As I explained above, you can adjust the "total" for checking "at most half". Depends on whether you want to count in halves or not.

Reminds me of that puzzle where siblings inherit a herd of horses.

A wealthy merchant at his death left his stable of 17 horses to his three sons. He specified that the eldest was to have one half of the horses, the next one-third, and the youngest one-ninth. The three young heirs were in despair, for they obviously could not divide 17 horses this way without calling in the butcher. They finally sought the advice of an old wise friend, who promised to help them.

She arrived at the stable the next day, leading one of her own horses. This she added to the 17 and directed the sons to make their choices. The eldest took one half of the 18, or nine; the next one-third of the 18, or six; and the youngest, one-ninth of 18, or two. When all 17 of the original horses had been chosen, the old woman took her own horse and departed.

The catch here is that the willed fractions didn't add up to 1, but to 17/18ths. Going with the "at most half" criteria, it doesn't say that both sides need to add to the total (assumption klaxon blaring). I don't see that it makes any difference whether to include, exclude, or split points on the line.