Let's analyze the problem step by step to determine the area of the shaded rectangle.

Problem Breakdown:

• Calvin has 12 identical rectangular tiles that are arranged around the edge of a shaded rectangle.
• The outer dimensions of the whole structure (tiles + shaded rectangle) are given as:
  • Length = 123 cm
  • Width = 67 cm

Our goal is to find the area of the shaded rectangle, which is inside the outer boundary formed by the tiles.

Approach:

1. Tile Dimensions: Since 12 tiles fit perfectly around the shaded rectangle, we need to estimate their dimensions. Let’s assume the tiles are placed symmetrically around the shaded rectangle. The number of tiles along the length and width will help us determine the reduction in size of the shaded rectangle compared to the outer dimensions.

2. Determine the dimensions of the shaded rectangle:

   • If the tiles are placed with half along each side, then the tiles along the longer side reduce the total length by twice the height of one tile.
   • Similarly, the tiles along the shorter side reduce the total width by twice the height of one tile.

3. Calculate the area of the shaded rectangle: Once we know the dimensions of the shaded rectangle (by subtracting the tile widths from the overall dimensions), we can compute its area using the formula:

   $\text{Area} = \text{Length} \times \text{Width}$

I will now calculate this.To calculate the area of the shaded rectangle, we need to first determine the dimensions of the shaded rectangle based on the arrangement of the tiles.

Step 1: Estimate the dimensions of the tiles

Since 12 tiles are placed around the shaded rectangle, we can infer the following:

• There are tiles along both the length and the width of the outer boundary.
• From the diagram, it seems that:
  • 4 tiles are placed along the length of the rectangle (2 on each side).
  • 2 tiles are placed along the width (1 on each side).

Let’s denote the dimensions of each tile as:

• Tile width (along the shorter side of the rectangle) = $x$
• Tile height (along the longer side of the rectangle) = $y$

Step 2: Calculate the reduction in dimensions

Now, to find the dimensions of the shaded rectangle:

• For the length of the shaded rectangle:

  • The overall length is $123 \, \text{cm}$.
  • Two tiles are placed at the ends (on both sides), so the length of the shaded rectangle is reduced by $2y$ (since the height of the tile is placed vertically along the length). $\text{Shaded Length} = 123 - 2y$

• For the width of the shaded rectangle:

  • The overall width is $67 \, \text{cm}$.
  • Two tiles are placed at the ends (on both sides), so the width of the shaded rectangle is reduced by $2x$ (since the width of the tile is placed horizontally along the width). $\text{Shaded Width} = 67 - 2x$

Step 3: Solve for tile dimensions

Given that the number of tiles is 12, and based on their placement:

• Along the length, 4 tiles span the total length (two placed horizontally along the top and bottom).
• Along the width, 2 tiles span the total width.

However, the problem doesn’t provide specific information about the individual tile dimensions. We might assume that the tiles are of a specific proportion and that the shaded rectangle can be determined by subtracting the tile dimensions from the total.

Would you like to explore these dimensions further, or should I assume a common tile size and continue to calculate?