Guess and Check Tables, Algebra

Date: 10/12/2002 at 00:45:58
From: B.b.
Subject: About Guess and Check Tables...

Hey Dr. Math! Can you explain this question for me? I don't understand 
it.

In science class, Hector decides to measure the mass of his math, 
science, and Spanish textbooks. He discovers that the mass of his 
Spanish text is five more than four times the mass of his math book, 
while the mass of his science text is 110 grams less than his Spanish 
book. Combined, the mass of the texts is 1,178 grams. Find the mass of 
each book.

A friend said to subtract 105 grams from the mass of the Spanish book. 
I didn't get it because the problem says '110 grams less.' She
calculated and finally, her answers were: The math book is 142 grams,
the Spanish book is 573 grams and the science book is 463 grams. Add
them all together and get 1,178 grams. Is she right?

Date: 10/12/2002 at 15:22:55
From: Doctor Ian
Subject: Re: About Guess and Check Tables...

Hi,

We can check her answer by comparing it to the original problem 
description. First, the weights _do_ add up to 1178 grams. So the 
answer is plausible.

If the math book is 142 grams, then the Spanish book should be 5 more 
than 4 times that, or 

  5 + 4(142) = 5 + 4(100 + 40 + 2)

             = 5 + 400 + 160 + 8

             = 573 grams

So it's looking pretty good. And the science book should be 110 grams 
less than this, or 473 grams. So it looks like she's right.

On the other hand, her instructions to you don't make sense to me, 
either.  

To solve this with a table, you would make places to guess the masses 
of the various books:

  Math   Spanish   Science   Total
  ----   -------   -------   -----

Suppose you guess that the math book is 50 grams.  

  Math   Spanish   Science   Total
  ----   -------   -------   -----
    50

Once you've chosen that, the mass of the Spanish book is fixed:

  Math   Spanish      Science   Total
  ----   ----------   -------   -----
    50   4*50 + 5
           ^
           |
       5 more than 4 times the
       mass of the math book

And if the mass of the Spanish book is known, the mass of the 
science book is also fixed:

  Math   Spanish      Science             Total
  ----   ----------   -----------------   -----
    50   4*50 + 5     (4*50 + 5) - 110    
                            ^
                            |
                    110 grams less than
                    the Spanish book

Now, suppose we add these all up.  We could go ahead and do the 
arithmetic, to get

    50 + 4*50 + 5 + 4*50 + 5 - 110

  = 50 + 200 + 5 + 200 + 5 - 110

  = 410 - 110

  = 300 

Which is way too low. But we can do something clever, which is to 
do only _some_ of the arithmetic:
  
    50 + 4*50 + 5 + 4*50 + 5 - 110

  = 50 + 4*50 + 4*50 + 5 + 5 - 110

  = 50*(1 + 4 + 4) + 10 - 110

  = 50*9 - 100

Now, this is interesting, because we know where the 50 came from. 
It was our original guess! So we could just use _this_ expression to 
check our other guesses:

  Math = 100, total = 100*9 - 100

                    = 800  (still too low)

  Math = 200, total = 200*9 - 100

                    = 1700 (too high)
and so on.  

Later on, when you get into algebra, you'll see how you can avoid 
guessing altogether by using a name instead of a number to work 
backward from the answer:

  guess + 4*guess + 5 + (4*guess + 5 - 110) = 1178

                              guess*9 - 100 = 1178

                                    guess*9 = 1278

                                      guess = 1278/9

                                            = 142

Don't worry if you couldn't follow what I just did there. It's sort of 
like a preview of a movie that you won't get to see until later.  

The main thing to remember now is that although you've normally been 
taught to combine numbers whenever you see them, sometimes there are 
good reasons to hold off doing that, especially if you're looking for 
some kind of pattern that you can use to generate smarter guesses. 

I hope this helps.  Write back if you'd like to talk more about 
this, or anything else. 

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

Date: 10/12/2002 at 16:40:59
From: B.b.
Subject: Thank you (About Guess and Check Tables...)

Hello, Dr. Ian! It's B.b. asking you about 'Guess and Check Tables'. 
I reread it over and over again and got it! Thanks a lot! :) You guys 
are a HUGE help to me!