Question

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.



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**.