Nested Squares
What proportion of the diagram is shaded?
Problem
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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.
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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$3 = 9 square inches.
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So the area of the shaded part is
9$-$4 = 5 square inches.
So the proportion of the area which is shaded is $\frac5{16}$.