2015 AMC 10B 第 7 题

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所有题目均经美国数学协会（MAA）官方合法授权使用。

7.

定义运算“减去倒数”为

a\diamond b=a-\frac{1}{b}

。求

((1\diamond2)\diamond3)-(1\diamond(2\diamond3))\text{？}

Consider the operation “minus the reciprocal of,” defined by

a\diamond b=a-\frac{1}{b}.

What is

((1\diamond2)\diamond3)-(1\diamond(2\diamond3))?

-\dfrac{7}{30}

-\dfrac{1}{6}

0

\dfrac{1}{6}

\dfrac{7}{30}

答案：A

知识点：自定义运算分数

难度评级：960

小提示：

先计算

1\diamond2

和

2\diamond3

。

Compute

1\diamond2

and

2\diamond3

first

大提示：

记住

a\diamond b=a-\frac1b

。

Remember

a\diamond b=a-\frac1b

解答：

\begin{aligned} &((1\diamond2)\diamond3)-(1\diamond(2\diamond3)) \\ &= \left(\dfrac 12- \dfrac 13\right) - \left(1 \diamond \dfrac 53\right)\\ &= \left(\dfrac 12 - \dfrac 13\right) - \left(1-\dfrac 35\right)\\ &= \dfrac 16 - \dfrac 25 \\&= -\dfrac{7}{30} \end{aligned}

所以正确答案是 A。

\begin{aligned} &((1\diamond2)\diamond3)-(1\diamond(2\diamond3)) \\ &= \left(\dfrac 12- \dfrac 13\right) - \left(1 \diamond \dfrac 53\right)\\ &= \left(\dfrac 12 - \dfrac 13\right) - \left(1-\dfrac 35\right)\\ &= \dfrac 16 - \dfrac 25 \\&= -\dfrac{7}{30} \end{aligned}

Thus, the correct answer is A .