# An escalator problem

My older son is working his way through Art of Problem Solving’s Introduction to Algebra book. He works through the problems in each section of the book and we discuss the ones that are giving him trouble. Today he had some difficulty with a problem about an escalator.

The problem is a twist on one of my favorite math contest problems from back when I was in high school – Problem 10 from the 1987 AIME: Here’s our discussion of the version of the of the problem in Chapter 7 of the AoPS’s Introduction to Algebra:

At the end of the last video my son was thinking about trying to find the speed of the sister running up the escalator. Here we turn our attention to a different quantity – how many steps does that sister take?

The main idea in this part was to try to gather together then information that we know in this problem. For example, we know the speed of the two escalators (because he assumed a speed), we know how many steps the escalator has, and we know how long it takes for the sister running up the escalator to get to the top.

To wrap up the project, I asked my son to summarize the solution. We had jumped around a little bit to get to the solution, so I wanted him to do this summary just to make sure that he’d understood the ideas that we’d discussed during the solution: