2025 AMC 8 Problems/Problem 14

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 $N$ ?

$\textbf{(A)}\ 7\qquad \textbf{(B)}\ 14\qquad \textbf{(C)}\ 20\qquad \textbf{(D)}\ 28\qquad \textbf{(E)}\ 34$

Solution

The median of the list is $7$ , so the mean of the new list will be $7 \cdot 2 = 14$ . Since there will be $6$ numbers in the new list, the sum of the $6$ numbers will be $14 \cdot 6 = 84$ . Therefore, $2+6+7+7+28+N = 84 \rightarrow N = \boxed{\text{(E)\ 34}}$

~Soupboy0

Solution 2 (Using answer choices)

We could use answer choices to solve this problem. The sum of the $5$ numbers is $50$ . If you add $7$ to the list, $57$ is not divisible by $6$ , therefore it will not work. Same thing applies to $14$ and $20$ . The only possible choices left are $28$ and $34$ . Now you check $28$ . You see that $28$ doesn't work because $(28+50) \div 6 = 13$ and $13$ is not twice of the median, which is still $7$ . Therefore, only choice left is $\boxed{\text{(E)\ 34}}$