AMC8 2023

AMC8 2023 · Q20

AMC8 2023 · Q20. It mainly tests Averages (mean), Logic puzzles.

Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?

将两个整数插入列表 $3, 3, 8, 11, 28$ 中，使其范围加倍。众数和中位数保持不变。两个附加数字的最大可能和是多少？

(A) 56 56

(B) 57 57

(D) 60 60

(E) 61 61

Answer

Correct choice: (D)

正确答案：（D）

Solution

To double the range, we must find the current range, which is $28 - 3 = 25$, to then double to: $2(25) = 50$. Since we do not want to change the median, we need to get a value less than $8$ (as $8$ would change the mode) for the smaller, which will be $7$. This fixes $53$ for the larger value. Anything less than $3$ is not beneficial to the optimization because you want to get the largest range without changing the mode. So, taking our optimal values of $7$ and $53$, we have an answer of $7 + 53 = \boxed{\textbf{(D)}\ 60}$.

要使范围加倍，我们必须找到当前范围，为 $28 - 3 = 25$，然后加倍为 $2(25) = 50$。由于我们不想改变中位数，我们需要小于 $8$ 的值（因为 $8$ 会改变众数）作为较小的值，为 $7$。这固定了较大的值为 $53$。小于 $3$ 的任何值都不利于优化，因为你希望在不改变众数的情况下获得最大范围。因此，取我们的最优值 $7$ 和 $53$，答案是 $7 + 53 = \boxed{\textbf{(D)}\ 60}$。

Topics