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

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

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.

2018 AMC 8 第 10 题

答案：C

难度评级：900

视频讲解：

文字解答：

其他年份的第 10 题