Mystery Operation: What Do These Strange Symbols Mean?
Date: 03/09/2010 at 09:15:02 From: Carol Subject: I have this question on a sample SAT test. Can you explain For all numbers a and b, a ⇒ b = 1/3a + 2/3b a ⇐ b = 2/3a + 1/3b What is the value of (3 ⇒ 2) ⇐ 6? I'm at a loss on this one! I don't know what the symbols mean; and without understanding them, I can't even attempt the question. I have tried asking all of the secondary math teachers in my middle school, and no one is able to explain it. I appreciate your help!
Date: 03/09/2010 at 10:46:08 From: Doctor Ian Subject: Re: I have this question on a sample SAT test. Can you explain Hi Carol, The problem itself _defines_ what the symbols mean. It's not something you'd be able to know in advance. The point of this kind of problem is to give you practice at reasoning abstractly, from definitions, rather than falling back on common sense. For example, suppose I tell you that a @ b = 2a + b It means that to evaluate the operation, you double the first number and add the second, so 3 @ 5 = 2(3) + 5 = 11 2 @ 9 = 2(2) + 9 = 13 -3 @ 7 = 2(-3) + 7 = 1 and so on. So maybe I define a couple of operations, a @ b = 2a + b and a % b = a^(b + 1) Then I might ask you to evaluate (3 @ 4) % 2 The normal rules for parentheses would apply, so we focus on '@' first: __________________________ This corresponds to a in the | definition of '@' (3 @ 4) % 2 |______________________ and this corresponds to b. We apply the definition of '@' to get _______ ___________________ Now this corresponds to a in the | | definition of '%' (2*3 + 4) % 2 |_______________ and this corresponds to b. Then we apply the definition of '%' to get (2*3 + 4)^(2 + 1) And now we have an expression that we can evaluate using the normal rules of arithmetic: (2*3 + 4)^(2 + 1) = (6 + 4)^3 = 10^3 = 1000 If we use ()'s to group differently, we get a different result: 3 @ (4 % 2) = 3 @ (4^(2 + 1)) = 2(3) + (4^(2 + 1)) = 6 + 4^3 = 6 + 64 = 70 Being able to do this kind of thing is a big step towards being able to reason about compositions of functions, and that's a big part of mathematics after basic algebra. So that's the ability that they're trying to assess, without testing your knowledge of functions directly. Does this make sense? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/
Date: 03/09/2010 at 21:48:32 From: Carol Subject: Thank you (I have this question on a sample SAT test. Can you explain) Now that I know I'm not looking for some obscure mathematical symbol, it makes much more sense. Thank you for laying out the steps so well, and I appreciate the logical reasoning behind the question. The answer to my problem is 32/9!
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