How much do the 5 bales weigh?

[Turtle]

OK, here's the scoop. A math tutor friend called me tonight with a problem brought in by a student. Without going too far into the details, we got off on a procedural discussion for using matrices to solve the problem and left off with no solution and each of us to work it over more.

So, I have worked it to death and by means other than matrices I have what I think is an answer, if not a solution. So, without further ado, here's the deal:

5 bales of hay are weighed two at a time, in each possible pairing, producing the following table. The question then posed is, "what is the weight, in pounds, of each of the 5 bales of hay?"

a+b=110 lbs.
a+c=112
a+d=113
a+e=114
b+c=115
b+d=116
b+e=117
c+d=118
c+e=120
d+e=121

i have concluded that the table is in error.

[Qfwfq]

The first useful thing to realize is that the weight of all 5 bales must be the sum of those 10 weights divided by 4. Let's call this total T. The weight of a single bale can be gotten by subtracting the weight of the other 4 from T, which can be done by subtracting the sum of the right two pairs. If we subtract a+b=110 and c+d=118 for instance, we are left with the weight of e.

[Qfwfq]

Hmmmmm... I tried a bit of checking, it seems as if the values given are not consistent (not just any set of 10 values will be). Can't check much better now though.

[Buffy]

Using simple substitution you can obtain:

a+b+c+2d+e=110+118+121
a+b+c+2d+e=349
a+b+c+2d+e-(a+b)=349-110
c+2d+e=239
c+2d+e-(c+e)=239-120
2d=119
d=59.5

a+d=113
a+59.5=113
a=113-59.5
a=53.5

b+d=116
b+59.5=116
b=56.5

c+d=118
c+59.5=118
c=58.5

d+e=121
59.5+e=121
e=61.5

But then substituting them back you get the following two inconsistencies:

a+e=115 (not 114)
b+e=118 (not 117)

So as Q suspected, this set of equations is inconsistent.

Will cause lots of puzzling I'm sure!

Warning: my arithmetic is totally sucky (that's why I went into computers!), so YMMV....

And so you have thirteen tens, And you take away seven, And that leaves five...Well, six actually, but the idea is the important thing,
Buffy

[Turtle]

Roger Q & B. If you highlight my first post you will find my concurrance at bottom right justified.

So, among our conclusions now we have that the table is purposefully in error and the student is expected to reply with our answer, or somewhere along the chain from instructor to student to tutor to me to you, the table was copied wrong.

Assuming a correct table, do you also agree with me that the Gaussian method of matrices is workable as well as back-substitution?

[CraigD]

Using matrix notation and simple Gauss–Jordan elimination,
transforms to

The 8th and 9th rows show the original system of equations to be inconsistent – that is, to have no real solution.

I suspect a misprint has occurred, and the last lines should be
c+e=119
d+e=120

The system is then consistent, with solutions: a=53.5, b=56.5, c=58.5, d=59.5, e=60.5

This problem is in the “too much information” family – you only need 5 independent equations to solve it, but are given 4 additional dependent ones. This can confuse many humans and computer programs, but can still be solved via the usual G-J reduction by simply ignoring rows consisting of all zeros, and concluding that the system of equations is inconsistent if any rows have zeros in all columns but the rightmost.

[Turtle]

Originally Posted by CraigD

Using matrix notation and simple Gauss–Jordan elimination,
1 1 0 0 0 -110
1 0 1 0 0 -112
1 0 0 1 0 -113
0 1 1 0 0 -115
0 1 0 1 0 -116
0 1 0 0 1 -117
0 0 1 1 0 -118
0 0 1 0 1 -120
0 0 0 1 1 -121
transforms to
1 0 0 0 0 -53.5
0 -1 0 0 0 56.5
0 0 -1 0 0 58.5
0 0 0 0 -1 60.5
0 0 0 0 0 0
0 0 0 1 0 -59.5
0 0 0 0 0 0
0 0 0 0 0 -1
0 0 0 0 0 -1

(Apologies for the ugly matrixes – I can’t seem to get them to render in LaTeX )

The 8th and 9th rows show the original system of equations to be inconsistent – that is, to have no real solution.

I suspect a misprint has occurred, and the last lines should be
c+e=119
d+e=120

The system is then consistent, with solutions: a=53.5, b=56.5, c=58.5, d=59.5, e=60.5

This problem is in the “too much information” family – you only need 5 independent equations to solve it, but are given 4 additional dependent ones. This can confuse many humans and computer programs, but can still be solved via the usual G-J reduction by simply ignoring rows consisting of all zeros, and concluding that the system of equations is inconsistent if any rows have zeros in all columns but the rightmost.

Roger. What we did with the matrices is to simply choose 5 of the 10 equations and then constructed a 5x6 augmented matrix to use in row operations. My tutor friend is out sick today so the student is on their own if today is the deadline.

[InfiniteNow]

Originally Posted by CraigD

(Apologies for the ugly matrixes – I can’t seem to get them to render in LaTeX )

You might try wrapping it in code tags like this [code]matrix in here[/code], which allows you to preserve the spaces you've typed when submitting the post.

Originally Posted by CraigD

Using matrix notation and simple Gauss–Jordan elimination,

1 1 0 0 0 -110
1 0 1 0 0 -112
1 0 0 1 0 -113
0 1 1 0 0 -115
0 1 0 1 0 -116
0 1 0 0 1 -117
0 0 1 1 0 -118
0 0 1 0 1 -120
0 0 0 1 1 -121
transforms to
1  0  0  0  0  -53.5
0 -1  0  0  0   56.5
0  0 -1  0  0   58.5
0  0  0  0 -1   60.5
0  0  0  0  0    0
0  0  0  1  0  -59.5
0  0  0  0  0    0
0  0  0  0  0   -1
0  0  0  0  0   -1

[Turtle]

Originally Posted by InfiniteNow

You might try wrapping it in code tags like this [code]matrix in here[/code], which allows you to preserve the spaces you've typed when submitting the post.

While arguably your post gives the appearence of aid, it reflects a lack of understanding of the problem and procedure inasmuch as those little 'dashes' (-) are not minus signs in the upper list right column. Written as you have it in the code tags, all 10 equations are wrong before any operations. Thanks for nothin'.

[InfiniteNow]

Originally Posted by Turtle

While arguably your post gives the appearence of aid, it reflects a lack of understanding of the problem and procedure inasmuch as those little 'dashes' (-) are not minus signs in the upper list right column. Written as you have it in the code tags, all 10 equations are wrong before any operations. Thanks for nothin'.

Always a pleasure Turtle. I do what I can. Your continued kindness and humility are appreciated.

Also, where is it wrong? As you can see, Craig has updated his post with latex and it looks like what I posted, except left to right instead of up to down. While being condescending is one thing, being able to teach others is quite another.