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?

Great Squares

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

Square Areas

Can you work out the area of the inner square and give an explanation of how you did it?

Guillotine

Stage: 3 Short Challenge 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.
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