A number $N$ is inserted into the list 2, 6, 7, 7, 28. The mean is now twice as great as the median. What is $N$?

A. 7

B. 14

C. 20

D. 28

E. 34

1. 2025 AMC 8 官方解析

Answer (E): The original list has a sum of $2 + 6 + 7 + 7 + 28 = 50$ and a mean of $\frac{1}{5} \cdot 50 = 10$. The new list will have a median between 6.5 and 7, inclusive, equal to the average of the middle two numbers when the list is ordered. Thus the new mean will be between 13 and 14, inclusive. Because the mean of the original list is 10 and the mean of the new list has increased, the new number $N$ must be greater than 10. This means the new median will be 7, the new mean will be $2 \cdot 7 = 14$, and the sum of the numbers will be $6 \cdot 14 = 84$. Therefore $N = 84 - 50 = 34$.