Cutting a Pipe

Date: 06/24/2003 at 12:48:31
From: Aaron
Subject: Math

You have a pipe that is 21 1/8" long. How many 2" segments can be made 
of this pipe if 1/8" waste is created from each cut?

Multiple choice answers were 9 or 10, and there were two others that 
were much higher or lower. I knew it was one of these answers. I 
figured 1/8 = .12.  10 * .12 = 1.2.  So 10 cuts = 1.2 waste.  If the 
pipe is 21.12" then the answer is 9 cuts right?  Because 1.2" (waste 
from 10 cuts) is greater than 1.12" (amount of pipe over 20") So does 
this mean .08" waste over for 10 cuts?  So the answer would be 
9 cuts?  Is there a more simple way to figure problems like this and 
am I on the right track and correct with my figures?  Thanks.

Date: 06/24/2003 at 13:14:53
From: Doctor Peterson
Subject: Re: Math

Hi, Aaron.

Your method is reasonable: make a good guess and then check it by 
multiplying. In real-life problems, that's often the best way - and 
the checking step is essential anyway. But there are several problems 
with your work.

My first concern is that you should not convert to decimal, which 
loses precision. Really, 1/8 is 0.125, so you are underestimating 
when you call it 0.12. Second, you confused the number of CUTS with 
the number of PIECES; with 10 cuts you make 11 pieces. Further, you 
showed that you can't get 10 cuts, but not that you CAN get 9 cuts. 
Here is a valid version of your approach:

Ignoring the kerf (the 1/8" waste), you can get ten 2-inch pieces from 
a 21 1/8 inch pipe (dividing by 2). That leaves 1 1/8 inches for 
waste, from 9 (not 10) cuts; but 9 times 1/8 inch is 9/8, or 1 1/8 
inches, which comes out exactly right. So we can get 10 pieces 
precisely.

I would approach it this way if I wanted to be a little more 
mathematical: Each piece, except the last, can be thought of as 2 1/8 
inches long, by including the kerf with the piece - that is, you are 
taking 2 1/8 inches off the pipe for each piece. The last piece, of 
course, needs no cut, and takes only 2 inches off the pipe. We can 
make up for that by imagining the pipe to be 1/8 inch longer than it 
is, as if we glued an extra kerf onto the end so we could make one 
last cut.

So we just have to divide 21 1/4 by 2 1/8 to find how many pieces we 
can get:

  21 1/4   85/4   85    8   85    8
  ------ = ---- = -- * -- = -- * --- = 5 * 2 = 10
   2 1/8   17/8    4   17   17    4

Or, you could do it in decimal, dividing 21.25 by 2.125 to get exactly 
10.

So we can make ten pieces, as before. I think your "guess-and-check" 
method, when done right, is better (quicker and more sure) than this 
more interesting way!

If you have any further questions, feel free to write back.

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