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Humphrey Huang
In 1982, the SAT was administered to 300,000 students. The test was divided into two sections: the verbal section, consisting of around 85-90 questions, and the math section, consisting of 60-70 questions. However, there was a single question which everyone answered incorrectly.
The question is as follows:
My first instinct was to divide the circumference of circle B by circle A. This approach felt logical, as I assumed that the distance circle A travels is equal to the circumference of circle B. By dividing the circumferences, I got an answer of three, which corresponded to choice B. This was the correct answer as marked on the 1982 SAT.
The problem is that three is not the correct answer. In fact, none of the answer choices on this SAT question are correct. But why is this the case?
Suppose you have one coin. If you roll one coin the length of its own circumference in a linear path, it would result in one rotation.
Now, let’s say you have two identical coins, You roll one coin around the other until it reaches its starting point. Using the same logic I previously used to solve the SAT problem, you would expect that the coin makes one rotation. But by rolling one quarter around another quarter, the result is actually two rotations. How can this paradox be?
In order to explain this phenomenon, one needs to understand the concept of relative motion: when one coin travels along the circumference of another, the path is a combination of linear and circular motion. This complex movement causes the coin to rotate more than anticipated from a simple linear perspective. The point of contact between the coins changes continuously while it rolls, contributing to extra rotations.
Let’s apply this concept using mathematical principles. Suppose there are two coins: coin A and coin B. If coin A rolls around coin B, the distance the center of coin A travels is the amount coin A rotates. This is because the tangential velocity of coin B relative to coin A is equal to the velocity of the center of coin A relative to coin B. Therefore, the number of revolutions of the outer coin corresponds to the ratio between the radius of the inner coin and the outer coin, increased by one.
Applying this reasoning, let’s solve the SAT question. As the expected number of rotations is three, our equation yields an answer of four rotations.
Three students identified this error in the SAT, and promptly reported it to the College Board. Following the acknowledgment of the mistake by the College Board, the question was subsequently removed from the SAT, causing a 10-point shift in the math section. Back in the 1980s, the SAT held significant weight in college admissions. A mere 10-point difference could determine acceptance or rejection from acclaimed universities, meaning numerous students were unable to gain admission to top-tier schools due to the adjustment in the SAT’s math section.
In recent years, the significance of the SAT in college decisions has been on the decline. With more than 80 percent of colleges being test optional as of 2023, the SAT’s importance as a sole determinant for admission has waned. Other factors, such as academic potential, extracurriculars, essays, and recommendation letters play a much bigger role in a college application. This multifaceted evaluation better measures a student’s success in college and beyond, rendering the SAT as one of several factors in the admissions process.
