Rearranging Numbers for an Additive Magic Square & Word

[geekily]

My math book gives me the following table:

Edit: Ooh, no, it doesn't look like the html is working. Let me just type this out then:
140, -56, -42, -28
-14, 70, 56, 28
42, 14, 0, 84
98, 112, 126, -70

It says, "switch two numbers to produce an additive magic square." Before moving anything around, I added up all the rows and columns and diagonals, finding that 140 was the most prevalent answer, and thus, probably the one I was trying to get. All but 5 of them equaled 140: The other answers were 266, 146, 14, 266, and 14. However, I'm not sure where to go from here. Just from playing around with the numbers I found that -56 + 70 + 84 + 42 = 140, for example, but that would require moving two numbers, which would effectively switch four numbers, and besides, it messes up the ones that already equal 140. Is there anyway to go about this other than what seems like a very frustrating, tedious process of guess and check?

The other problem I have is, "Two towns A and B are 3 miles apart. It is proposed to build a new school to serve 200 students in town A and 100 students in town B. How far from town A should the school be built if the total distance traveled by all 300 students is to be as small as possible?" I don't even know where to start with that one, especially because the way I'm interpreting it, the distance for all the students would be as small as possible if it were 1.5 miles away from both towns, but I know that's too easy. The only thing I can think of is that since there are 100 students in town B and 200 in town A, the school should be located 1 mile away from town A, because since town A makes up 2/3 of the school, it should only have to travel 1/3 of the distance. I have no idea if that's right, though, and even if it is, no idea how I would set it up algebraically.

I'd really appreciate your input on either one of these. Thanks so much!

 

[galactus]

For the second problem, it would appear the distance is smallest when they are equal. Set 200x=100(3-x) and solve for x. I hope I am interpreting the problem correctly.

 

[geekily]

Okay, thank you! That makes sense. And it looks like my answer is right, then!

Thanks!

 

[soroban]

Re: Rearranging Numbers for an Additive Magic Square & W

Hello, geekily!

  *-----*-----*-----*-----*
  | 140 | -56 | -42 | -28 |   14
  *-----*-----*-----*-----*
  | -14 |  70 |  56 | .28 |  140
  *-----*-----*-----*-----*
  |  42 |  14 |   0 |  84 |  140
  *-----*-----*-----*-----*
  |  98 | 112 | 126 | -70 |  266
  *-----*-----*-----*-----*
    266   140   140   140

Switch two numbers to produce an additive magic square.

Your reasoning leading to "140" seems to be logical.
Now the incorrect totals appear in the first and fourth rows and the first and fourth columns.
I suspected that swtiching the "corner numbers" might work.

And, sure enough, switching the "-28" and the "98 " is the solution.

Two towns A and B are 3 miles apart.
It is proposed to build a new school to serve 200 students in town A and 100 students in town B.
How far from town A should the school be built if the total distance
traveled by all 300 students is to be as small as possible?

      * - - - - * - - - - - *
      A    x    S    3-x    B

The school (S) is x miles from town A and 3-x miles from town B.

The 200 students in town A will travel a total of \(\displaystyle 200x\) miles.
The 100 students in town B will travel a total of \(\displaystyle 100(3\,-\,x)\) miles.

The total distance is: \(\displaystyle \ \:=\:200x\,+\,100(3\,-\,x)\:=\:100x\,+\,300\)

Obviously, the total distance is a minimum when \(\displaystyle x\,=\,0\)

The school should be built in town A.

 

[geekily]

Thank you so much! I really appreciate all your help. So is the answer to the second question really 0? I guess that makes sense, but the answer of 1 also seemed to make sense. (Following the formula in galactus's answer.)

Thanks again, especially for the first one - I never could have figured that out on my own!

 

[Denis]

Thanks again, especially for the first one - I never could have figured that out on my own!

Well no; you said: "The other answers were 266, 146, 14, 266, and 14."
There is no 146; should be 140.