To find the total number of tiles in the border of a rectangular table defined by lengths
The total number of tiles is made up of the tiles around the perimeter. The formula to calculate the perimeter of a rectangle is
-
The outer dimensions include the border, which adds one row of tiles on each side. Therefore, the new dimensions with the border add 2 tiles to the length and 2 tiles to the width.
This gives us the adjusted dimensions as:
- Outer length =
- Outer width =
- Outer length =
-
The total number of tiles including the inner rectangle (the center part without the border) is:
Since we only want the number of tiles in the border, we subtract the number of tiles in the center rectangle:
-
Expanding the expression:
Therefore, the number of tiles in the border becomes:
So the expression that represents the total number of tiles in the border can be simplified to
Now, we must check which options (I, II, III, IV) match this expression.
Assuming the options provided correspond to:
I.
II.
III.
IV.
The valid representations of the border tiles are:
- I is correct since it equals
. - II is incorrect since it represents the area of the inner rectangle without the border.
- III is also correct, as expanding
gives . - IV is incorrect since it doesn’t account for the border, as it’s simply
.
Thus, the correct options are I and III.
Therefore, the final answer is: A. I only and D. III and IV only relates to inner tiles only.
Given these observations, and assuming I only indicates