Question

What is the sum of the positive integers k such that k/27 is greater than 2/3 and less than 8/9?

Answer 1

so we want to make them have the same denomenator so

k/27
2/3 times 9/9=18/27
8/9 times 3/3=24/27

so k/27>18/27 and k/27<24/27 or
18/27<k/27<24/27

the possible numbers for k are 19,20,21,22,23
so the sum of the positive integers could be any positive whole numbers (1,2,3,4, etc) that add up to 19,20,21,22,23 exg 10,9 10,10 10,11 20,2 21,2 etc

Answer 2

`(k)/(27)>(2)/(3) \wedge (k)/(27)<(8)/(9)\\ k>18 \wedge k<24\\ k\in(18,24)\\\\ k\in(18,24) \wedge k\in \mathbb{Z}\\ k\in\{19,20,21,22,23\}\\\\ 19+20+21+22+23=\boxed{105}`