Today's math problem

[pokerfan]

Posted this in Kali's engineering problem thread on GD but no bites...

Burt works in the city and lives in the suburbs with Ernie. Every afternoon, Burt gets on a train which arrives at the suburban station at exactly 5pm. Ernie leaves the house and drives so as to arrive at the station at exactly 5pm to pick up Burt. The route that Ernie drives never changes and his driving speed is constant.

One day Burt gets to leave early and catches an earlier train arriving at the suburban station at 4pm instead of 5. Instead of calling Ernie to ask for an earlier pick-up, Burt decides to get some exercise and begins walking home along the route that Ernie drives knowing that eventually he will meet Ernie and they will ride home together in the car.

This is what happens and Burt ends up arriving at home ten minutes earlier than normal. Assuming that Burt's walking speed is constant and no time is lost on the pick-up, for how many minutes does Burt walk?

[NightWatcher]

1. why are you prying into Burt and Ernie's private life? it's really none of our business

[pokerfan]

2. LOL

[freshwest]

3. Calvin and I think alike. I hated those kinds of questions.

[pokerfan]

6. I absolutely loved them

I assumed that was the purpose of mathematics. I was an odd kid. Despite my DU avatar, I was/am much more like Oliver Wendell Jones from Outland than Calvin. I aspire to be more Calvin-like.

[freshwest]

24. I always think that there are about three answers to every question, which isn't good for numbers.

Extremely verbal.

[pokerfan]

26. Yeah, there's that

and sometimes I would exploit that. If I was facing a problem on an exam that had me stuck, I would sometimes "interpret" the problem into one I could solve. I would at least score some partial credit because after all it was just an honest mistake.

Another little trick: suppose I didn't know how to get from A to B. I would work down the page from A and up the page from B on the bottom into a jumbled, confused mess in the middle and hope for the best. Hope whoever's grading the test is in a hurry. :)

[madinmaryland]

4. Burt works?? I never knew that...

In fact, I never knew that they even leave their apartment.

:shrug:

[pokerfan]

5. You must be thinking of Bert and Ernie

This is Burt and Ernie. Completely different couple. :rofl:

[southerncrone]

15. LOL!

Good. I'd hate to work w/Bert!
:rofl:

[pokerfan]

18. I don't think Ernie would be much of a picnic either

http://www.youtube.com/watch?v=NShgvtEro7I (R rated for language)

[krispos42]

32. Oh my!

:rofl:

[rug]

7. Here it is

[pokerfan]

9. You know, that's actually part of the answer

and a triangle is involved, though not necessarily a right triangle.

[pintobean]

8. That's a fargin' trick question!

[fishwax]

10. I'd say he walks for

55 minutes.

[pokerfan]

12. Check out the big brain on fishwax

You're a smart motherfucker. That's right. Fifty-five minutes. Can you show your work?

http://2.bp.blogspot.com/-E_545fSoEFk/TVtMbbMeNYI/AAAAAAAABzI/U1W-gnnVMtw/s400/Samuel+L+Jackson.jpg

[fishwax]

17. well, my work isn't conventional

I just figured that the easiest way to solve it was if he met Ernie halfway. I guess my work would look something like this, though I intuited most of it:

Normal Driving time = x
Walking time = y
walk/drive = z

Z = 60 + (x-10)
z = 50 + x

Assuming they met halfway, then z = y +.5x; therefore,

y + .5x = 50 + x
y - 50 = .5x

That means 50 minutes is too short and obviously 60 minutes is too long, since then they wouldn't meet at all, so 55 minutes makes sense. I tested that with a few other scenarios and it seemed to work.

I'm sure there's a better way, though, so I'd be happy if you let me know :) :hi:

[pokerfan]

19. Correctamundo

Though there is a non-algebraic method that I thought was really clever:

http://www.democraticunderground.com/discuss/duboard.php?az=show_mesg&forum=105&topic_id=9746092&mesg_id=9746224

[Kali]

11. hmmm

50 minutes? no that doesn't allow for any driving time

he gets to the station an hour early, but only gets home 10 minutes early


arrrggghhhh (I actually used to like word problems too, but I am old now)

[pokerfan]

13. did you try drawing a graph?

Distance (home/station) on the vertical and time (4pm/5pm) on the horizontal?

[Kali]

14. yes I tried

but it didn't help (I went with T and T-10) for the times so obviously I was doing something wrong

It's 55? why? I would think it has to be less than 50.

[pokerfan]

16. Here's the answer from the Paul Zeitz DVD

ETA: fixed some typos.

I made some screen grabs for illustrative purposes. It's from this video series by Paul Zeitz but was originally posed years ago by Martin Gardner, Scientific American.

http://www.amazon.com/Art-Craft-Mathematical-Problem-So...

Well worth it. Anyway, while I couldn't find a clip to share (except on bit torrent) I managed to find a transcript. The names were changed to foil the googlers...

Now, the question is: Is there even enough data? We don't know the speed that Ernie is going. We don't know the speed that Burt walks. Well, it turns out it doesn't matter, and one way to organize the little data that we have is by drawing a picture. Let's try adding a picture to this problem, and the natural thing to do is a distance-time graph. So here's a graph where the horizontal axis is time; I'm labeling it 4:00 and 5:00. The vertical axis is position; H stands for home, and S stands for the station. Here's a typical day. This is what Ernie does. Ernie leaves some time before 5:00. We don't know when, so I start at an arbitrary starting time. Then Ernie travels in a straight line because it's constant speed from H to S, arriving at S, the station, at exactly 5:00 and then reverses the path. So what you get is something geometrically nice, an isosceles triangle. We don't know the slope of the triangle. If Ernie went fast, it would be very steep. We just don't know that. Now, on the day of the power failure, Burt has arrived at the station at 4:00, and Burt begins walking toward home at a much lower speed than a car, so it's some kind of a shallower sloped line. The green line is Burt's progress, and those red lines are Ernie's. Burt will walk until intercepting with Ernie. Once that occurs, then they head back, going at the car speed. That blue line is the new path, which is parallel to the original red path. That's the picture. So pictorially that's what's happened, and what we know is when that blue line finally hits the bottom of the graph, the time axis, that would mean that Ernie and Burt have gotten home. What we know for sure, is that that is going to be 10 minutes earlier than the normal arrival time. What we need to find out is how much time Burt has walked, and so what we need to know is how much before 5:00 it was when the interception occurred



So let's blow up the picture and look at the top of that triangle. If you look at it, that thick, black line stands for the 10 minutes before the normal time of getting home. We drop a perpendicular down, and it bisects that because it's an isosceles triangle. It bisects that thick, black line, and so we know that's going to be 5 minutes.



Yet if we drop that down, we see that the interception time had to have been 5 minutes before 5:00. So what we deduce is that Burt walked for 55 minutes. This was done without algebra, just done with pictures.



Now, the pictures are a little complicated, but they're fun. The key thing is that we've been able to convert a word problem into the geometry of parallel and perpendicular lines. Now, is it always worth it to ask the question: Can we draw a picture and if so, how? Maybe it's not always worth it, but in general, it's a good idea to look at a problem and think is there some way we can force this into a visual mode because at least it will loosen you up.

[fishwax]

20. that's quite interesting

:thumbsup:

[Kali]

23. some how I intuitively got a triangle for the regular car travel

but not any of the rest of it - pretty cool (and in my ranch/imperfect world 45, 50, 55 are all pretty close, nevermind the flat tire Ernie had to change at the last minute:rofl: )

[Ptah]

25. The sentence that made all the difference for me:

"The vertical axis is position"

[JVS]

21. I've seen this problem before but my memory is dim. 55 minutes.

If Burt gets home 10 minutes early, that means that he has been picked up 5 car-minutes (car minutes is a distance, not unlike light-year) from the station (because if he had not been walking the car would spend 5 minutes more getting to the stations and 5 minutes more on the trip back in order to total the normal time. But that doesn't tell us how long he has been walking without figuring a bit more.

If he's moved 5-car minutes, that means that he's getting picked up at 4:55. Thus he has been walking for 55 minutes.

[pokerfan]

22. Yep

See post #16 for the pretty pictures. Basically the way it works is 5 is half the 10 minute wide base of an isosceles triangle. Isosceles, because Ernie drives the same speed in both directions.

[pokerfan]

30. yeah

...if he had not been walking the car would spend 5 minutes more getting to the stations and 5 minutes more on the trip back in order to total the normal time.

And that symmetry is expressed in the tip of the isosceles triangle. It's so elegant to see it illustrated graphically.

[FloridaJudy]

27. So he walks for 55 minutes to get home 10 minutes early?

Fuck it. Stay at the station and order a beer.

[JVS]

29. The first time I saw the problem it was about a man getting picked up by his wife...

and the instructor said that the problem had been cleaned up from being un-PC. My buddies and I took it upon ourselves to figure out what had been un-PC about it. Our conclusion was that either the wife drove like the proverbial "bat out of hell" or that the man walking was either crippled, slowed by obesity, or possibly belonging to an ethnicity considered lazy.

[HopeHoops]

28. You left out a crucial piece of information - how many bars are on Burt's route?

[krispos42]

31. If B&E get home 10 minutes earlier, then Bert's walk saves...

...5 minutes of car travel to the station and 5 minutes of car travel back home.

Therefore, Ernie picks up Bert at 4:55pm instead of 5:00pm. Since Bert left the station at 4pm, Bert walks for 55 minutes.