1997 AMC 8 Problem 14
Attempt Problem 14 of the 1997 AMC 8 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1997 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
14.
There is a set of five positive integers whose average (mean) is whose median is and whose only mode is What is the difference between the largest and smallest integers in the set?
Answer: D
Small hint:
The mode must appear exactly twice
Big hint:
Use the mean to get the total sum
Solution:
The sum of all the numbers in the list is
In increasing order, the middle number is . Because is the only mode, the last two numbers must both be ; there cannot be three s, since then the median would be .
These three known numbers sum to so the two smaller numbers must add to They must be distinct positive integers; otherwise a second value would tie as a mode. Thus they are and .
The desired difference is then
Thus, D is the correct answer.
Problem 14 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
- 1988 AMC 8
- 1989 AMC 8
- 1990 AMC 8
- 1991 AMC 8
- 1992 AMC 8
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