Question 4

Step 1 · Find the Area of the Big Square and Components

Let the side length of each small square be $x$.

Side length of the big square $= 2x$

$\text{Area of big square} = (2x) \times (2x) = 4x^2$ $\text{Area of each small square} = x \times x = x^2$

Step 2 · Calculate the Shaded Area and Fraction

The shaded region is divided into three parts:

Part 1 (Bottom-left triangle): $\text{Area of Part 1} = \dfrac{1}{2} \times x \times x = \dfrac{x^2}{2}$

Part 2 (Top-right triangle): $\text{Area of Part 2} = \dfrac{1}{2} \times x \times x = \dfrac{x^2}{2}$

Part 3 (Middle parallelogram): Dividing the parallelogram into two equal triangles with base $x$ and height $\dfrac{x}{2}$: $\text{Area of one triangle} = \dfrac{1}{2} \times x \times \dfrac{x}{2} = \dfrac{x^2}{4}$ $\text{Area of parallelogram} = \dfrac{x^2}{4} + \dfrac{x^2}{4} = \dfrac{2x^2}{4} = \dfrac{x^2}{2}$

Total shaded area: $\text{Total shaded area} = \dfrac{x^2}{2} + \dfrac{x^2}{2} + \dfrac{x^2}{2} = \dfrac{3x^2}{2}$

Fraction of the big square:

Step 1 · Find the Area of the Top-Left Sub-Square

Let the side length of the top-left square be $L$.

Side length of the big square $= 2L$

$\text{Area of big square} = (2L) \times (2L) = 4L^2$ $\text{Area of top-left square} = L \times L = L^2$

$\text{Fraction of big square occupied by top-left square} = \dfrac{L^2}{4L^2} = \dfrac{1}{4}$

Step 2 · Calculate Shaded Area and Total Fraction

The top-left square is divided into $8$ identical triangles using its diagonals and medians.

The shaded region occupies $2$ out of these $8$ triangles: $\text{Fraction in top-left square} = \dfrac{2}{8} = \dfrac{1}{4}$

Total fraction of the big square: $\text{Total fraction} = \dfrac{1}{4} \times \dfrac{1}{4} = \dfrac{1}{16}$