Nested squares
What proportion of the diagram is shaded?
Problem
Image
The diagram shows three nested squares.
What proportion of the diagram is shaded?
Student Solutions
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
Image
Image
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$4 = 16 square inches.
Image
The shaded part is contained within a 3 by 3 square, so has area 3$\times$3 = 9 square inches.
Image
So the area of the shaded part is
9$-$4 = 5 square inches.
So the proportion of the area which is shaded is $\frac5{16}$.