2025 AMC 8 第 9 题

Question

Ningli 看时钟上

6

对正对着的数字。她求每一对数字的平均数。所得

6

个数的平均数是多少？

Ningli looks at the

6

pairs of numbers directly across from each other on a clock. She takes the average of each pair of numbers. What is the average of the resulting

6

numbers?

5

6.5

8

9.5

12

答案：B

知识点：平均数等差数列

难度评级：1070

小提示：

这些成对平均数的平均数等于全部

12

个数字的平均数

The average of pair-averages equals the average of all

12

numbers

大提示：

1

到

12

的平均数是两个端点的中点

The average of

1

through

12

is halfway between the endpoints

视频讲解：

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文字解答：

答案等于全部

12

个数字的平均数。一般地，把一组数两两配对，先求每对平均，再求这些平均数的平均，结果仍是所有原数的平均。

例如看较小的

6

个数

a

、

b

、

c

、

d

、

e

、

f

。这些成对平均数的平均为

\frac{1}{3} \left( \frac{a+b}{2} + \frac{c+d}{2} + \frac{e+f}{2} \right)\text{，}

化简后等于

\frac{a+b+c+d+e+f}{6}\text{。}

对

12

个数也同理。

由于这

12

个数字等差排列，平均数等于首尾两项的平均，即

\frac{1+12}{2} = \frac{13}{2} = 6.5\text{，}

所以选 B。

The answer is the same as the average of all

12

numbers. To understand why this is the case, it is actually generally true that if a whole bunch of numbers is split up into pairs, and each pair is averaged, and then all those pair-averages are averaged, the answer is the average of all the numbers.

To see why this is true, consider a smaller example of

6

numbers

a,

b,

c,

d,

e,

and

f.

The average of the pair-averages is:

\frac{1}{3} \left( \frac{a+b}{2} + \frac{c+d}{2} + \frac{e+f}{2} \right)

and that is equal to

\frac{a+b+c+d+e+f}{6}.

The same type of simplification happens with

12

numbers.

Since the

12

numbers are equally spaced (in an arithmetic progression), the answer is the same as the average of the first and last number, which is

\frac{1+12}{2} = \frac{13}{2} = 6.5,

or choice B.