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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, 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.