Give these Word Problems a Try

For the discussion of math. Duh.

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taisuke
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Give these Word Problems a Try

Postby taisuke » Wed Jan 20, 2010 5:04 pm UTC

Ok, so we had this quiz. At first it didn't seem difficult (it's just good ol algebra) but these problems are actually not a piece of cake. Somehow, I got a passing mark for this :mrgreen: . Let's see you guys answer these word problems, hehehe :D . I'm sure you can do it but really, it's not as easy as they seem. Just put the representations, equations and answers, it might be too difficult to encode your complete solutions :D

Image

Goodluck!

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Re: Give these Word Problems a Try

Postby skeptical scientist » Wed Jan 20, 2010 5:22 pm UTC

Two words: Gaussian elimination. These problems are all extremely straightforward if you know the algorithm: write down an equation for each condition, and use a simple algorithm to solve the resulting system of linear equations. There are actually some ancient puzzles which are of exactly this sort. For example, from Mahavira:
Three merchants find a purse lying in the road. One merchant says, "If I keep the purse, I shall have twice as much money as the two of you together."
"Give me the purse, and I shall have three times as much as the two of you together," said the second merchant.
The third merchant said, "I shall be much better off than either of you if I keep the purse; I shall have five times as much as the two of you together."
If there are 60 coins of equal value in the purse, how much money does each merchant have?

Puzzles of this type can be quite tricky when you first encounter them, but once you know how to solve them, they are all equally simple.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

achan1058
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Re: Give these Word Problems a Try

Postby achan1058 » Wed Jan 20, 2010 6:09 pm UTC

I will go with my 1 word answer of "Maple". Failing that, Gaussian elimination is indeed the way to go.

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Re: Give these Word Problems a Try

Postby skeptical scientist » Wed Jan 20, 2010 6:39 pm UTC

achan1058 wrote:I will go with my 1 word answer of "Maple". Failing that, Gaussian elimination is indeed the way to go.

That's not really solving them though - it's asking a computer to solve them for you.
I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

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Re: Give these Word Problems a Try

Postby DavCrav » Wed Jan 20, 2010 7:49 pm UTC

skeptical scientist wrote:
achan1058 wrote:I will go with my 1 word answer of "Maple". Failing that, Gaussian elimination is indeed the way to go.

That's not really solving them though - it's asking a computer to solve them for you.


Gaussian elimination is the same, whether Maple does it or you do it. Computers are a wonderful invention in that they meant that I don't have to perform algorithms again. :D

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Re: Give these Word Problems a Try

Postby achan1058 » Wed Jan 20, 2010 8:12 pm UTC

skeptical scientist wrote:
achan1058 wrote:I will go with my 1 word answer of "Maple". Failing that, Gaussian elimination is indeed the way to go.

That's not really solving them though - it's asking a computer to solve them for you.
It's no problem if I program Gaussian myself, no? :wink:

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Re: Give these Word Problems a Try

Postby Yakk » Wed Jan 20, 2010 9:28 pm UTC

skeptical scientist wrote:
achan1058 wrote:I will go with my 1 word answer of "Maple". Failing that, Gaussian elimination is indeed the way to go.

That's not really solving them though - it's asking a computer to solve them for you.

True, using tools someone else made is cheating.
So no using a computer. And paper and pencil that someone else built? Pretty cheesy.
Come to think of it, most flat surfaces and sharp objects where manipulated/created by something else. So you'll have to learn how to make stone tools.
Stone, however, only exists because our sun collapsed out of interstellar matter. So some stellar engineering is needed.
The raw materials that went into the solar system come from an earlier nova, which star formed because of galaxy formation.
The galaxy was formed by flux left over from cosmic expansion in the big bang.
Hmm. Big bang.

In order to solve an equation from scratch, you must first create the universe.
One of the painful things about our time is that those who feel certainty are stupid, and those with any imagination and understanding are filled with doubt and indecision - BR

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Syrin
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Re: Give these Word Problems a Try

Postby Syrin » Wed Jan 20, 2010 9:44 pm UTC

I thought that only applied to apple pies.

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Re: Give these Word Problems a Try

Postby squareroot1 » Wed Jan 20, 2010 11:35 pm UTC

Ohh, the universe exists because someone wanted apple pie.

It all makes sense now. *Does some math.*

I know have a 3 line proof of how to square a circle, but have to go check on some apple pie in the oven.

Mmmm, pie.

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Re: Give these Word Problems a Try

Postby the tree » Thu Jan 21, 2010 11:10 am UTC

achan1058 wrote:
skeptical scientist wrote:
achan1058 wrote:I will go with my 1 word answer of "Maple". Failing that, Gaussian elimination is indeed the way to go.

That's not really solving them though - it's asking a computer to solve them for you.
It's no problem if I program Gaussian myself, no? :wink:
The issue is whether you understand the algorithm or not, if you can honestly say that given enough time and Red Bull you could do exactly what the computer is doing (which you could, this is not a big* system we're talking about) then I don't see the problem. All you're doing is using a really big calculator and that's okay. Bear in mind that Maple does all kinds of different stuff depending on what it thinks it's looking at, behind the scenes it decides whether it's dealing with integers or floating points and assigns memory appropriately, then it pivots the matrix before getting down with the elimination - even then there are subtly different algorithms to pick from.

If you're only just learning then a decent hybrid between getting a computer to do it for you is putting the matrix into maple and then doing individual row operations (the command is something like RowOperation - check the help file) so you don't have to get bogged down by adding up numbers and whatever but it's still you at the helm.

*big isn't the right word is it? urm... convoluted? what's a word for having many parts?

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Re: Give these Word Problems a Try

Postby jestingrabbit » Thu Jan 21, 2010 12:04 pm UTC

If you slavishly follow the gaussian elimination algorithm when you row reduce, you'll take a lot longer to get there than someone who's done a few dozen examples by hand. Its one thing to know the algorithm, its another to know how it works so that you can arrive at the destination by a better path.

<pompous> In the exam, will you have a big enough calculator? You will... if your brain is your calculator! </pompous>
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Re: Give these Word Problems a Try

Postby DavCrav » Thu Jan 21, 2010 12:23 pm UTC

jestingrabbit wrote:<pompous> In the exam, will you have a big enough calculator? You will... if your brain is your calculator! </pompous>


That's why you should take pure mathematics linear algebra, rather than applied! :)

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Re: Give these Word Problems a Try

Postby jestingrabbit » Thu Jan 21, 2010 2:53 pm UTC

DavCrav wrote:
jestingrabbit wrote:<pompous> In the exam, will you have a big enough calculator? You will... if your brain is your calculator! </pompous>


That's why you should take pure mathematics linear algebra, rather than applied! :)


Goddammit, what sort of a wuss are you? I was doing gaussian elimination when I was in highschool. Not because I was some sort of wunderkind, but because that is what the class was doing. Its piss easy.

And, you know, hands on experience with matrices led me to being able to readily identify sub algebras, have a real feeling for KAN decomposition, and a bunch of other things that were very much pure.

/rant

its not hard, its not something to avoid at all costs, just do it and stop wussing out.

okay now we're /rant
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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Re: Give these Word Problems a Try

Postby the tree » Thu Jan 21, 2010 3:29 pm UTC

jestingrabbit wrote:If you slavishly follow the gaussian elimination algorithm when you row reduce, you'll take a lot longer to get there than someone who's done a few dozen examples by hand. Its one thing to know the algorithm, its another to know how it works so that you can arrive at the destination by a better path.
That's kind of why I suggested using a computer but only a little bit - you need to see the algorithm working.

jestingrabbit wrote:<pompous> In the exam, will you have a big enough calculator? You will... if your brain is your calculator! </pompous>
But outside of an exam situation you may as well use all the resources at your disposable to eliminate (heh) any possible errors. Also knowing how to use a computer is a fairly important skill as well.

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Re: Give these Word Problems a Try

Postby jestingrabbit » Thu Jan 21, 2010 11:46 pm UTC

the tree wrote:But outside of an exam situation you may as well use all the resources at your disposable to eliminate (heh) any possible errors.


If you've got 100+ variables, or really 5+, then sure, use the machine, but these are two or three. You should be able to scribble out a solution in the time it takes for you to confirm syntax, input the equations and interpret the results.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.

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Re: Give these Word Problems a Try

Postby achan1058 » Fri Jan 22, 2010 1:49 am UTC

jestingrabbit wrote:
the tree wrote:But outside of an exam situation you may as well use all the resources at your disposable to eliminate (heh) any possible errors.


If you've got 100+ variables, or really 5+, then sure, use the machine, but these are two or three. You should be able to scribble out a solution in the time it takes for you to confirm syntax, input the equations and interpret the results.
Try doing this for a class of 50, with 1/2 of them getting it wrong.

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Re: Give these Word Problems a Try

Postby DavCrav » Fri Jan 22, 2010 2:07 am UTC

jestingrabbit wrote:
DavCrav wrote:
jestingrabbit wrote:<pompous> In the exam, will you have a big enough calculator? You will... if your brain is your calculator! </pompous>


That's why you should take pure mathematics linear algebra, rather than applied! :)


Goddammit, what sort of a wuss are you? I was doing gaussian elimination when I was in highschool. Not because I was some sort of wunderkind, but because that is what the class was doing. Its piss easy.

And, you know, hands on experience with matrices led me to being able to readily identify sub algebras, have a real feeling for KAN decomposition, and a bunch of other things that were very much pure.

/rant

its not hard, its not something to avoid at all costs, just do it and stop wussing out.

okay now we're /rant


1) If you are solving a 2-dimensional system, you should just be able to look at it and work out the answer.
2) For three variables, not being a monkey and using Gaussian elimination is normally quicker, because you can generally use elimination of variables to do things much more speedily.
3) For strictly between 3 and infinitely many variables, I would use a computer. I would probably use one for three, because I have done Gaussian elimination in school, and don't need to prove to anybody else that I can do it. Besides, it is deeply boring, like implementing all algorithms by hand, and has very little going for it in the way of being illuminating, once you have got everything out of undergraduate linear algebra that there is to get.

jestingrabbit wrote:You should be able to scribble out a solution in the time it takes for you to confirm syntax, input the equations and interpret the results.


Seriously? I think it really wouldn't take me very long to type the matrix into Magma, invert it, and multiply it by a vector. I think I could probably type the matrix into Magma at close to the same speed as you can write it down yourself. I did Q4 on the sheet in exactly the same amount of time on the computer as it took me to construct the system, simply by typing it into the computer rather than writing it down.

At some point, the old-school mathmos need to realize that computers are game-changing inventions. Look at what happened to art after the advent of photography. The thing about maths is that we can already do things that computers cannot do, but we need to start delegating computations to computers. Think about it this way: if you performed a Gaussian elimination, and a computer did it, and you got different answers, which one would I believe?

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Re: Give these Word Problems a Try

Postby achan1058 » Fri Jan 22, 2010 2:15 am UTC

DavCrav wrote:At some point, the old-school mathmos need to realize that computers are game-changing inventions. Look at what happened to art after the advent of photography. The thing about maths is that we can already do things that computers cannot do, but we need to start delegating computations to computers. Think about it this way: if you performed a Gaussian elimination, and a computer did it, and you got different answers, which one would I believe?
Depends on who programmed the Gaussian elimination. :wink: I would believe most likely believe neither if I did the work, and I programmed the Gaussian.

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Re: Give these Word Problems a Try

Postby Qaanol » Fri Jan 22, 2010 6:22 am UTC

Make the necessary representations of the unknown quantities and formulate a system of linear equations for each problem and find the solution set.

1. A man has 3 sums of money invested, one at 4%, one at 5%, and one at 3%. His total yearly income from the three investments is P1,400. The last investment yields P100 more per year than the other two combined. If he could receive 1% more on each investment, his annual income would be increased by P400. How much does he have invested at each rate?

2. I have a positive two-digit number. I then reverse the digits to obtain a new number. I then get the sum of these two numbers, and divide the result by the sum of the digits of either number but not both. a) What is the quotient? b) What is/are the possible original number/s?

3. In a PTA fund raising activity, couples who came were charged P40, however, if only one parent came, he/she was charged P25. Fifty-nine people attended the said activity and P1,215 was raised. How many couples attended the activity?

4. A water tank is supplied buy pipes A and B and emptied by pipe C. If the pipes are open, the tank can be filled in 15 hours. If the tank is full and pipes A and C are open, the tank can be emptied in 10 hours. If the tank is full and pipes B and C are open, the tank can be emptied in 6 hours. If pipe A supplies 50 more liters a minute than pipe B and the capacity of the tank is 45,000 liters, find the number of liters that pass through each pipe per minute.

5. The sum of the reciprocals of two numbers is 9. Three times the reciprocal of the first is one-third greater than four times the reciprocal of the second. Find the numbers.

6. In an algebra test, 30 more students passed than failed. In the next test, 7 students who passed in the first test failed and one-third of those who failed passed. As a result, 31 more students passed the second test than failed on it. What was the record of passing or failing on the first test?

7. A contractor worked for 7 days after which he sought the help of his son and together they finished a painting job in 5 days. Had he sought the help of his son 3 days after he started working, they would have finished the job in 7 days. How long would it take each to finish the job alone?
[sic]

I choose to disregard the initial instructions.

Spoiler:
1: Clearly 1% of the total amount invested is 400, so the total is 40,000. The total income is 1,400 and the last investment makes 100 more than the other two combined, so its earnings are 750 and the other two sum to 650. Since 750 is earned at 3%, a total of 25,000 is invested there, leaving 15,000 split between the other two. Each of those earns at least 4%, so that's 600 base income. The remaining 50 has to come from the extra income on the 5% account, and thus constitutes 1% of the total there, which is then 5,000. That leaves 10,000 for the 4% account.

2: 10A + B becomes 10B + A when reversed, the digits of either of which sum to A+B. The sum of the number and its reverse is 11(A+B), which leaves a quotient of 11 when divided by the sum of digits of the original number or its reverse. The digit B can be anything 0-9, while A must be 1-9, not 0, since 10A + B is a 2-digit number. Thus the original number is an integer at least 10 and no more than 99.

3: Each person contributes at least 20, which for 59 people makes 1,180. That leaves another 35 that must come from the singles, at 5 a pop, meaning 7 people were not in a couple. Thus 52 people were in couples, so there were 26 couples, assuming there were only singles and couples contributing to the total.

4: A+B-C=45,000/(15*60)=50, A-C=-45,000/(10*60) = -75, B-C = -45,000/(6*60) = -125, A = B+50, gives us 4 equations in 3 variables. Luckily the fourth one follows directly by subtracting the second from the third. Anyway, subtracting the second equation from the first gives B = 125. Then C = 250, and A = 175. Units are liters per minute as requested.

5: Let A and B be the reciprocals of the desired numbers. A+B=9 and 3A=(4/3)*4B, so 9(9-B)=16B, meaning B = 81/25. Then A = 144/25, meaning the numbers are 25/144 and 25/81.

6: We do not know how many more or fewer students took the second test than took the first, whether by adding or dropping the course, or missing a test. Incomplete problem statement means there is not a unique correct answer.

7: Assuming the job is perfectly parallelizable, 7A + 5(A+B) = 3A + 7(A+B) = C, so 2A = 2B, meaning they work at the same rate. They would each take 17 days to finish the job alone.
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Fluid_Dynamic
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Re: Give these Word Problems a Try

Postby Fluid_Dynamic » Sat Jan 23, 2010 6:16 pm UTC

Qaanol wrote:
I choose to disregard the initial instructions.

Spoiler:
6: We do not know how many more or fewer students took the second test than took the first, whether by adding or dropping the course, or missing a test. Incomplete problem statement means there is not a unique correct answer.


Spoiler:
6:Your answer makes more sense than if you assume the same number of students took the exam.


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