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word problems and turning them into algebraic equation to slove

A tile company makes rectangles tiles. Mary lays the center white tiles. JIM comes along and boarders the area with gray tiles. ( example would be 1x3 area. Mary would lay the 3 center tiles and Jim would lay the 12 boarder tiles) Suppose each tile is 1 square meter.
*How many tiles are needed for a 4x8 area? (To be bordered)
*how many for a 28x56 area? (bordered)
*HOW MANY FOR n x m area?
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1 Answer

How many tiles are needed for a 4x8 area? (to be bordered)
 
This is asking for the number of center white tiles. To do this, simply multiply 4x8=32
 
How many for a 28x56 area? (bordered)
 
This is asking for the number of tiles for a 28x56 area, plus the bordering.
28x56 = 8x56 + 20x56 = 448 + 1120 = 1568 is the number of center tiles.
To get the bordering, a simple trick for purely rectangular or square areas (in number of tiles) is:
 
Take the dimensions of the area (in the example, 1x3), take each number and double it (1x2 and 3x2), add them together (2+6=8) then add 4 (because the corner tiles are added and are not directly along the sides of any of the 4 sides, as shown below
 
C = Corner
S = Side of Center
B = Border of Each Side
For the example,
C B B B C
B S S S B
C B B B C
 
Looking at this, you can see that a 1x3 area has 3 border panels on the top and bottom, and 1 on the left and right sides, with 4 corner tiles.
 
Thus 1x2+3x2+4=12
 
Going back to the 28x56 area,
 
28x2+56x2+4=56+112+4=172
 
172 border tiles + 1568 center tiles = 1740 total tiles.
 
Alternatively, you can simply increase the dimensions to include the border tiles by 1 on each side.
So instead of 28x56 you would have 30x58=1740. Doing it this way does not require the addition of 4 corner tiles because you have expanded the area to include the side AND corner border tiles.
 
How many for n x m area?
 
Without the border tiles, simply multiply the two numbers. With the border tiles, you can use either method shown for the 2nd part, either taking the center and adding the border or simply extending the center to include the border.
So...
 
Option 1:
n x m = nm
+ nx2 + mx2 + 4
= nm +2n + 2m + 4
 
Option 2 is simply:
(n+2) x (m+2)
 
 
Hope this helps!