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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!
Associated Topics:
High School Functions

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