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?

wiskdls
Mar 17, 2018

#1**+2 **

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

CPhill
Mar 18, 2018