Financial aid letters are designed to make the gift component look larger than it is. The first job of any family receiving an offer is to decode it into its actual components.
Five categories of money

Every offer typically contains some mix of grants (gift aid based on need), scholarships (gift aid based on merit), federal loans (debt), private loans (debt), and work-study (potential earned income). Only grants and scholarships are aid in the sense most families mean. Loans and work-study are obligations or possibilities, not gifts.

The actual net price

Net price is the published cost of attendance minus grants and scholarships only. Loans and work-study should be calculated separately. Two offers that present similar “total aid” figures at the top of the page can produce dramatically different actual net prices once loans are subtracted.

Scholarship names are wrappers

The New York Times has documented cases of single students receiving multiple simultaneously-named scholarships from the same school — a Distinguished Scholar Award, a Chancellor’s Award, a Personal Distinction Award — all from one institution, all reflecting one underlying decision: the school wanted to discount tuition by a specific dollar amount to enroll that student. The names are marketing. The dollar amount is the price.

Cross-check against the school’s average

The Common Data Set, which every American college publishes annually, reports the school’s average non-need-based aid award. Compare your student’s offer to that average. If it’s in the same range, your student is being treated as a typical merit recipient and the offer is unlikely to move. If it’s materially below, there’s room for an appeal. If it’s materially above, the school is signaling unusual interest.

The cleanest single rule

Net price equals sticker price minus grants and scholarships only. Every other figure on the letter is either a wrapper for that number or a separate financial obligation being presented as if it were aid. Once you have the actual net prices for every school, the comparison becomes possible — and the question of whether to appeal becomes answerable.