I am currently studying approximation algorithms and I have run into an issue with a study problem.

The approximation algorithm is for the general Knapsack problem, and it proposes a greedy approach, where it sorts by the value/weight ratio, and picks the first item in this list that pushes the weight over the limit, and then picks either all the previous items or this particular item, whichever has a greater value.

The claim is that there is at least one case where this algorithm provides a value half that of the optimal value, but after many fruitless attempts I don't see a way to do this.

I have tried many value/weight pairs of 1/1 with a last pair of 5/6, resulting in a ratio of 5/9. Close, but even if I continue this process I am only going to approach a ratio of 1/2, but never achieve it.

Any hints as to how to proceed?