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

1 answer

To determine which expression represents the total number of tiles in the border of a rectangular table with \( l \) tiles along the length and \( w \) tiles along the width, we need to analyze the layout of the tiles.

The total number of tiles making up the entire rectangle, including both the inner area (or the area inside the border) and the border itself, can be calculated as follows:

  1. The total area of the rectangle is \( l \times w \).
  2. The border is formed by the tiles that outline the rectangle, which consists of:
    • Two lengths of tiles along the longer sides (lengths): 2 × \( l \)
    • Two widths of tiles along the shorter sides (widths): 2 × \( w \)

However, simply adding these values gives us the total perimeter of the rectangle. The border counts each tile distinctly.

So the total number of tiles in the border alone (assuming the border is one tile wide) can be computed as follows:

The total number of tiles in the border can be represented by:

  • Total tiles: \( l \times w \)
  • Total interior tiles (excluding the border): \( (l - 2) \times (w - 2) \)

The border tiles can then be calculated as the total tiles minus the interior tiles: \[ \text{Border tiles} = (l \times w) - ((l - 2) \times (w - 2)) \]

Simplifying that gives: \[ \text{Border tiles} = (l \times w) - (lw - 2l - 2w + 4) \] \[ = lw - lw + 2l + 2w - 4 \] \[ = 2l + 2w - 4 \]

This can be factored as: \[ = 2(l + w - 2) \]

Thus, the expression representing the total number of tiles in the border is: \[ 2(l + w - 2) \]

Given the options, we would have to compare this derived expression with the provided choices (I, II, III, IV) to find the correct match.

Without having the specific expressions I, II, III, IV presented in the problem, I can't definitively identify which choice corresponds to \( 2(l + w - 2) \). But based on the explanation above, the correct answer would include that expression.

If any of the expressions I, II, III, IV resonated or simplified to \( 2(l + w - 2) \), then the correct answer choice would identify that.

If you provide the specific expressions from the options I, II, III, IV, I can further help you identify which ones are correct.