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?
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 $x=72$.