22. The diagram shows a shaded region inside a large square. The shaded region is divided into small squares.
What fraction of the area of the large square is shaded?
A $\frac{3}{10}$
B $\frac{1}{3}$
C $\frac{3}{8}$
D $\frac{2}{5}$
E $\frac{3}{7}$
22. B Let the length of the sides of each small square be $x$. Then the shaded area is $24 x^{2}$. Let the perimeter of the square be divided into eight line segments, each of length $y$, and four line segments of length $z$. Some of these are labelled in the diagram. By Pythagoras' Theorem in triangle $A B C$ : $y^{2}+y^{2}=(5 x)^{2}$, that is $2 y^{2}=25 x^{2}$.

So $y=\frac{5}{\sqrt{2}} x=\frac{5 \sqrt{2}}{2} x$. Similarly, in the triangle with hypotenuse $C D$ : $x^{2}+x^{2}=z^{2}$, that is $2 x^{2}=z^{2}$. So $z=\sqrt{2} x$. Therefore the length of the side of the large square is $2 y+z=5 \sqrt{2} x+\sqrt{2} x=6 \sqrt{2} x$. So the area of the large square is $(6 \sqrt{2} x)^{2}=72 x^{2}$. Hence the required fraction is $\frac{24 x^{2}}{72 x^{2}}=\frac{1}{3}$.