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LostInParadise's avatar

Can you solve this math problem in your head?

Asked by LostInParadise (29137points) March 29th, 2021
8 responses
“Great Question” (0points)

Two people a mile apart from each other work out the following plan for meeting each other. One person starts by walking half the distance between them. Then the second person walks half the new distance between them. They keep alternating like this. At what point along the road will they meet?

From the symmetry of the problem, it is tempting to say that they each walk the same distance, but this is obviously incorrect, since the first person starts off by walking half the total distance.

No algebra is required. If you look at the problem the right way, the answer should leap out at you.

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Answers

Zaku's avatar

Yeah, but it’s a trick problem. I won’t give it away.

I will say that I have always felt hostility towards math problems worded as if they are a natural situation, but then want you to do math that fails to model the situation they describe in a reasonable or accurate way. This is a pretty bad case of that, in my opinion.

Tropical_Willie's avatar

OOH I know I know . . . . .

Kropotkin's avatar

Will they be socially distanced?

Kardamom's avatar

I would think never, because each person only walks half of whatever distance there is, and never walks the full distance between them. The halving of the distances make each section tinier and tinier, but theoretically they would never meet.

Reality is another thing. Because humans are not microscopic, the distance would be met as soon as one of the halved distances was smaller than the size of one or both of their feet.

flutherother's avatar

I think I’ve got it but it took a few minutes.

LostInParadise's avatar

@Zaku , I did my best to create some context for the problem.

Here is the reasoning I used to solve it. The first person starts by traveling ½ a mile. The distance is now half a mile and the second person travels ½ of a ½ mile, which is half the distance that the first person traveled. For each round, we can use similar reasoning to show that the second person always travels an additional amount that is half the distance that the first person just traveled. Therefore the total distance that the second person travels is half the total distance that the first person travels. The ratio of the distances is 2:1, so they will meet ⅔ of a mile from the first person.

Zaku's avatar

Ok, but by wording such a problem as if it is about people, I think the problem mis-represents what it says the problem is. As with many math “story problems”, there is a mathematical problem that the author really wants the reader to translate it into. But the wording could be translated differently, in ways that that seem to me as much (or more) reasonable or realistic for the story given.

Also common is to not very clearly state what mathematical aspect the question wants answered, because again it’s hidden by describing the question as if it were natural. In this example, maybe the question is thinking of infinitesimal points and micro-distances, and is going to tell the reader that they never meet, because they’re always only moving half the distance, never the whole distance (even though in reality, people would declare each other to have “met” when they reached whatever distance they considered having met).

And as typical, the question is about behavior and curiosity that almost no one would ever have any real reason or way to do or to ask. Zero effort is put into saying why anyone would ever even think to move half the distance toward another person, nor how they would know the distance of someone a mile away, nor how they’d ever measure or even perceive how far they went (I guess those things are more possible now with smart phones), or why they would care how far each had gone.

Not that the any of the actual math around it isn’t interesting (it is, to me). It’s the gap between what I think when I read about the situation, and what the author expected me to think about the situation, and the displacement of responsibility of communicating that from the author to the student, that bothers me.

Several times in school and university math and physics classes, I tried answering with something like “I am confident that this problem expects me to so standard math problem type 14 here, but the description of the problem I would actually model like [description of a more accurate way to model the problem], which I solve like [elaborate solving of that problem]”, which almost always resulted in zero credit from the teacher.

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