The SAT exam allows for about two minutes to solve each math problem. The key to conquering the math section of the test is knowing how to break down a deliberately confusing question and sift through unnecessary details to quickly find the answer.
In a Quora thread of the most difficult SAT math problems, one question emerged as "the meanest test problem ever."
In a class of p students, the average (arithmetic mean) of the test scores is 70.
In another class of n students, the average of the scores for the same test is 92.
When the scores of the two classes are combined, the average of the test scores is 86.
What is the value of p/n?
Can you figure out how to solve it?
If not, don't fear — Presh Talwalkar, a math whiz who wrote the book "The Joy of Game Theory: An Introduction to Strategic Thinking" and tackles math questions and riddles on his YouTube channel and blog, both called MindYourDecisions, shared a step-by-step solution to this notoriously tough problem.
There are a few ways to solve it, but Talwalkar presents a simple shortcut.
The first class had an average of 70. That's 16 points below the average score of 86. In other words, 86 - 70 = 16. Since there are p students in the class, the difference from the average is 16p.
The second class had an average of 92. That's 6 points more than the average of 86. In other words, 92 - 86 = 6. There are n students in this class, so the difference from the average is 6n.
Because these classes average out together — as the problem says "when the scores of the two classes are combined" — the deficit of points has to be equal to the surplus of points. Therefore, 16p is equal to 6n.
Turning that into an equation, we can easily figure out what p/n is:
16p = 6n
p/n = 6/16, or 3/8
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