A set of five integers {17, 12, 10, 21, n} has the property that the median is equal to their arithmetic mean.

Determine all possible values of n.


Solution to the Problem:


The possible answers for n are 0, 15, and 25.

The median is the value that's exactly in the middle of a dataset when it is ordered.
This set can be ordered in 5 possible ways:
{n, 10, 12, 17, 21}
{10, n, 12, 17, 21}
{10, 12, n, 17, 21}
{10, 12, 17, n, 21}
{10, 12, 17, 21, n}

Observe that the arithmetic mean of the set is equal to:
(10+12+n+17+21)/5 = (60+n)/5

There are three possible distinct values of the middle element of each set.
These are 12, n, and 17 which must then correspond to the possibilities for medians.

Case 1: When the median is 12, it follows that n<=12.
Then, (n+60)/5=12, which yields n=0, which agrees with n<=12.   Accordingly, n=0 is a solution.

Case 2: When the median is n, it follows that 12<=n<=17.
Then, (n+60)/5=n, so that: n=15.   This agrees with 12<=n<=17.   Accordingly, n=15 is a solution.

Case 3: When the median is 17, it follows that n>=17.
Then, (n+60)/5=17, so that: n=25.   This agrees with n>=17.   Accordingly, n=25 is a solution.

Consequently, there are precisely three possible values for n and these are 0, 15, and 25.

Renee Markey's 4th Period Algebra 1 students at the Muckleshoot Tribal School in Washington did some collaboration on the problem:


Click here to see the work from the individual students at Muckleshoot Tribal School.

Correctly solved by:

1. Colin (Yowie) Bowey Beechworth, Victoria, Australia
2. Ritwik Chaudhuri Santiniketan, West Bengal, India
3. Ivy Joseph Pune, Maharashtra, India
4. Benny Varghese Cochin, Kerala, India
5. Dr. Hari Kishan D.N. College,
Meerut, Uttar Pradesh, India
6. Davit Banana Istanbul, Turkey
6. Paul Muckleshoot Tribal School,
Auburn, Washington
7. Cameron W. Muckleshoot Tribal School,
Auburn, Washington
8. Tabor Muckleshoot Tribal School,
Auburn, Washington
9. Austin Hale Central High School,
Grand Junction, Colorado