A rectangular table, like the rectangle shown, has a border of same-sized square tiles. The rectangle has 10 tiles along the length and 6 tiles along the width for a total of 28 total tiles. Notice that 10 + 10 + 6 + 6 ≠ 28.

Question

If the rectangular table has l tiles along the length and w tiles along the width, which of these expressions represents the total number of tiles in the border?

I.
II.
III.
IV.
A.
I only
B.
III only
C.
II and IV only
D.
III and IV only

Answer 1

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:

1. 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 = $w+2$

2. 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: $$\text{Number of border tiles}=(l+2)(w+2)-lw$$

3. 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.