A set of numbers {10, 21, 17, 12, x} has the property that the mean is equal to the median.
What could the value of x be?
(In reply to
Answer by K Sengupta)
The set can be ordered in 5 possible ways:
1. {10 ,12, 17, 21, x}
2. {10,12,17, x,21}
3. {10,12,x,17,21}
4. {10, x, 12, 17, 21}
5. {x, 10, 12, 17, 21}
At the outset, we notice that:
The (arithmetic) mean of the set is:
(10+12+x+17+21)/5 = (60+x)/5
There are seemingly three values corresponding to the middle element of each set. These are: 17, x and 12. These values must then correspond to the possibilities for the medians.
When the median is 17, it follows that: x>=17. Then, we have: (x+60)/5=17
=> x=25, which is in consonance with x>=17
When the median is x, it follows that: 12<= x<=17
Then, if follows that: (x+60)/5 =x => x= 15, which is in consonance with
12<=x<=17. Therefore, x=15 is a solution.
When the median is 12, it follows that x<=12
Then, (x+60)/5 =12, so that: x=0 which is in consonance with x<=12. Yherefore x=0 is a solution.
Consequently, there are precisely three distinct values for x, and these are: 0, 15, and 25.
Edited on September 12, 2022, 12:46 am
Explanation to Puzzle Answer
