AMC8 2018

AMC8 2018 · Q10

AMC8 2018 · Q10. It mainly tests Fractions, Averages (mean).

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?

一组非零数的调和平均数是这些数的倒数的平均数的倒数。1、2和4的调和平均数是多少？

(A) 3 3

(B) $\frac{7}{12}$ $\frac{7}{12}$

(D) $\frac{7}{4}$ $\frac{7}{4}$

(E) $\frac{7}{3}$ $\frac{7}{3}$

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The reciprocals of $1$, $2$, and $4$ are $1$, $\frac{1}{2}$, and $\frac{1}{4}$, and their average is $\frac{1+\frac{1}{2}+\frac{1}{4}}{3}=\frac{7}{12}$. So the harmonic mean of $1$, $2$, and $4$ is the reciprocal of $\frac{7}{12}$, which is $\frac{12}{7}$.

答案（C）：$1$、$2$、$4$ 的倒数分别是 $1$、$\frac{1}{2}$、$\frac{1}{4}$，它们的平均值为 $\frac{1+\frac{1}{2}+\frac{1}{4}}{3}=\frac{7}{12}$。因此，$1$、$2$、$4$ 的调和平均数是 $\frac{7}{12}$ 的倒数，即 $\frac{12}{7}$。

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