To find the total number of tiles in the border of a rectangular table defined by lengths \( l \) (length) and \( w \) (width), we need to consider how the border is constructed.
The total number of tiles is made up of the tiles around the perimeter. The formula to calculate the perimeter of a rectangle is \( 2l + 2w \). However, since the tiles form a border around the tiles that make up the inner rectangle, we need to calculate it as follows:
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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 = \( l + 2 \)
- Outer width = \( w + 2 \)
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The total number of tiles including the inner rectangle (the center part without the border) is: \[ \text{Total Tiles} = (l + 2)(w + 2) \]
Since we only want the number of tiles in the border, we subtract the number of tiles in the center rectangle: \[ \text{Number of border tiles} = (l + 2)(w + 2) - lw \]
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Expanding the expression: \[ (l + 2)(w + 2) = lw + 2l + 2w + 4 \] Therefore, the number of tiles in the border becomes: \[ \text{Border Tiles} = (lw + 2l + 2w + 4) - lw = 2l + 2w + 4 \]
So the expression that represents the total number of tiles in the border can be simplified to \( 2l + 2w + 4 \).
Now, we must check which options (I, II, III, IV) match this expression.
Assuming the options provided correspond to:
I. \( 2l + 2w + 4 \)
II. \( lw \)
III. \( 2(l + w + 2) \)
IV. \( 2l + 2w \)
The valid representations of the border tiles are:
- I is correct since it equals \( 2l + 2w + 4 \).
- II is incorrect since it represents the area of the inner rectangle without the border.
- III is also correct, as expanding \( 2(l + w + 2) \) gives \( 2l + 2w + 4 \).
- IV is incorrect since it doesn’t account for the border, as it’s simply \( 2l + 2w \).
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 \( 2l + 2w + 4 \), the best choice is A.