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.