In the diagram, all triangles are equilateral. What fraction of $\triangle A B C$ is coloured black?
(A) $\frac{3}{4}$
(B) $\frac{1}{2}$
(C) $\frac{9}{16}$
(D) $\frac{4}{9}$
(E) $\frac{27}{64}$

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-154.jpg?height=488&width=471&top_left_y=203&top_left_x=1399)
Solution

In the diagram, there are 27 black triangles. If the entire diagram was divided into the smallest size equilateral triangles, there would be $8+2(7)+2(6)+2(5)+2(4)+2(3)+2(2)+2(1)=64$ (counting by rows). Thus, $\frac{27}{64}$ of $\triangle A B C$ is coloured black. ANSWER: (E)

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