AMC8 1998

AMC8 1998 · Q23

AMC8 1998 · Q23. It mainly tests Fractions, Basic counting (rules of product/sum).

If the pattern in the diagram continues, what fraction of the interior would be shaded in the eighth triangle?

如果图中的模式继续，第八个三角形的内部有几分之几被涂黑？

(A) \frac{3}{8} \frac{3}{8}

(B) \frac{5}{27} \frac{5}{27}

(D) \frac{9}{16} \frac{9}{16}

(E) \frac{11}{45} \frac{11}{45}

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): \[ \begin{array}{c c c} \text{Step} & \text{Number of triangles} & \text{Number of shaded triangles} \\ 1 & 1 & 0 \\ 2 & 4 & 0+1=1 \\ 3 & 9 & 1+2=3 \\ 4 & 16 & 1+2+3=6 \\ 5 & 25 & 1+2+3+4=10 \\ 6 & 36 & 1+2+3+4+5=15 \\ 7 & 49 & 1+2+3+4+5+6=21 \\ 8 & 64 & 1+2+3+4+5+6+7=28 \\ \end{array} \] The ratio at step eight: $\frac{28}{64}=\frac{7}{16}$.

答案（C）： \[ \begin{array}{c c c} \text{步骤} & \text{三角形的数量} & \text{涂色三角形的数量} \\ 1 & 1 & 0 \\ 2 & 4 & 0+1=1 \\ 3 & 9 & 1+2=3 \\ 4 & 16 & 1+2+3=6 \\ 5 & 25 & 1+2+3+4=10 \\ 6 & 36 & 1+2+3+4+5=15 \\ 7 & 49 & 1+2+3+4+5+6=21 \\ 8 & 64 & 1+2+3+4+5+6+7=28 \\ \end{array} \] 第八步的比值：$\frac{28}{64}=\frac{7}{16}$。

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