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.

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.