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Got a Math question

calyxman

calyxman

macrumors 6502a
Original poster
Apr 17, 2005
610
0
While tutoring at the local community college, I ran into a word problem today that I found rather interesting. Here's the question (and I'm paraphrasing the problem to the best of my ability):

A train station during morning rush hour can fill up in as little as 30 minutes. Fortunately, the trains leaving during rush hour are able to empty the station in 40 minutes. At the same rate of rush hour traffic, how long would it take until the station was completely filled?
Click to expand...

Just would like to know the methodology you would use to solve this. Thanks!
 
calyxman said:
While tutoring at the local community college, I ran into a word problem today that I found rather interesting. Here's the question (and I'm paraphrasing the problem to the best of my ability):



Just would like to know the methodology you would use to solve this. Thanks!
Click to expand...

Ok, since it doesn't mention how often trains depart, let us just assume continual departures that would empty a full station in 40 minutes. Also, assume station is initially empty.

Thus, the amount of people who have entered the station by time t (as a percentage of the station capacity) is 100t/30 [Check that at time t=30, the station is 100% full, as described]

Similarly, the amount of people who have left the station on trains by time t (as a percentage of the station capacity) is 100t/40 [Check that at t=40, 100% of the capacity has been removed, as described]

Thus, at time t, the amount of people left in the station (as a percentage of total capacity) is 100t/30 - 100t/40 [People in - people out]

When does the station get full? whenever the above = 100%

thus, solving 100t/30 - 100t/40 = 100

implies that 4t - 3t = 120

Thus t =120 minutes = 2hours
 
If I understand the question correctly (and assuming you start from an empty station), here's how I look at it. (It also includes some assumptions about the rates of ingress and egress remaining constant, which isn't entirely valid, especially in the egress rate.)

Say the capacity of the station is x. In 30 minutes, x people will enter the station. But it take 40 minutes for x people to exit the station, so .75x people exit in 30 minutes. So after 30 minutes, x people have come in and .75x people have left, leaving the station with .25x people inside.

After an hour, you have a total of 2x people having entered, with 1.5x people having left, giving .5x left in the station after an hour. Continuing with this train (pun entirely intended) of thought, the station would be full after 2 hours.
 
WildCowboy said:
If I understand the question correctly (and assuming you start from an empty station), here's how I look at it. (It also includes some assumptions about the rates of ingress and egress remaining constant, which isn't entirely valid, especially in the egress rate.)

Say the capacity of the station is x. In 30 minutes, x people will enter the station. But it take 40 minutes for x people to exit the station, so .75x people exit in 30 minutes. So after 30 minutes, x people have come in and .75x people have left, leaving the station with .25x people inside.

After an hour, you have a total of 2x people having entered, with 1.5x people having left, giving .5x left in the station after an hour. Continuing with this train (pun entirely intended) of thought, the station would be full after 2 hours.
Click to expand...

Seems like we understood the question in the same way :)
 
calyxman said:
While tutoring at the local community college, I ran into a word problem today that I found rather interesting. Here's the question (and I'm paraphrasing the problem to the best of my ability):



Just would like to know the methodology you would use to solve this. Thanks!
Click to expand...

OK, there are some assumptions you have to make:

1. The station would be full in 30 minutes.
2. The first train will be 10 minutes late but because the government changed the leeway on arrival times it still counts as "on time."
3. If the trains are now all late, they will get progressively later.
4. The train broke down on a day when a sporting event was occurring.

So, seeing as all four of those are correct, the train station will in fact fill up in 15 minutes and won't be empty until 4:55pm - the train arriving at 4:55pm actually being the one due to arrive at 3:42pm but seeing as how there is actually a train due at 4:55pm (although that train won't arrive till 6:35pm) everything is now considered "on time."
 
Cool, so I got the same answer. By the way, the assumption is that the station starts off empty.

The method I used was the following: 1/30 - 1/40 = 1/x
 
Chundles said:
OK, there are some assumptions you have to make:

1. The station would be full in 30 minutes.
2. The first train will be 10 minutes late but because the government changed the leeway on arrival times it still counts as "on time."
3. If the trains are now all late, they will get progressively later.
4. The train broke down on a day when a sporting event was occurring.

So, seeing as all four of those are correct, the train station will in fact fill up in 15 minutes and won't be empty until 4:55pm - the train arriving at 4:55pm actually being the one due to arrive at 3:42pm but seeing as how there is actually a train due at 4:55pm (although that train won't arrive till 6:35pm) everything is now considered "on time."
Click to expand...

Hmm, seems like you have managed to get hold of a copy of British Rail's secret operating handbook :)
 
Chundles said:
OK, there are some assumptions you have to make:

1. The station would be full in 30 minutes.
2. The first train will be 10 minutes late but because the government changed the leeway on arrival times it still counts as "on time."
3. If the trains are now all late, they will get progressively later.
4. The train broke down on a day when a sporting event was occurring.

So, seeing as all four of those are correct, the train station will in fact fill up in 15 minutes and won't be empty until 4:55pm - the train arriving at 4:55pm actually being the one due to arrive at 3:42pm but seeing as how there is actually a train due at 4:55pm (although that train won't arrive till 6:35pm) everything is now considered "on time."
Click to expand...

very nice

so anywhere from-30 mins to 2 hours...
 
Chundles said:
OK, there are some assumptions you have to make:

1. The station would be full in 30 minutes.
2. The first train will be 10 minutes late but because the government changed the leeway on arrival times it still counts as "on time."
3. If the trains are now all late, they will get progressively later.
4. The train broke down on a day when a sporting event was occurring.

So, seeing as all four of those are correct, the train station will in fact fill up in 15 minutes and won't be empty until 4:55pm - the train arriving at 4:55pm actually being the one due to arrive at 3:42pm but seeing as how there is actually a train due at 4:55pm (although that train won't arrive till 6:35pm) everything is now considered "on time."
Click to expand...

LOL. I bet you were the one who wrote the answer to this question:

math3.jpg
 
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