The sum of five different positive integers is 320. The sum of the greatest three integers in this set is 283. The sum of the greatest and least integers is 119. If x is the greatest integer in the set, what is the positive difference between the greatest possible value and least possible value for x?
I'm using "e" for "x"
Let a < b < c < d < e
So we have......
a + b + c + d + e = 320 (1)
Where "a" is the smallest integer and "e" is the largest
And
c + d + e = 283 (2)
a + e = 119 (3)
Sub (2) into (1)
a + b + 283 = 320
a + b = 37
Let a = 1
Then.....as large as "e" can be is
1 + e = 119
e = 118
Let a = 36
Then...... a possibility for as small as "e" can be is
36 + e = 119
e = 83
However c < d < e
And c + d + e = 283
So....e will be minimized when the sum of c and d are as large as possible
If c = d = e = 95
95 * 3 = 285....but c and d are < e.....so....c and d < 95 and c < d
So....the largest that the sum of c and d can be is 93 + 94 = 187
But.....
93 + 94 + 95 = 282 ......too small
Then e cannot be < 96 because 93 + 94 + 96 = 283
So.....as small as e can be is 96 and as large as e can be is 118
So.....the positive difference between the greatest positive value of e and the least positive value of e is 118 - 96 = 22