Problem 14 · 2025 AMC 8
Medium
Arithmetic & Operations
work-backwardsubstitution
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?
Show answer
Answer: E — 34.
Show hints
Hint 1 of 2
Peek at the answer choices before doing any algebra: every option is at least 7. The sorted list already has 7, 7 sitting dead center — what does inserting a big number do to the median?
Still stuck? Show hint 2 →
Hint 2 of 2
With 6 numbers the median is the average of the middle two, and those stay 7 and 7. So the median is locked at 7 — turn that into the mean and back-solve the sum.
Show solution
Approach: let the answer choices pin the median, then back-solve the mean
- Glance at the choices first: all are ≥ 7. The starting list sorts to 2, 6, 7, 7, 28, and inserting any number that big keeps 7 and 7 as the middle two of the six. So the median is locked at 7 — no algebra needed for it.
- "Mean is twice the median" then forces mean = 2 × 7 = 14, so the six numbers sum to 6 × 14 = 84.
- The original five sum to 2 + 6 + 7 + 7 + 28 = 50, so N = 84 − 50 = 34.
- Why this transfers: reading the answer choices can collapse the hard part of a problem — here they guarantee where N lands, freezing the median so you only have to chase the mean.
Mark:
· log in to save