Hayate is just pretending to be an idiot
She and Nagi are identical twin sisters. It's normal for twins to try acting differently, it's not normal for them to actually have different base abilities. Intelligence is definitely something they'd share.
- Arguable. Hayate is very bad at schoolwork, barely passing first-year finals; lacks common sense due to her being raised in a dojo/orphanage; and even Nagi, her own twin, thinks her normal behavior is something like this◊. One possible explanation would be that both Nagi and Hayate are idiots, though it manifests differently for each of them — Nagi has No Sense of Direction and is actually kind of awkward when it comes to interpersonal relationships despite her politeness to strangers.
- There are different kinds of intelligence, and habits - effort - matters quite a lot too. Do we know that Nagi is any better at schoolwork, for that matter? Nagi's mental picture of Hayate is pretty much exactly what I expected, but Hayate has shown some jarring flashes of intelligence at times.. until I can actually get hold of those books myself, I'm going to persist in suspecting she's (possibly they're) playing us all for fools. It's funnier that way.