1

I was going through old past papers and came across this question with only the answer and no explanation. Here is the question:

An escalator moves up at a constant rate. Chloe takes 90 steps to walk down the escalator. Jacob takes 45 steps to walk up the escalator. Given that Chloe walks 3 times as fast as Jacob, how many steps has the escalator?

I’ve tried to use their speed ratios of $3:1$ and apply it to how many steps they take, but I just realized I’m using the variables $c$, $j$, and $r$ for the speeds of Chloe, Jacob, and the rate of the escalator respectively.

I also tried to assign constants $t_1$ and $t_2$ to signify the time taken to go up the escalator for Chloe and Jacob, but that’s too many unknown variables.

The answer was 72. How do I get to this answer?

Tyrcnex
  • 572

1 Answers1

1

We let the total number of steps on the escalator be $x$, the speed of the escalator be $e$ and the speed of Jacob be $j$.

In the time it took Jacob to climb up the escalator he saw 45 steps and also climbed the entire escalator. Therefore the contribution of the escalator is an additional $x - 45$ steps. Since Jacob and the escalator were both moving at a constant speed over the time it took Jacob to climb, the ratio of their distances covered is equivalent to the ratio of their speeds, so $\frac{e}{j} = \frac{x - 45}{45}$.

Similarly, in the time it took Chloe to walk down the escalator she saw 90 steps, so the escalator must have moved $90 - x$ steps in that time. Thus $\frac{e}{3j} = \frac{90 - x}{90}$

Equating the two values of $\frac{e}{j}$, we have $\frac{x - 45}{45}=\frac{90 - x}{30}$, so $x=72$.

meta_warrior
  • 3,288
  • 18
  • 34
  • 1
    Thanks for your answer! However, I don’t really get the part when you say that Jacob “saw” 45 steps and Chloe “saw” 90 steps. How can the two people “see” a different amount of steps on the same escalator? – Tyrcnex Jul 15 '21 at 09:30
  • 1
    The meaning of "seeing" in this context is the number of steps which they perceived to take for travelling to the other end of the escalator. – meta_warrior Jul 15 '21 at 09:32
  • 1
    So, the meaning of “seeing” in this context is the number of steps they “see” while travelling the escalator? Not at the start? – Tyrcnex Jul 15 '21 at 09:36
  • 1
    @Tyrcnex, yes correct! – meta_warrior Jul 15 '21 at 23:25
  • 1
    Anyway this seems like an old Math Olympiad question to me! – meta_warrior Jul 16 '21 at 00:25
  • Good judgement! It is a Math Olympiad past paper! – Tyrcnex Jul 16 '21 at 10:40