But Operation Varsity Blues, the racketeering investigation that led to charges against 50 people, is one of the most audacious schemes yet: Prosecutors say proctors were bribed to fake scores, test takers were hired to impersonate students and at least one family was encouraged to falsely claim their son had a disability.

[Read more on the Justice Department’s largest ever college admissions prosecution.]

The SAT and ACT are not aptitude or IQ tests. They are intended to assess how well students have mastered standard high school reading and math concepts.

Though tutoring companies often brag about how much they can boost scores, there is no clear consensus on how much test preparation actually changes the final result, or whether test prep warps the results or merely demonstrates a student’s diligence.

Dennis Yim, the director of academics at Kaplan, one of the country’s largest test prep companies, said prep wasn’t to blame for an unequal system. “Studying for these tests isn’t about doing something that is circumventing the test, it’s about learning the content,” he said.

Kaplan now livestreams some prep sessions for free, and some school districts provide free prep courses. In 2017, the College Board, which administers the SAT, and Khan Academy, an education app, announced that students who used a free service saw gains on their scores.

But for most families, drilling for the exams remains an expensive proposition, ranging from $299 for a self-guided study course at Kaplan to hundreds of dollars an hour for the priciest private tutors. The strongest predictor of a student’s score “is affluence of parents and education of parents,” said Steve Syverson, a vice chancellor for enrollment at the University of Washington Bothell.

The anxiety surrounding the tests, Mr. Syverson said, is premised on the myth that there are only a few schools worth fighting — and even cheating — for.