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$\frac{5}{12}$
Guillotine
Age 11 to 14
ShortChallenge Level 
$\frac{5}{12}$
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 $\frac{5}{12}$.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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