Nested Squares

Splitting the diagram into one-inch squares

The largest square is 4 squares long, and the smallest square in the middle is 2 squares long.

Then, we can put the shaded pieces together to make whole square inches.

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So there are 5 shaded squares, out of 16 squares in total.

So the proportion of the area which is shaded is $\frac5{16}$.

Finding the areas of the squares

The whole diagram is a 4 inch by 4 inch square, so has area 4$\times$4 = 16 square inches.

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The shaded part is contained within a 3 by 3 square, so has area 3$\times$3 = 9 square inches.

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 There is a hole in the shaded part, which is a 2 by 2 square. So the area of the hole is 2$\times$2 = 4 square inches.

So the area of the shaded part is

9$-$4 = 5 square inches.

So the proportion of the area which is shaded is $\frac5{16}$.