Carpet and Room Areas

Date: 10/26/2001 at 01:41:43
From: Matt Moore
Subject: Carpet area

A man buys a roll of carpet 9 ft. wide and 12 ft. long to fit a room  
10 ft. x 10 ft. When the carpet is unrolled, a rectangular hole 1 ft. 
wide and 8 ft. long is discovered in the exact middle of the carpet.  
The man decides that since the area of the roll is 108 sq.ft. and 
the area of the hole is 8 sq. ft., the carpet will still fit in the 
100 sq. ft. room. How can he cut the roll of carpet with the hole in 
the middle of it into to two pieces that will fit into the 100 sq. ft. 
room when the two pieces are put together?  

I don't know where to start. Please help me.

Thank you,          
Matt Moore

Date: 10/26/2001 at 12:10:14
From: Doctor Rob
Subject: Re: Carpet area

Thanks for writing to Ask Dr. Math, Matt.

Let's start with a smaller case:  a 4-by-4 room and a 3-by-6 carpet,
with a 1-by-2 hole in the middle:

   o---+---+---+---+---+---o
   |                       |
   +       o---+---o       +
   |       |       |       |
   +       o---+---o       +
   |                       |
   o---+---+---+---+---+---o

One should cut the carpet thus:

   o---+---o---+---+---+---o
   |       |               |
   +       o---+---o       +
   |       |       |       |
   +       o---+---o       +
   |               |       |
   o---+---+---+---o---+---o

Then move the right-hand piece 1 foot up and 2 feet to the left
(to make the result 1 foot taller and 2 feet narrower), and you'll
have

   o---+---+---+---o
   |               |
   o---+---o       +
   |       |       |
   +       o       +
   |       |       |
   +       o---+---o
   |               |
   o---+---+---+---o

Ta-daaaa!

Now the next larger case, a 6-by-6 room and a 5-by-8 carpet with a
1-by-4 hole in the middle:

   o---+---+---+---+---+---+---+---o
   |                               |
   +                               +
   |                               |
   +       o---+---+---+---o       +
   |       |               |       |
   +       o---+---+---+---o       +
   |                               |
   +                               +
   |                               |
   o---+---+---+---+---+---+---+---o

The cuts should look like this:

   o---+---o---+---+---+---+---+---o
   |       |                       |
   +       o---+---o               +
   |               |               |
   +       o---+---o---+---o       +
   |       |               |       |
   +       o---+---o---+---o       +
   |               |               |
   +               o---+---o       +
   |                       |       |
   o---+---+---+---+---+---o---+---o

Then move the right-hand piece 1 foot up and 2 feet to the left (to 
make the result 1 foot taller and 2 feet narrower), and you'll have

   o---+---+---+---+---+---o
   |                       |
   o---+---+               +
   |       |               |
   +       o---+---o       +
   |               |       |
   +       o---+---o       +
   |       |               |
   +       o---+---o       +
   |               |       |
   +               o---+---o
   |                       |
   o---+---+---+---+---+---o

Now you try the same ideas on the 10-by-10 room and 9-by-12 carpet
with a 1-by-8 hole in the middle.

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