AMC 8 1991 Problem 25

AMC8 1991 年第 25 题

比与比例★★★★☆

An equilateral triangle is originally painted black. Each time the triangle is changed, the middle fourth of each black triangle turns white. After five changes, what fractional part of the original area of the black triangle remains black?

一个等边三角形最初被涂成黑色。每变换一次，每个黑色三角形中间的四分之一变成白色。经过五次变换后，原等边三角形的面积还剩几分之几是黑色的？

A. $\displaystyle \frac{1}{1024}$
B. $\displaystyle \frac{15}{64}$
C. $\displaystyle \frac{243}{1024}$
D. $\displaystyle \frac{1}{4}$
E. $\displaystyle \frac{81}{256}$

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 逐次乘 3/4 每次把每个黑三角分成 4 个小三角、中间那个变白，剩 3 个黑，黑色面积变为原来的

\tfrac34

。5 次后为

\left(\tfrac34\right)^5=\dfrac{243}{1024}

。

Solution 1

Elementary · multiply by 3/4 each step At each change, the middle fourth of every black triangle turns white, leaving 3 of the 4 sub-triangles black, so the black area becomes

\tfrac34

of before. After five changes the black fraction is

\left(\tfrac34\right)^5=\dfrac{243}{1024}

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