Open Middle®
Directions: Create a set of five positive integers from 1 to 20 that have the same mean, median, and range.
Hint
How do you find the mean/median/range of a set of five integers?
Answer
There are many answers including
{2, 3, 4, 5, 6} mean, median, range = 4
{3, 4, 6, 8, 9 } mean, median, range = 6
{3, 5, 6, 7, 9} mean, median, range = 6
{4, 6, 8, 10, 12} mean, median, range = 8
{4, 5, 8, 11, 12} mean, median, range = 8
{5, 8, 10, 12, 15} mean, median, range = 10
{6, 8, 12, 16, 20} mean, median, range = 12
{7, 8, 13, 17, 20} mean, median, range = 13
{6, 11, 14, 19, 20} mean, median, range = 14
{2, 3, 4, 5, 6} mean, median, range = 4
{3, 4, 6, 8, 9 } mean, median, range = 6
{3, 5, 6, 7, 9} mean, median, range = 6
{4, 6, 8, 10, 12} mean, median, range = 8
{4, 5, 8, 11, 12} mean, median, range = 8
{5, 8, 10, 12, 15} mean, median, range = 10
{6, 8, 12, 16, 20} mean, median, range = 12
{7, 8, 13, 17, 20} mean, median, range = 13
{6, 11, 14, 19, 20} mean, median, range = 14
Source: Eric Berchtold and Melissa Minnix
I had a student come up with 2, 3, 4, 5, 6.
Thanks. I added this to the answer.
Additional solutions:
{4, 6, 8, 10, 12} mean, median, range = 8
{7, 8, 13, 17, 20} mean, median, range = 13
{6, 11, 14, 19, 20} mean, median, range = 14
{4, 5, 8, 11, 12} mean, median, range = 8
{3, 4, 6, 8, 9 } mean, median, range = 6
{6, 8, 12, 16, 20} mean, median, range = 12
{3, 5, 6, 7, 9} mean, median, range = 6
Thanks Eric and Dan for the extra solution sets.
Additional solutions:
{4, 5, 7, 8, 11} mean, median, range = 7
{4, 7, 8, 9, 12} mean, median, range = 8
{6, 12, 13, 15, 19} mean, median, range = 13
{5, 9, 10, 11, 15} mean, median, range = 10
One of the answers included is incorrect.
{6, 8, 12, 16, 20} mean= 12.4 median= 12 range= 14
The problem does not state that the whole numbers need to be unique whole numbers, so can a set include a mode? The set {3,3,5,6,8} contains 5 whole numbers whose mean, median, and range are 5.
I don’t see why not Tracey. That would be an interesting addition. I think the only reason it is not mentioned is because mode is not a Common Core State Standard.
Some of mine did this too – it’s a fair solution.
Nrich has a similar problem set which includes mode if some want to include it: https://nrich.maths.org/11281
2 plus 2 is 4 minus 1 is three quick maths
My students found the following:
6,7,10,11,16
2,4,5,7,7
5,15,15,20,20
5,6,10,14,15
4,9,10,13,14
5,7,10,13,15
4,8,9,11,13
{2, 3, 4, 5, 6}=4 {3, 4, 6, 8, 9 } = 6 {3, 5, 6, 7, 9} = 6 {4, 6, 8, 10, 12} mean, median, range = 8
{4, 5, 8, 11, 12} = 8 {5, 8, 10, 12, 15} = 10 {6, 8, 12, 16, 20}= 12 {7, 8, 13, 17, 20} = 13
{6, 11, 14, 19, 20}= 14
2,3,6,9,3-mean4.6-median3-range7
I got 2,2,2,4,4, mean, median, range=2
whoops! I got mean & mode mixed up
now I got 1,2,2,2,3, mean, median, range=2
Wouldn’t 4, 4, 6, 6, 10 be an answer?
Unless you can’t do modes.
Wouldn’t 4, 4, 6, 6, 10 be an answer?
Unless you can’t do modes
Oops I sent it twice
6,8,12,16,20
{2, 3, 4, 5, 6} mean, median, range = 4
{6, 8, 12, 16, 20} mean, median, range = 12
3, 5, 6, 7, 9} mean, median, range = 6
There are many answers including
{2, 3, 4, 5, 6} mean, median, range = 4
{3, 4, 6, 8, 9 } mean, median, range = 6
{3, 5, 6, 7, 9} mean, median, range = 6
{4, 6, 8, 10, 12} mean, median, range = 8
{4, 5, 8, 11, 12} mean, median, range = 8
{5, 8, 10, 12, 15} mean, median, range = 10
{6, 8, 12, 16, 20} mean, median, range = 12
{7, 8, 13, 17, 20} mean, median, range = 13
{6, 11, 14, 19, 20} mean, median, range = 14
{6,11,14,19,20}mean,median,range=14
{4,5,8,11,12} mean,median,range=8
{2, 3, 4, 5, 6} mean, median, range = 4
{3, 4, 6, 8, 9 } mean, median, range = 6
{3, 5, 6, 7, 9} mean, median, range = 6
{4, 6, 8, 10, 12} mean, median, range = 8
{4, 5, 8, 11, 12} mean, median, range = 8
{5, 8, 10, 12, 15} mean, median, range = 10
{6, 8, 12, 16, 20} mean, median, range = 12
{7, 8, 13, 17, 20} mean, median, range = 13
{6, 11, 14, 19, 20} mean, median, range = 14
{2, 3, 4, 5, 6} mean, median, range = 4
{3, 4, 6, 8, 9 } mean, median, range = 6
{3, 5, 6, 7, 9} mean, median, range = 6
{4, 6, 8, 10, 12} mean, median, range = 8
{4, 5, 8, 11, 12} mean, median, range = 8
{5, 8, 10, 12, 15} mean, median, range = 10
{6, 8, 12, 16, 20} mean, median, range = 12
{7, 8, 13, 17, 20} mean, median, range = 13
{6, 11, 14, 19, 20} mean, median, range = 14
6, 5, 6, 9, 7, 10, 6, 6, 8, 7
I can’t find the mean , mode and median of this.
{3, 6, 7, 9, 10} – mean, median and range =7
My students came up with 3,4,5,5,8 and not only are the mean, median and range equal to 5, but the mode is too.
2,3,4,5,6
Mean: 2+3+4+5+6= 20
20/5=4
Median: 4
Range: 6-2=4
We used repeats and came up with 5, 15, 15, 20, 20 to have them all be 15.
i had 2 2 4 6 6
There are 45 solutions with no repeating numbers:
[7, 11, 13, 14, 20]
[7, 10, 13, 15, 20]
[7, 9, 13, 16, 20]
[7, 9, 12, 13, 19]
[7, 8, 13, 17, 20]
[7, 8, 12, 14, 19]
[6, 13, 14, 17, 20]
[6, 12, 14, 18, 20]
[6, 12, 13, 15, 19]
[6, 11, 14, 19, 20]
[6, 11, 13, 16, 19]
[6, 11, 12, 13, 18]
[6, 10, 13, 17, 19]
[6, 10, 12, 14, 18]
[6, 9, 13, 18, 19]
[6, 9, 12, 15, 18]
[6, 9, 11, 12, 17]
[6, 8, 12, 16, 18]
[6, 8, 11, 13, 17]
[6, 7, 12, 17, 18]
[6, 7, 11, 14, 17]
[6, 7, 10, 11, 16]
[5, 12, 13, 17, 18]
[5, 11, 12, 15, 17]
[5, 10, 12, 16, 17]
[5, 10, 11, 13, 16]
[5, 9, 11, 14, 16]
[5, 9, 10, 11, 15]
[5, 8, 11, 15, 16]
[5, 8, 10, 12, 15]
[5, 7, 10, 13, 15]
[5, 7, 9, 10, 14]
[5, 6, 10, 14, 15]
[5, 6, 9, 11, 14]
[4, 9, 10, 13, 14]
[4, 8, 9, 11, 13]
[4, 7, 9, 12, 13]
[4, 7, 8, 9, 12]
[4, 6, 8, 10, 12]
[4, 5, 8, 11, 12]
[4, 5, 7, 8, 11]
[3, 6, 7, 9, 10]
[3, 5, 6, 7, 9]
[3, 4, 6, 8, 9]
[2, 3, 4, 5, 6]
Plus another 63 with repeating numbers:
[2, 2, 4, 6, 6]
[2, 2, 3, 3, 5]
[1, 3, 3, 4, 4]
[1, 2, 2, 2, 3]
[1, 1, 2, 3, 3]
[8, 8, 12, 12, 20]
[7, 12, 13, 13, 20]
[7, 10, 12, 12, 19]
[7, 8, 11, 11, 18]
[7, 7, 13, 18, 20]
[7, 7, 12, 15, 19]
[7, 7, 11, 12, 18]
[6, 14, 14, 16, 20]
[6, 13, 13, 14, 19]
[6, 12, 12, 12, 18]
[6, 10, 14, 20, 20]
[6, 10, 11, 11, 17]
[6, 8, 13, 19, 19]
[6, 8, 10, 10, 16]
[6, 6, 12, 18, 18]
[6, 6, 11, 15, 17]
[6, 6, 10, 12, 16]
[6, 6, 9, 9, 15]
[5, 15, 15, 20, 20]
[5, 14, 14, 18, 19]
[5, 13, 13, 16, 18]
[5, 13, 14, 19, 19]
[5, 12, 12, 14, 17]
[5, 11, 11, 12, 16]
[5, 11, 13, 18, 18]
[5, 10, 10, 10, 15]
[5, 9, 12, 17, 17]
[5, 8, 9, 9, 14]
[5, 7, 11, 16, 16]
[5, 6, 8, 8, 13]
[5, 5, 10, 15, 15]
[5, 5, 9, 12, 14]
[5, 5, 8, 9, 13]
[4, 12, 12, 16, 16]
[4, 11, 11, 14, 15]
[4, 10, 11, 15, 15]
[4, 10, 10, 12, 14]
[4, 9, 9, 10, 13]
[4, 8, 10, 14, 14]
[4, 8, 8, 8, 12]
[4, 6, 9, 13, 13]
[4, 6, 7, 7, 11]
[4, 4, 8, 12, 12]
[4, 4, 7, 9, 11]
[4, 4, 6, 6, 10]
[3, 9, 9, 12, 12]
[3, 8, 8, 10, 11]
[3, 7, 8, 11, 11]
[3, 7, 7, 8, 10]
[3, 6, 6, 6, 9]
[3, 5, 7, 10, 10]
[3, 4, 5, 5, 8]
[3, 3, 6, 9, 9]
[3, 3, 5, 6, 8]
[2, 6, 6, 8, 8]
[2, 5, 5, 6, 7]
[2, 4, 5, 7, 7]
[2, 4, 4, 4, 6]
If we add the number 0, the only additional solution is [0, 0, 0, 0, 0].
Note: The solution set given by the creator contains an error:
The median and mean of {6, 8, 12, 16, 20} is 12, but the range is not 12, it’s 14.
By the way: 26 of the solutions with repeating numbers have a unique mode that is also equal to the range, the mean and the median:
[1, 2, 2, 2, 3]
[2, 4, 4, 4, 6]
[2, 5, 5, 6, 7]
[3, 4, 5, 5, 8]
[3, 6, 6, 6, 9]
[3, 7, 7, 8, 10]
[3, 8, 8, 10, 11]
[4, 10, 10, 12, 14]
[4, 6, 7, 7, 11]
[4, 8, 8, 8, 12]
[4, 9, 9, 10, 13]
[5, 10, 10, 10, 15]
[5, 11, 11, 12, 16]
[5, 12, 12, 14, 17]
[5, 13, 13, 16, 18]
[5, 14, 14, 18, 19]
[5, 6, 8, 8, 13]
[5, 8, 9, 9, 14]
[6, 10, 11, 11, 17]
[6, 12, 12, 12, 18]
[6, 13, 13, 14, 19]
[6, 14, 14, 16, 20]
[6, 8, 10, 10, 16]
[7, 10, 12, 12, 19]
[7, 12, 13, 13, 20]
[7, 8, 11, 11, 18]
Another 9 (plus the 45 without repeated numbers) have the median=mean=range equalling one of the modes:
[1, 3, 3, 4, 4]
[2, 6, 6, 8, 8]
[3, 9, 9, 12, 12]
[4, 11, 11, 14, 15]
[4, 12, 12, 16, 16]
[4, 4, 6, 6, 10]
[5, 15, 15, 20, 20]
[6, 6, 9, 9, 15]
[8, 8, 12, 12, 20]