Borrowing and Returning Books

Date: 03/26/2001 at 17:16:57
From: Samantha Worthington
Subject: Working Backward

In her classroom, Mrs. Guelker has a shelf of books for students to 
read. During the day, four students borrow books and eight students 
return books. At the end of the day, there are 27 books on the shelf. 
How many books were there at the beginning of the day? 

I do not understand any of the problem, and need to figure out how you 
get the answer.

Date: 03/27/2001 at 14:00:39
From: Doctor Ian
Subject: Re: Working Backward

Hi Samantha,

There is no way to solve the problem unless you assume that each 
student borrows or returns exactly one book. If you assume that, then 
you can solve the problem in the following way:

Let's use '?' to represent the number of books on the shelf at the 
beginning of the day. Then after the four students borrow books, the 
number of books on the shelf is:

     ? - 4

And after the eight students borrow books, the number of books on the 
shelf is:

     ? - 4 + 8

We're told that this number is 27. So we know that:

     ? - 4 + 8 = 27

Can you find a number to substitute for '?' that will make this 
equation true?

If you think of the equation as a balance beam,

      ? - 4 + 8         27
     \_________/      \_____/
         |_______________|
                 ^

we can add to or remove anything from both sides without throwing it 
out of balance, right? So we can add 4 to each side, which cancels out 
the -4 
on the left:

      ? + 8            27 + 4
     \_________/      \______/
         |_______________|
                 ^

We can subtract 8 from each side, to cancel out the 8 on the left.

And that will leave us with '?' balancing some number - which will be 
the number of books at the beginning of the day.

Does this help? Write back if you'd like to talk about this some more, 
or if you have any other questions.

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