Question

There is a set of five positive integers whose average (mean) is $5,$ whose median is $5,$ and whose only mode is $8.$ What is the difference between the largest and smallest integers in the set?

$3$

$5$

$6$

$7$

$8$

Answer: D

Concepts:mean median (data)mode

Difficulty rating: 1270

Solution:

The sum of all the numbers in the list is $5 \cdot 5 = 25.$

In increasing order, the middle number is $5$ . Because $8$ is the only mode, the last two numbers must both be $8$ ; there cannot be three $8$ s, since then the median would be $8$ .

These three known numbers sum to $5 + 8 + 8 = 21,$ so the two smaller numbers must add to $25 - 21 = 4.$ They must be distinct positive integers; otherwise a second value would tie $8$ as a mode. Thus they are $1$ and $3$ .

The desired difference is then $8 - 1 = 7.$

Thus, D is the correct answer.

1997 AMC 8 Problem 14

Answer: D

Difficulty rating: 1270

Solution:

Problem 14 in Other Years