Question

四个数排成一行。前两个数的平均数是 $21$ ，中间两个数的平均数是 $26$ ，后两个数的平均数是 $30$ 。第一个数和最后一个数的平均数是多少？

Four numbers are written in a row. The average of the first two is $21,$ the average of the middle two is $26,$ and the average of the last two is $30.$ What is the average of the first and last of the numbers?

$24$

$25$

$26$

$27$

$28$

答案：B

知识点：平均数方程组

难度评级：1100

解答：

设四个数依次为 $a,b,c,d$ 。

因为 $a$ 和 $b$ 的平均数是 $21$ ，所以 $21\cdot2 = 42$ 。因为 $c$ 和 $d$ 的平均数是 $30$ ，所以 $30\cdot2 = 60$ 。也就是 $a+b = 42$ ， $c+d = 60$ ，从而 $\begin{align*} a+b+c+d &= 42 + 60\\ &= 102 \end{align*}$

中间两个数 $b$ 和 $c$ 的平均数是 $26$ ，所以 $26\cdot2=52$ 。因此 $b+c = 52$ 。

相减得 $a+d =50$ 。所以第一个和最后一个数的平均数是 $\dfrac{a+d}{2}=25$ ，答案为 $25$ 。

正确答案是 B。

Let the numbers be $a,b,c,d$ in order.

Since the average of $a$ and $b$ is $21,$ we know their sum is $21\cdot2 = 42.$
Since the average of $c$ and $d$ is $30,$ we know their sum is $30\cdot2 = 60.$
Since $a+b = 42$ and $c+d = 60 ,$ we know $\begin{align*} a+b+c+d &= 42 + 60\\ &= 102 \end{align*}$

Now, with the average of $b$ and $c$ being $26,$ we know their sum is $26\cdot2=52.$ This means $b+c = 52.$

Subtracting this result from the sum of all the terms yields $a+d =50.$
Since $\dfrac{a+d}{2}=25,$ our answer is $25.$

Thus, the correct answer is B.

2022 AMC 8 第 16 题

答案：B

难度评级：1100

解答：

其他年份的第 16 题