Cutting Equal Strips

Date: 09/08/2005 at 09:21:33
From: george
Subject: Dimensions

Given five 4 ft by 8 ft rectangular sheets of plywood, you are 
assigned to cut strips that are 7 1/2 inches wide and 8 feet long. 
Each saw cut eliminates 1/16 inch of the plywood as sawdust.  You are
not able to glue any strips together.

Ignoring the sawdust that results, show how many 7 1/2 inch by 8 foot 
strips you will get from the five sheets of plywood and give 
dimensions of the leftover strips of plywood.

I have tried to work it out many different ways but it all comes out 
wrong.

Date: 09/08/2005 at 10:02:33
From: Doctor Peterson
Subject: Re: Dimensions

Hi, George.

Let's simplify the numbers so you can get a sense of what's happening, 
and then you can apply that understanding to solving the actual 
problem.

Suppose you want to cut 3-inch strips from a 14-inch wide board, but 
you have a really bad saw that turns 1 inch of wood into sawdust.  You 
cut 3 inches from the left side and take out the next inch, like this:

  +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
   <---------> <->
    strip 1    saw

So the next strip will begin at what was the 4-inch mark on the 
original board, and take up the next 3 inches:

  +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
   <---------> <-> <---------> <->
    strip 1    saw   strip 2   saw

Repeat again, and we have

  +---+---+---+---+---+---+---+---+---+---+---+---+---+---+
   <---------> <-> <---------> <-> <---------> <->
    strip 1    saw   strip 2   saw   strip 3   saw

So we have a 2-inch strip left that is too small to make another 3-
inch strip from.

How could we have calculated this?  Well, each strip uses up not just 
3 inches, but 4 inches of wood; so we can divide 14 by 4 to find how 
many times we can repeat the process--that is, in cutting three times, 
we used up 3 times 4 inches of wood, counting both the strips we made 
and the sawdust that was produced.

There is just one little detail to worry about.  What if we had had 
a 15-inch board to start with, so that the last strip was exactly 3 
inches wide, and we didn't have to make the last cut?

  +---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
   <---------> <-> <---------> <-> <---------> <-> <--------->
    strip 1    saw   strip 2   saw   strip 3   saw   strip 4

Dividing 15 by 4 gives 3 with a remainder of 3, and we'd have to 
recognize that the remainder is enough to leave one more strip.  
There's a neat trick I can see that would allow us always to find the 
number of strips we can make by dividing; it involves pretending that 
the board started out 1 inch wider than it is, so that we would need 
that one last cut and have nothing left.  But you don't need that 
trick in order to solve the problem.

If you need more help, please write back and show me how far you got 
in solving the real problem.


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