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Pebbles

Place four pebbles on the sand in the form of a square. Keep adding as few pebbles as necessary to double the area. How many extra pebbles are added each time?

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Great Squares

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

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Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

Guillotine

Stage: 3 and 4 Short Challenge Level: Challenge Level:1

$\frac{5}{12}$

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.

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