More equation practice
Early Train Word Problem Fun word problem that is almost a brain teaser.
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- This is another problem that Kortaggio sent me,
- and I like these problems a lot
- because they seem fairly simple on the surface,
- and actually the solutions are pretty simple.
- But just the nature of the lack of information they give,
- it becomes kind of hard to even get started with these problems.
- And I'll be frank, this problem, I kind of stumbled
- around with it for a couple of minutes until I finally realized
- what they were asking for or how to solve for it.
- So in this problem, we have Bev,
- and she takes a train home at 4 o'clock.
- She arrives at the station at 6 o'clock.
- Every day, driving at the same rate,
- her husband meets her at the station.
- Fair enough.
- One day, she takes the train an hour early and arrives at 5:00.
- Her husband leaves home to meet her at the same usual time.
- Husband leaves home to meet her at the usual time.
- So maybe an interesting thing is to think about
- when does the husband normally leave?
- So let's say that this is a normal scenario.
- I'll do that in green.
- At 6 o'clock, she arrives.
- We got there.
- She arrives at the station at 6 o'clock.
- I don't know if this 4 o'clock is useful yet.
- And the husband, he gets there right when she gets there.
- So he traveled-- he gets there, and I don't know if I'm--
- maybe I should draw a more--
- well, let me just draw it this way.
- I think this is how my brain kind of handles it.
- So he gets there.
- And when did he have to leave?
- What was his time?
- So let's say it takes him t minutes to get there,
- whatever t is given in.
- So if he gets there at 6 o'clock,
- that means he left at 6 o'clock minus t.
- If t is 30 minutes, it would be 5:30.
- If t is one hour, it would be 5 o'clock.
- So leaves at 6 o'clock minus t.
- And t, whatever units t happens to be.
- Then he picks her up, and what does he do?
- He goes back home, right?
- He goes back home.
- Let me do that with a-- I'll do that with a skinnier line.
- He goes back home, same distance.
- We assume that the car just turns around immediately,
- and that there's no time devoted to picking her up.
- She just kind of jumps on the car as he spins around.
- So it should take him the same amount of time, right, t.
- So when does he arrive back at home?
- So if it took him-- he left t minutes before 6 o'clock and
- got there, so when does he get back?
- Well, it's going to take him t minutes to get back,
- so he's going to arrive at 6 o'clock plus t.
- And so what was the total amount of time
- that he traveled?
- And this is almost superficially easy.
- Well, if you-- 6 o'clock plus t minus 6 o'clock minus t,
- the difference in time is just 2t.
- He traveled a distance of-- or a total time of 2t minutes,
- and that's almost obvious.
- If it takes him t minutes to go and t minutes to come back,
- he traveled t minutes.
- Fair enough.
- Now, what happened on this day?
- It says that he leaves-- her husband leaves home
- to meet her at the usual time.
- So once again, he is going to leave-- he leaves--
- I'll do this in red.
- He leaves at 6 o'clock minus t, right?
- And then, she arrived early.
- So she's going to be walking back.
- So there's going to be some smaller time
- than it takes him to reach her.
- So let's just leave that unlabeled right now.
- So if I have a line here--
- so he's going to travel a shorter distance,
- and, of course, it's going to take him less time
- because she started walking.
- That's what the problem tells us, right?
- Today she arrives at 5 o'clock, and she begins to walk home.
- So she's going to make some distance up
- so he's not going to have to travel quite as far.
- And then whatever time that was,
- he goes back the other way.
- It's going to be the same amount of time.
- It's the same amount of time.
- And what does it tell us?
- He meets her on the way,
- and then they arrive home 20 minutes early.
- So normally, they arrive at 6 o'clock plus t, right?
- That's normal.
- So today, they're going to arrive 20 minutes early.
- So they're going to arrive at, you could say, 6 o'clock
- plus t-- that's just that-- minus 20 minutes, right?
- They're going to arrive 20 minutes early.
- So what's the total amount of time
- that he would have traveled on that day?
- The total amount of time?
- Well, he essentially travels 20 minutes less than
- he does on a normal day, right?
- He leaves at the same time and he gets there 20 minutes early.
- And if you take the difference between 6+t-20 and 6-t,
- you're going to get 2t-20.
- And that's almost--
- I probably didn't even have to draw all of this.
- You could just say, well, you know, on a regular day,
- it takes them two-- if t is the amount of time,
- it normally takes him to go to the station,
- on a normal day, he travels 2t minutes.
- Today, he leaves at the same time, gets 20 minutes early,
- so he travels 2t minus 20 minutes.
- Fair enough.
- Now, what does that tell us?
- Well, how long did it take him to get to her?
- How long did this take her?
- Well, to go to pick her up takes the same amount of time
- as to come back, so he must have taken half of this time to go
- and half of this time to come back.
- So it must have been t minus 10 to go,
- and then t minus 10 to come back.
- So it took him 10 minutes less to reach her this time.
- So what time does he pick her up?
- And this is the key.
- So normally, when it takes him t minutes,
- he gets there at 6 o'clock.
- This time she started walking because she got there early,
- and he gets there 10 minutes--
- he reaches her 10 minutes earlier than
- he normally would have reached her, right? t minus 10.
- Normally, he reaches her at 6 o'clock.
- Today he reaches her 10 minutes earlier.
- So he reaches her at 5:50.
- Now, the question is, how many minutes did Bev walk?
- How many minutes did Bev walk?
- Remember, she arrived an hour early and arrives at 5 o'clock,
- and then she starts walking.
- She arrives at 5:00, 5 o'clock, and just starts walking.
- And when does he pick her up?
- He picks her up at 5:50.
- So she walked for 50 minutes.
- That's a neat problem.
- Because on some level, it's very easy,
- but on a lot of other levels,
- they give you all of this other information that's not necessary.
- For example, you don't have to know that
- she leaves at 4 o'clock. That's actually probably
- the most unnecessary piece of information.
- But everything else is kind of left abstract.
- But even though you can--
- they're not giving a lot of details.
- You can actually figure out how far she walked
- without knowing how fast she walks,
- or how fast her husband drives,
- or how far they are from the train station,
- or how far the house is from the train station, or any of that.
- You're still able to figure out how far she walked.
- Anyway, neat problem.
- Thanks again to Kortaggio.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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