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Mean = median (Posted on 2014-02-20) Difficulty: 2 of 5
Add a fifth number to the set {3,7,9,10} so that the mean is equal to the median of the set.

Show whether or not it is possible to create a set of numbers where the same problem yields more solutions than the above.

No Solution Yet Submitted by Jer    
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Solution solution | Comment 2 of 4 |

If we are to make the 7 the median and mean, then we need to make the total 35. The total is already 29 so we need to add a 6.

If we are to make the 9 the median and mean, then we need to make the total 45. The total is already 29 so we need to add a 16.

This allows all integral members of the set. Ady has shown a way of making the new member be the mean and median, by using a non-integer.

In all cases, the maximum is three possibilities: either of the two existing central numbers or the newly added member.

Some cases where a set of four can have three numbers serve as answer and all be integers are:

original numbers         possible add-ons
3  5  7  9                  1  11  6
3  6  8  11                 2  12  7
3  6  9  10                 2  17  7
3  8  10  15                4  14  9
3  8  11  14                4  19  9
3  8  12  13                4  24  9
3  9  11  17                5  15  10
3  9  12  16                5  20  10
3  9  12  20                1  16  11
3  9  13  15                5  25  10
3  9  13  19                1  21  11
3  9  14  18                1  26  11
3  9  15  17                1  31  11
3  10  12  19               6  16  11
3  10  13  18               6  21  11
3  10  13  22               2  17  12
3  10  14  17               6  26  11
3  10  14  21               2  22  12
3  10  15  16               6  31  11
3  10  15  20               2  27  12
3  10  16  19               2  32  12
3  10  17  18               2  37  12
3  11  13  21               7  17  12
3  11  14  20               7  22  12
3  11  15  19               7  27  12
3  11  16  18               7  32  12
3  12  14  23               8  18  13
3  12  15  22               8  23  13
3  12  15  26               4  19  14
3  12  16  21               8  28  13
3  12  16  25               4  24  14
3  12  17  20               8  33  13
3  12  17  24               4  29  14
3  12  18  19               8  38  13
3  12  18  23               4  34  14
3  12  19  22               4  39  14
3  12  20  21               4  44  14
3  13  15  25               9  19  14
3  13  16  28               5  20  15
3  13  17  23               9  29  14
3  13  17  27               5  25  15
3  13  17  31               1  21  16
3  13  18  22               9  34  14
3  13  18  26               5  30  15
3  13  18  30               1  26  16

from

CLS
FOR a = 3 TO 10
FOR b = a + 1 TO 20
FOR c = b + 1 TO 40
FOR d = c + 1 TO 60
   sum = a + b + c + d
   avg = sum / 4
   IF avg = INT(avg) THEN
     IF avg > b AND avg < c THEN
       sum1 = 5 * b: sum2 = 5 * c
       new1 = sum1 - sum: new2 = sum2 - sum
       IF new1 <> a AND new1 <> b AND new1 > 0 AND new2 <> c AND new2 <> d THEN
         PRINT a; b; c; d, new1; new2; avg
         lct = lct + 1
       END IF
     END IF
   END IF
   IF lct > 44 THEN lct = 0: PRINT : STOP
NEXT
NEXT
NEXT
NEXT


 


  Posted by Charlie on 2014-02-20 16:07:36
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