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How Shady?

Age 11 to 14 Short
Challenge Level

Answer: $\dfrac{127}{256}$


Largest shaded triangle: $\dfrac14$ of the whole triangle




Next largest shaded triangle: occupies $\dfrac14$ of $\frac14$ $\therefore$ area $=\frac1{16}$
 

That means the shaded triangles of this size occupy $3\times\dfrac1{16}=\dfrac3{16}$ of the whole triangle.


Next smallest shaded triangles:  $\dfrac14$ of $\dfrac1{16}=\dfrac1{64}$
    

That means the shaded triangles of this size occupy $3\times\dfrac1{16}=\dfrac3{16}$ of the whole triangle.


Smallest shaded triangles: $\dfrac14\times\dfrac1{64}=\dfrac1{256}$
 

That means the shaded triangles of this size occupy $3\times\dfrac1{256}=\dfrac3{256}$ of the whole triangle.


So the total shaded area is $\dfrac14+\dfrac3{16}+\dfrac3{64}+\dfrac{3}{256}$

$=\dfrac{64+3\times16+3\times4+3\times1}{256}\\

=\dfrac{127}{256}$











You can find more short problems, arranged by curriculum topic, in our short problems collection.