2000 AMC 8 第 17 题

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17.

对所有非零数，运算

\otimes

定义为

a\otimes b =\dfrac{a^{2}}{b}\text{。}

求

[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)]\text{。}

The operation

\otimes

is defined for all nonzero numbers by

a\otimes b =\dfrac{a^{2}}{b}.

Determine

[(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)].

-\dfrac{2}{3}

-\dfrac{1}{4}

0

\dfrac{1}{4}

\dfrac{2}{3}

答案：A

知识点：自定义运算分数

难度评级：1410

小提示：

从内到外计算每个

\otimes

表达式。

Evaluate each

\otimes

expression from the inside out

大提示：

保留括号；这个运算不满足结合律。

Keep the parentheses; the operation is not associative

解答：

计算如下：

\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(\frac{1}{2})^2}{3} - \dfrac{1^2}{(\frac{4}{3})} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3} \end{gather*}\text{。}

所以正确答案是 A。

We can calculate it as follows.

\begin{gather*} [(1\otimes 2)\otimes 3]-[1\otimes (2\otimes 3)] \\ = [\dfrac{1^2}{2} \otimes 3] - [1 \otimes \dfrac{2^2}{3}] \\ = [\dfrac{1}{2} \otimes 3] - [1 \otimes \dfrac{4}{3}] \\ = \dfrac{(\frac{1}{2})^2}{3} - \dfrac{1^2}{(\frac{4}{3})} \\ = \dfrac{1}{4} \cdot \dfrac{1}{3} - \dfrac{3}{4} \\ = \dfrac{1}{12} - \dfrac{3}{4} \\= -\dfrac{2}{3}. \end{gather*}

Thus, A is the correct answer.