timing word problem

Beer soaked ramblings and correction follows.
jonah2.0 said:
Let ...
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Let's put it in a real figure.
Let the distance from Lake Geneva to Chicago be 83 miles.

distance from lake geneva and chicago - Google Search

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At 5 minutes, the Speedy Coach bus that left 5 min. earlier that Travis must first pass has a headstart of (5/60)*83/(90/60) or 4 11/18 miles. Since Leisure Tour buses take 60 minutes to reach Chicago, it follows that its speed or rate is 83/(60/60) or 83 mph. The first Speedy Coach bus that Travis passes is at the point when (5 min. headstart) + 83/(90/60)*x = 83*x where x is the time when Travis is side by side with the bus with the 5 min. headstart. This is the time when both buses have covered the same distance which is 13 5/6 miles.

The 2nd Speedy Coach bus that Travis must pass has a head start of (5 + 10) min. or 13 5/6 miles. They will be side by side at the point when (5 + 10) min headstart + 83/(90/60)*x = 83*x. This is the time when both buses have covered the same distance which is 41.5 miles (halfway through Chicago).

The 3rd Speedy Coach bus that Travis must pass has a head start of (5 + 10*2) min. or 23 1/18 miles. They will be side by side at the point when (5 + 10*2) min. headstart + 83/(90/60)*x = 83*x. This is the time when both buses have covered the same distance which is 69 1/6 miles (5/6 of the way through Chicago).

Travis cannot pass the 4th bus which has a headstart of (5 + 10*4) min. or 32 5/18 miles since it can cover the remaining 50 13/18 miles (83 - 32 5/18) in 33 22/27 minutes [(50 13/18)/(90/60)] while Travis is still at the 46 1259/1620 miles point in 33 22/27 minutes.
 
JeffM said:
The buses leave every 10 minutes

0
10
20
30
40

How many are higher than 0 but lower than 35?
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i'm with you with the 10 minute interval setup. but you lost me with "How many are higher than 0 but lower than 35?" how did you know to make 35?
 
if i wanted to practice more problems like this, what particular word problem would this be called? it's not the usual algebra problems i'm used to (like systems of equations) so it threw me for a loop.
 
eric beans said:
if i wanted to practice more problems like this, what particular word problem would this be called? it's not the usual algebra problems i'm used to (like systems of equations) so it threw me for a loop.
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Where did you find it?
 
eric beans said:
if i wanted to practice more problems like this, what particular word problem would this be called? it's not the usual algebra problems i'm used to (like systems of equations) so it threw me for a loop.
Click to expand...
Great question.

Your original question was how to think about solving problems. And learning how to solve problems is the value of learning mathematics for everyone who is not a mathematician (me for example).

It is a fact that there are frequently many ways to solve a problem. Some people find one way more natural; others find a different way more natural.

But before plunging into equations, etc, it is always valuable to think. That helps you determine what tools may be valuable. Lev thought about relative speeds, and that led to algebraic tools. I thought about relative times and that led to counting. Both were successful. Neither is better in the abstract.

The quantitative problems you will face in practice do not come with labels like “simultaneous equations” or “counting.” You will have no clue what techniques may be helpful (or even whether the problem is soluble by any technique you already know). What you will always know is what do you need to find out and what information do you already have. It is always helpful to spend some time thinking about how you might be able to get from the known information to the desired information.

Obviously, virtually any problem assigned in an algebra class can be solved by some UNKNOWN combination of algebraic techniques. But the way to start thinking about any mathematical problem is to grasp what is known and what needs to be known and a plan to go from one to the other. I admit that a great way to do well on tests is to memorize a technique that always works on a problem of a recognizable type, but it does not work whenever you run across a type you do not recognize.

True story. About four or five months ago, my son introduced me to a game. After working at it for a few hours, I said that there must be some kind of algebra to solve the game easily. He did not know one, but agreed that one must exist. A day or so ago, I found out that Claude Shannon had figured out in the 1930’s that Boolean algebra, developed in the early 1800’s, was the correct algebra for such problems. Without being skilled in the algebra and without knowing that it applied, I still succeeded at the game. It just took me a long time, over days in fact, to do so. Recognition of patterns is great, but not needing to rely on patterns to solve problems is even greater.

I fear that too much of math education suggests that doing math is to: (1) recognize the type of problem , (2) memorize one technique for each type, and (3) plug and chug. That is not math. That is not problem solving.
 
Beer soaked correction follows.
jonah2.0 said:
...
Travis cannot pass the 4th bus which has a headstart of (5 + 10*4) min. or 32 5/18 miles since it can cover the remaining 50 13/18 miles (83 - 32 5/18) in 33 22/27 minutes [(50 13/18)/(90/60)] while Travis is still at the 46 1259/1620 miles point in 33 22/27 minutes.
Click to expand...
That should have been
...
Travis cannot pass the 4th bus which has a headstart of (5 + 10*3) min. or 32 5/18 miles since it can cover the remaining 50 13/18 miles (83 - 32 5/18) in 55 minutes
{i.e., [83-(5+10*3)/60*r]/r*60 where r=83/(90/60)}
while Travis is still at the (55/60)*83 or 76 1/12 miles point in 55 minutes.
 
Beer inspired ramblings follow.
jonah2.0 said:
That ...
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Generalizing, with [MATH]d>0[/MATH], [MATH]n\ge1[/MATH] (nth bus to be passed), and [MATH]x[/MATH] is a fraction of an hour and `0<x\le1`, we have

`(5+10(n-1))/60*d/(90/60)+d/(90/60)*x=d*x`

Or

`(5+10(n-1))/60*(2*d)/3+(2*d)/3*x=d*x``Leftrightarrow``x=(2*n-1)/6`

n=1 `\implies` x=1/6 or 10 minutes (Travis passes 1st bus with 5 min. headstart in 10 minutes)
n=2 `\implies` x=1/2 or 30 minutes (Travis passes 2nd bus with 15 min. headstart in 30 minutes)
n=3 `\implies` x=5/6 or 50 minutes (Travis passes 3rd bus with 25 min. headstart in 50 minutes)
n=4 `\implies` x=7/6 or 70 minutes (Travis cannot pass the 4th bus with 35 min. headstart as it will be on the finish line or Chicago in 55 minutes while Travis is still at the `55/60``d` point in 55 minutes)
 
JeffM said:
I believe students are way too quick to look for formulas and equations. The hardest part of a problem usually involves understanding what we want to know and how to use what we do know to get there.
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Yep.
 
easy problems are easy because i can see how to get to the end. it's like trying to cross a small stream and you see some stones you can walk over to get to the other side without getting your feet wet.

hard problems seem impossible because i can't see any possible way to get to the other side. it's like trying to cross a massive river rapid with no foreseeable path to the other side. at the onset it looks next to impossible. no matter how hard i swim, i get washed away. it's so frustrating because i'm going about it all wrong and i don't know why.

eventually, the beauty is when i see a hard problem become an easy problem because i see the moves to get to the other side of the river no matter how scary the river looks. those are the most enjoyable.

sometimes those moves are crazy complicated though.

my question is how do you learn to look at an impossible river that seems to have no possible way to cross it and find a way to cross it? how do you do it without picking up bad habits that work against your thinking and growth? how do you learn properly? surely some ways are worse than others? what are the good habits to help me find ways to the solution?



JeffM said:
Great question.

Your original question was how to think about solving problems. And learning how to solve problems is the value of learning mathematics for everyone who is not a mathematician (me for example).

It is a fact that there are frequently many ways to solve a problem. Some people find one way more natural; others find a different way more natural.

But before plunging into equations, etc, it is always valuable to think. That helps you determine what tools may be valuable. Lev thought about relative speeds, and that led to algebraic tools. I thought about relative times and that led to counting. Both were successful. Neither is better in the abstract.

The quantitative problems you will face in practice do not come with labels like “simultaneous equations” or “counting.” You will have no clue what techniques may be helpful (or even whether the problem is soluble by any technique you already know). What you will always know is what do you need to find out and what information do you already have. It is always helpful to spend some time thinking about how you might be able to get from the known information to the desired information.

Obviously, virtually any problem assigned in an algebra class can be solved by some UNKNOWN combination of algebraic techniques. But the way to start thinking about any mathematical problem is to grasp what is known and what needs to be known and a plan to go from one to the other. I admit that a great way to do well on tests is to memorize a technique that always works on a problem of a recognizable type, but it does not work whenever you run across a type you do not recognize.

True story. About four or five months ago, my son introduced me to a game. After working at it for a few hours, I said that there must be some kind of algebra to solve the game easily. He did not know one, but agreed that one must exist. A day or so ago, I found out that Claude Shannon had figured out in the 1930’s that Boolean algebra, developed in the early 1800’s, was the correct algebra for such problems. Without being skilled in the algebra and without knowing that it applied, I still succeeded at the game. It just took me a long time, over days in fact, to do so. Recognition of patterns is great, but not needing to rely on patterns to solve problems is even greater.

I fear that too much of math education suggests that doing math is to: (1) recognize the type of problem , (2) memorize one technique for each type, and (3) plug and chug. That is not math. That is not problem solving.
Click to expand...
 
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First, some problems may be impossible to solve; the Goldbach conjecture may be such. Certainly Euclid’s fifth postulate was impossible to “solve.”

Second, some problems are very hard to solve. It took mathematicians over 300 years to solve Fermat’s Last Theorem.

So there is no reason to beat yourself up if you can’t solve a problem.

Third, there are two techniques that are frequently helpful. One is to make up and solve much simpler but much simpler problems in the hope of finding a general solution to problems of that type. Another is to ask yourself what would I need to know to make this problem easy to solve. You then work backwards

It would be easy to cross this river if I knew where there was a ford.

OK. How might I locate a ford? There would be a road leading to it.

How might I find the nearest road? Ask that shepherd driving his sheep.

And so on.

But my main point was and is to think about what you do know and what you want to know before writing down formulas and equations
 
i'm not very good at math but i enjoy it. i had a rough time with chi square. if i like graphing lines and parabolas and pictures, what branch of math would you suggest i go into?


JeffM said:
First, some problems may be impossible to solve; the Goldbach conjecture may be such. Certainly Euclid’s fifth postulate was impossible to “solve.”

Second, some problems are very hard to solve. It took mathematicians over 300 years to solve Fermat’s Last Theorem.

So there is no reason to beat yourself up if you can’t solve a problem.

Third, there are two techniques that are frequently helpful. One is to make up and solve much simpler but much simpler problems in the hope of finding a general solution to problems of that type. Another is to ask yourself what would I need to know to make this problem easy to solve. You then work backwards

It would be easy to cross this river if I knew where there was a ford.

OK. How might I locate a ford? There would be a road leading to it.

How might I find the nearest road? Ask that shepherd driving his sheep.

And so on.

But my main point was and is to think about what you do know and what you want to know before writing down formulas and equations
Click to expand...
 
I understand what this question is trying to ask, but I feel like the reader is supposed to make several assumptions that aren't explicitly stated. Is this the exact phrasing of the question, or is there more?
 
I'm also a little confused about the phrasing "as he arrives in Chicago" because that implies to me that we're looking at who is he passing upon his arrival, not on his whole journey.
 
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