Broken Escalator question A kid is at a mall where there are 3 escalators: an up escalator, a down escalator, and a broken escalator. The up and down escalators move at the same speed but in opposite directions. It takes the kid a total of 6 seconds to run up the escalator moving upward. It takes the kid 30 seconds to run up the downward escalator. How long will it take the kid to run up the broken escalator? A 5th grade student asked me this problem and I'm not sure where to even begin. I'm confident that it involves rate, time, and distance, but I'm not certain how. It seems logical to average the times to get 18 seconds but I feel like that is an incorrect solution. Your help is greatly appreciated!

Let $L$ be the unknown length of an escalator, and $s$ the unknown speed of a (working) escalator. And let $q$ be the kid's speed.

Then

\begin{align} \frac{L}{q + s} = 6 \\\ \frac{L}{q - s} = 30 \end{align}

and we want to know the value of \begin{align} \frac{L}{q}. \end{align} Simplifying those first equations gives

\begin{align} L &= 6q + 6s \\\ L &= 30q - 30s \end{align} Multiplying the first by 5 and adding, we get \begin{align} 5L &= 30q + 30s \\\ L &= 30q - 30s \\\ 6L &= 60 q \\\ L &= 10q \\\ \frac{L}{q} &= 10. \end{align}

It'll take 10 seconds.