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 Mean = median (Posted on 2014-02-20)
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 No Rating

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 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-ons3  5  7  9                  1  11  63  6  8  11                 2  12  73  6  9  10                 2  17  73  8  10  15                4  14  93  8  11  14                4  19  93  8  12  13                4  24  93  9  11  17                5  15  103  9  12  16                5  20  103  9  12  20                1  16  113  9  13  15                5  25  103  9  13  19                1  21  113  9  14  18                1  26  113  9  15  17                1  31  113  10  12  19               6  16  113  10  13  18               6  21  113  10  13  22               2  17  123  10  14  17               6  26  113  10  14  21               2  22  123  10  15  16               6  31  113  10  15  20               2  27  123  10  16  19               2  32  123  10  17  18               2  37  123  11  13  21               7  17  123  11  14  20               7  22  123  11  15  19               7  27  123  11  16  18               7  32  123  12  14  23               8  18  133  12  15  22               8  23  133  12  15  26               4  19  143  12  16  21               8  28  133  12  16  25               4  24  143  12  17  20               8  33  133  12  17  24               4  29  143  12  18  19               8  38  133  12  18  23               4  34  143  12  19  22               4  39  143  12  20  21               4  44  143  13  15  25               9  19  143  13  16  28               5  20  153  13  17  23               9  29  143  13  17  27               5  25  153  13  17  31               1  21  163  13  18  22               9  34  143  13  18  26               5  30  153  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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