Nested Squares
What proportion of the diagram is shaded?
Image
![Nested Squares Nested Squares](/sites/default/files/styles/large/public/thumbnails/content-id-13774-original%252520diagram.png?itok=PuS3Y45f)
The diagram shows three nested squares.
What proportion of the diagram is shaded?
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.
Image
![Nested Squares Nested Squares](/sites/default/files/styles/large/public/thumbnails/content-id-13774-inches%252520grid.png?itok=GICPUnFF)
Image
![Nested Squares Nested Squares](/sites/default/files/styles/large/public/thumbnails/content-id-13774-coloured%252520parts.png?itok=cP4sHwFW)
Image
![Nested Squares Nested Squares](/sites/default/files/styles/large/public/thumbnails/content-id-13774-whole%252520square%252520inches.png?itok=UeF2twu3)
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$4 = 16 square inches.
Image
![Nested Squares Nested Squares](/sites/default/files/styles/large/public/thumbnails/content-id-13774-path6128.png?itok=hSuNbNp7)
The shaded part is contained within a 3 by 3 square, so has area 3$\times$3 = 9 square inches.
Image
![Nested Squares Nested Squares](/sites/default/files/styles/large/public/thumbnails/content-id-13774-hole.png?itok=WY8eTLt1)
So the area of the shaded part is
9$-$4 = 5 square inches.
So the proportion of the area which is shaded is $\frac5{16}$.