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Age 11 to 14 ShortChallenge 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.