalator?

Carlo’s Solution:

Let N = total number of steps between floors.
E = the number of steps the escalator moves down for every step that I make when
moving down

As I am going down, the moving escalator “helps” to improve my rate such that after only 10 steps, I would have reached the bottom of the escalator. This means that for every step I make, the escalator also “steps down” by E steps and thus enabling me to descend a total of 1/10 of the way or N/10, each time.

1 + E = N/10

Running up against the downward-moving escalator, I have to sprint back 5 times faster than my original speed to overcome the opposing motion of the escalator. But since I am 5 times faster than initially, the escalator could only take away E/5 number of steps for every stride of mine. After 25 steps I would have reached the top which means that for every step taken, I am able to go up a net of 1/25 of the flight or N/25:

1 – E/5 = N/25

From these two equations, one would easily find that N = 20 steps.

QED

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That was elegant! How could you make it so easy, Carlo? Any questions, Class? Hehehe...

- icarus