The problem involves determining the total shaded area after repeatedly subdividing a square into smaller squares and shading the central square in each iteration. Let us analyze the situation and calculate step by step.

Problem Breakdown

1. Initial Square Dimensions: The original square has side length $27$ inches. Its total area is: $A_{\text{original}} = 27^2 = 729 \, \text{square inches}.$

2. Step 1: The square is divided into $9$ smaller squares, each with side length $\frac{27}{3} = 9$ inches. The central square (shaded) has an area: $A_{\text{shaded, step 1}} = 9^2 = 81 \, \text{square inches}.$

3. Step 2: The central square is again subdivided into $9$ smaller squares. Each of these smaller squares now has a side length of $\frac{9}{3} = 3$ inches. The shaded square in this step has an area: $A_{\text{shaded, step 2}} = 3^2 = 9 \, \text{square inches}.$

4. Step 3: The process repeats. The shaded square is again subdivided, with each smaller square having side length $\frac{3}{3} = 1$ inch. The shaded square in this step has an area: $A_{\text{shaded, step 3}} = 1^2 = 1 \, \text{square inch}.$

5. Step 4: Finally, the shaded square is subdivided once more. Each of these smaller squares has side length $\frac{1}{3}$ inch, and the shaded square's area is: $A_{\text{shaded, step 4}} = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \, \text{square inch}.$

Total Shaded Area

The total shaded area is the sum of all shaded regions across the four steps: $A_{\text{total shaded}} = A_{\text{shaded, step 1}} + A_{\text{shaded, step 2}} + A_{\text{shaded, step 3}} + A_{\text{shaded, step 4}}.$

Substituting the values: $A_{\text{total shaded}} = 81 + 9 + 1 + \frac{1}{9}.$

Combine terms: $A_{\text{total shaded}} = 91 + \frac{1}{9} = \frac{819}{9} + \frac{1}{9} = \frac{820}{9}.$

Convert to a decimal (if required): $A_{\text{total shaded}} \approx 91.11 \, \text{square inches}.$

Final Answer

The total shaded area after four iterations is: $\boxed{\frac{820}{9} \, \text{square inches} \, (\text{or approximately } 91.11 \, \text{square inches})}.$

Would you like further clarification or additional examples related to similar problems?

Related Questions

1. What would be the shaded area if the process was repeated one more time?
2. How does the total shaded area change if the initial square's side length is doubled?
3. What is the sum of the shaded areas after infinite iterations?
4. Can you derive a formula for the total shaded area after $n$ steps?
5. How would the problem change if the square was divided into $16$ smaller squares instead of $9$?

Tip

For problems involving recursive subdivisions, geometric series can often simplify calculations when the process continues indefinitely.