AMC12 2024 A

AMC12 2024 A · Q5

AMC12 2024 A · Q5. It mainly tests Averages (mean), Arithmetic misc.

A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set?

一个包含 $20$ 个数字的数据集，其中有些是 $6$，其平均数为 $45$。当所有 $6$ 被移除后，数据集的平均数为 $66$。原来数据集中有多少个 $6$？

(A) 4 4

(B) 5 5

(D) 7 7

(E) 8 8

Answer

Correct choice: (D)

正确答案：（D）

Solution

Because the set has $20$ numbers and mean $45$, the sum of the terms in the set is $45\cdot 20=900$. Let there be $s$ sixes in the set. Then, the mean of the set without the sixes is $\frac{900-6s}{20-s}$. Equating this expression to $66$ and solving yields $s=7$, so we choose answer choice $\boxed{\textbf{(D) }7}$.

数据集有 $20$ 个数字，平均数 $45$，所以总和为 $45\cdot20=900$。设有 $s$ 个 $6$。移除 $6$ 后的平均数为 $\frac{900-6s}{20-s}$。设此等于 $66$ 并解得 $s=7$，所以选择答案 $\boxed{\textbf{(D) }7}$。

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