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 $x$

17

Step-by-step explanation:

Let a, b, c, d, x represent the five integers in Ascending order.

Given;

The sum of five different positive integers is 320

a+b+c+d+x = 320 ... 1

The sum of the greatest three integers in this set is 283

c+d+x = 283 ... 2

The sum of the greatest and least integers is 119

a+x = 119 ... 3

For the largest possible value;

From equation 3, if a is as low as 1,

x = 119-1 = 118

x = 118

For the least possible value;

Subtract equation 2 from 1

a+b = 320-283 = 37

We know that a cannot be higher than b, so the highest possible value of a is;

a = 37-b

a = 37 - 19 = 37-19

a = 18

Substituting into equation 3

x = 119-18 = 101

Difference;

∆x = 118-101 = 17