A8. The diagram shows a square made from five rectangles. Each of these rectangles has the same perimeter.
What is the ratio of the area of a shaded rectangle to the area of an unshaded rectangle?
SolUtion
$3: 7$

Let the width and length of a shaded rectangle be $x$ and $3 y$ respectively and the length of an unshaded rectangle be $l$. Since each rectangle has the same perimeter we have $2(l+y)=2(x+3 y)$ and hence $l+y=x+3 y$, which gives $l=x+2 y$. Since the original shape is a square we have $2 x+l=3 y$, which gives $l=3 y-2 x$. Therefore, $x+2 y=3 y-2 x$ and hence $y=3 x$. The area of a shaded rectangle is $3 x y$ and the area of an unshaded rectangle is $l y=y(x+2 y)=y(x+6 x)=7 x y$. Therefore, the ratio of the area of a shaded rectangle to the area of an unshaded rectangle is $3 x y: 7 x y=3: 7$.