Solving a Math Poem

Date: 05/24/2000 at 14:04:06
From: Amber
Subject: Math poem

   Take five times which plus half of what
   And make the square of what you've got.
   Divide by one and thirty square
   To get just four - that's right, it's there.
   Now two more points I must impress:
   Both of which and what are fractionless,
   And what less which is not a lot:
   Just two or three. So now, what's what?

My teacher gave me this problem to try and figure out but every time I 
look at it, it doesn't make sense. I get stuck at the beginning. 
Please help me!

Thank-you very much for your time and effort,
Amber

Date: 05/25/2000 at 11:24:01
From: Doctor TWE
Subject: Re: Math poem

Hi Amber - thanks for writing to Dr. Math.

When I get a problem like this, the first thing I do is read it over 
several times, trying to understand a little bit more of it each time. 
(I had to read this one over three times before I understood what it 
was asking, and then I had to reread parts of it again as I worked it 
out!)

To solve this, you need to realize that 'what' and 'where' are 
numbers, which we'll represent with variables. I'll use X = what and 
Y = where.

Now it helps to write the word problem (poem) in the form of an 
algebraic equation. The problem is that English - especially when used 
poetically - is not as precise as algebra, so some parts are open to 
multiple interpretations. I'll get you started:

     "Take five times which plus half of what,"

This is easy; algebraically it's: 5*Y + (1/2)X

    "...and make the square of what you've got."

Here we have to decide if the word 'what' means the variable X or is 
just part of the expression "what you've got." If we interpret the 
"and make the square of what" as X^2, the "you've got" doesn't make 
any sense. So I interpret this as meaning take the square of the 
expression so far. That's: [5*Y + (1/2)X]^2

     "Divide by one and thirty square,"

Again, this is not clear. This could mean divide by 31^2 = 961, or it 
could mean divide by 1 + 30^2 = 901. I'll leave it to you to figure 
out which interpretation is correct (only one works). Now we have 
either:

     [5*Y + (1/2)X]^2 / (31^2)
or   [5*Y + (1/2)X]^2 / (1+30^2)

     "...to get just four-that's right, it's there."

This completes the equation. We have either:

     [5*Y + (1/2)X]^2 / (31^2) = 4
or   [5*Y + (1/2)X]^2 / (1+30^2) = 4

     "... both of which and what are fractionless,"

This means that X and Y are integers. This is useful information, 
especially if you have to resort to trial-and-error, since you only 
have to try integer combinations.

     "...and what less which is not a lot: just two or three."

This translates to: X - Y = 2 or 3. This gives us four possible 
combinations to try:

     [5*Y+(1/2)X]^2 / (31^2) = 4     and   X - Y = 2
     [5*Y+(1/2)X]^2 / (31^2) = 4     and   X - Y = 3
     [5*Y+(1/2)X]^2 / (1+30^2) = 4   and   X - Y = 2
     [5*Y+(1/2)X]^2 / (1+30^2) = 4   and   X - Y = 3

     "So now, what's what?"

This tells us that they're looking for the value of X. Can you take it 
from there?

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/