1994 AMC 12 Problem 24

Question

A sample consisting of five observations has an arithmetic mean of

10

and a median of

12.

The smallest value that the range (largest observation minus smallest) can assume for such a sample is

2

3

5

7

10

Order the observations as

a\le b\le12\le d\le e

Use the total sum

50

, then test whether a range below

5

is possible

The sample

7,7,12,12,12

has sum

50,

median

12,

and range

5,

so range

5

is possible. Suppose the range were at most

4.

If the largest value were at most

12,

then the top three values would all be

12,

leaving the bottom two with sum

14;

yet each would be at least

8,

a contradiction. If the largest value exceeded

12,

then the smallest would exceed

8,

making the total exceed

8+8+12+12+12=52.

Thus no smaller range works. The correct answer is C.

Problem 24 in other years