Math CentralQuandaries & Queries


Subject: Math
Name: Patrick
Who are you: Parent


I was helping my son with a homework problem and came across a very similar on your site:

We are trying to understand the answer shown. There seems to be leap, in the answer that isn't explained or we can't see it. Specifically in the answer it says:

"We know that the woman only walks 21 steps to reach the top, and thus in the time she walks 7 more steps the escalator also goes up 7 steps. Hence the woman and the escalator are travelling at the same rate."

We understand that she still has 7 steps left, but how is it that you can conclude from that (or other factors) that during those 7 steps the escalator will also travel 7 steps.

Thanks in advance for any help you can offer.


Hi Pat.

At this point (the point where the man steps off), she has taken 14 steps and you are given that when she has taken 21 steps, she is finished, so she takes 7 steps more (21-14).

However we also concluded that when she has taken the first 14 steps, she is still 14 steps from the top of the escalator. So there are 14 steps above her. She will take 7 more steps, so 14-7 must be the number of steps the escalator moves her in that same time.

If she moves 7 steps and the escalator moves 7 steps in the same time, they are going the same speed.

Hope this helps,
Stephen La Rocque.


We know that the elevator traveled 7 steps because the woman was 14 steps from the top (where her husband was), but walked only 7 of them; so the missing 7 steps are those that disappeared under the top floor as the motor thingy rolled the steps upwards.

All in all, what makes the problem abstract is that the escalator is moving, so we not only keep track of the relative motion between the husband and wife, but also of their relative motion to the ground floor, the top floor, the initial step (painted red) and also between the red step and the ground floor. So perhaps the problem would be simpler if we stopped the escalator.

So now, we have a good old staircase. The man and his wife start at the bottom at the same time, and instead of the steps moving up, it is the top floor which is coming down towards them. Hmmmm... Come to think of it, this is still a bit too weird. Instead of the top floor, let's use a real person. Furthermore, let's call this person Bob:

''A man walks twice as fast as his wife. They start together at the bottom of a staircase, and at the same Bob starts at the top. When the man meets Bob, he has taken 28 steps. When the woman meets Bob, she has taken 21 steps. How many steps does the staircase have? (Assuming they all walk at constant speeds.)''

Is it easier to work out this way?


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