2018 AMC 8 第 10 题

Question

一组非零数的调和平均数定义为这些数的倒数的平均数再取倒数。

1

、

2

和

4

的调和平均数是多少？

The harmonic mean of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of

1,

2,

and

4?

\dfrac{3}{7}

\dfrac{7}{12}

\dfrac{12}{7}

\dfrac{7}{4}

\dfrac{7}{3}

答案：C

知识点：调和平均数分数

难度评级：900

小提示：

先求

1,2

和

4

的倒数的平均数

Average the reciprocals of

1,2,

and

4

.

大提示：

调和平均数就是这个平均数的倒数

The harmonic mean is the reciprocal of that average.

视频讲解：

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文字解答：

1

、

2

和

4

的倒数分别是

\dfrac11

、

\dfrac12

和

\dfrac14

。它们的平均数为

\begin{align*}\dfrac{\left(1 + \dfrac12 + \dfrac14\right)}{3} &= \dfrac{\left(\dfrac{7}{4}\right)}{3} \\&= \dfrac{7}{12} \end{align*}\text{。}

调和平均数是这些倒数的平均数的倒数；刚才算出的平均数是

\dfrac{7}{12}

，所以调和平均数为

\dfrac{12}{7}

。

正确答案是 C。

The reciprocals of

1

,

2

, and

4

are

\dfrac11

,

\dfrac12

, and

\dfrac14

, respectively. The average of these reciprocals is

\begin{align*}\dfrac{\left(1 + \dfrac12 + \dfrac14\right)}{3} &= \dfrac{\left(\dfrac{7}{4}\right)}{3} \\&= \dfrac{7}{12}. \end{align*}

As the harmonic mean is the reciprocal of the average of the reciprocals of the numbers (which we just calculated to be

\dfrac{7}{12}

), we conclude that the harmonic mean is

\dfrac{12}{7}.

Thus, the correct answer is C.