Do I just eat June to August on a nine month lease?

Both structures exist and they're priced differently, so the comparison only works annualized.

A 12 month by-the-bed lease means each student signs for their own bed for a full year, usually with a parent signing as guarantor, and pays every month including summer. That's the standard in purpose-built student housing. An academic-year lease covers roughly mid-August to mid-May and the bed is yours again after that. The monthly rent on the nine month version is normally higher to make up some of the gap, and the operator either takes the summer vacancy or fills it with interns, grad students, summer session, or conference groups if the campus generates any.

So run it flat. A bed at $700 for twelve months is $8,400. A bed at $850 for nine months is $7,650, and anything you collect in summer sits on top. Whichever number is bigger after turn costs is your answer, and turn costs are real, because on a nine month lease you're repainting and re-carpeting every single unit in the same two weeks in August.

The part that catches people out is pre-leasing. Student leasing runs on a calendar, and at most campuses the following August is largely signed by the previous winter or early spring. If you buy in June with beds unsigned, you're not late by a few weeks, you've likely missed the season and you're chasing the leftovers. Ask the seller for a signed-bed count for the coming term before you agree on a price, in writing, not verbally.