Guillotine
Weekly Problem 9 - 2006
What fraction of the area of the rectangle is shaded?
What fraction of the area of the rectangle is shaded?
Age
11 to 14
Challenge level
Problem
Image
$ABCD$ is a rectangle. $P$ is the midpoint of $AD$; the length of $BQ$ is one third the length of $BC$. What fraction of the area of the rectangle is the area of the shaded quadrilateral $ABQP$?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
$\frac{5}{12}$
Let $AB$ and $AD$ be of length $b$ and $h$ respectively. Then
the area of $ABCD = bh$ and the area of $$ABQP =
\frac{1}{2}b\left(\frac{h}{2} + \frac{h}{3}\right)=
\frac{5bh}{12}$$
Thus the required ratio is $\frac{5}{12}$.