Question 1210665: the mean of 1,2,x,11,y,14 is 8 and the median is 9. the value of x and y
Found 3 solutions by greenestamps, ikleyn, KMST:
Answer by greenestamps(13374)   (Show Source):
Answer by ikleyn(53996)  (Show Source):
You can put this solution on YOUR website! .
the mean of 1,2,x,11,y,14 is 8 and the median is 9. the value of x and y
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As I read this incoming post, I do not like the style in which it is written.
Professional mathematical composers NEVER write in this style.
Below I explain WHY.
According to the context, this problem is a standard problem on average and median.
It does not assume considering many different variations/versions.
Tutor @greenestamps writes that "For a set with an even number of elements, the median is the average
of the two in the middle."
It is not precisely correct. The precisely correct formulation is THIS:
For a set with an even number of elements, the median is the average
of the two in the middle after sorting the set in ascending or descending order."
Do you see the difference ? This difference is  .
So, if it is assumed that the given set is just ordered, then the solution by @greenestamps is correct.
But if it is NOT ORDERED from the very beginning, then all the construction of the problem and of its solution
hangs in the air without support on the ground - - - until some expert will come and will edit everything from scratch.
A correct problem's formulation is THIS
"The set {1, 2, x, 11, y, 14} is an ordered set.
Its average is 8 and the median is 9. Find the value of x and y."
Without the assumption that the set is ordered from the very beginning, the problem's formulation is a lame horse
with three legs.
It is not a style of writing mathematical problems, where every assumption should be properly expressed.
So, this incoming post is EITHER a copy from a bad source OR is inappropriate composition.
I write it to a person who created/posted this formulation in order for he/she
would know/understand the true price of this composition: it is not a style writing mathematical problems.
According to statistics, every post at this forum is read by other visitors about 15 times.
(It was so before the era of artificial intelligence. And now, when this era just came,
every post/solution is distributed by artificial intelligence in uncounted number of copies).
So we, the tutors, should take care editing defective posts to support the forum at a certain level.
Answer by KMST(5422)  (Show Source):
You can put this solution on YOUR website! We all agree that  , and that {7,13} could be the missing number x, and y, as greenstamps showed.
I agree that the problem is formulated in a way that suggests but does not state that the list 1,2,x,11,y,14 is an ordered list.
Would {7,13} be the only solution if the numbers in the list were not ordered from least to greatest?
I asked myself that question, answered it to my own satisfaction, and concluded that
either the person who proposed the problem did not realize the wording suggested a list ordered from least to greatest value,
or it was meant as an evil trick question, where answers would lose points by not proving that {7,13} is the only possible solution.
Hats off to iklein for not letting that slide! It was worth pointing out that.
When the numbers are ordered by value from least to greatest,
the average of the two numbers in the middle is 9.
That means the two missing numbers must add to  .
What could be the set of missing numbers with value in the middle?
If the numbers of the pair did not include any of the listed numbers,
they would be x, and y, and add to 20, instead.
The pair in the middle of the ordered list must include one of the listed numbers;
 ,  ,  , or  .
 , but {1,17} , could not be the pair of missing numbers in the middle of the re-ordered list,
because 1 is not a value equal or greater than 2, and 17 is not a value lesser or equal than 11.
The list ordered by value would be 1,1,2,11,14,17, with a median of
 .
 , but {2,16} does not work because 16 is not a value lesser or equal than 11.
The list ordered by value would be 1,2,2,11,14,16 , with a median of {{(2+11)/2=6.5}}} .
 , but {4,14} does not work because 14 is not a value lesser or equal than 11.
The list ordered by value would be 1,2,4,11,14,14, with a median of {{(4+11)/2=7.5}}} .
 , and {7,11} is the only possible pair of middle numbers in an ordered list of 6 numbers including 1, 2, 11, and 14.
The list ordered by value would be 1,2,7,11,14, with a median of {{(7+11)/2=9}}} .
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