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Early train word problem

Video transcript
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, and 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 that 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 plus t minus 20 and 6 minus t, you're going to get 2t minus 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.