hikstac0

2022-10-08

Renata walks down an escalator that moves up and counts 150 steps. Her sister Fernanda climbs the same escalator and counts 75 steps. If the speed of Renata (in steps per time unit) is three times the speed of Fernanda, determine how many steps are visible on the escalator at any time.

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Marcel Mccullough

Expert

Let:
- $x$ be the number of visible steps.
- $y$ be the speed of the escalator (in steps per time unit).
- $z$ be the speed of Fernanda (in steps per time unit), relative to the escalator.
Consider Renata's trip. Notice that in the time that Renata takes to finish her trip, she has travelled $150$ steps at a speed of $3z$, while the escalator has travelled $150-x$ steps at a speed of $y$. This yields:
$\frac{150}{3z}=\frac{150-x}{y}$
Now consider Fernanda's trip. Notice that in the time that Fernanda takes to finish her trip, she has travelled $75$ steps at a speed of $z$, while the escalator has travelled $x-75$ steps at a speed of $y$. This yields:
$\frac{75}{z}=\frac{x-75}{y}$
Dividing the first equation by the second yields:
$\begin{array}{rl}\frac{\frac{150}{75}}{\frac{3z}{z}}& =\frac{\frac{150-x}{x-75}}{\frac{y}{y}}\\ \frac{2}{3}& =\frac{150-x}{x-75}\\ 2\left(x-75\right)& =3\left(150-x\right)\\ 2x-150& =450-3x\\ 5x& =600\\ x& =120\end{array}$
as desired

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