Is the 2 Related to the Numbers in Parentheses?

Date: 10/05/2001 at 11:13:49
From: Tressie Grant
Subject: Order of operations

I'm having difficulty understanding the correct procedure for an 
order of operations 6th grade math problem. My math-oriented family 
disagrees with the teacher. Which is the correct way to solve this 
problem: 36/2(8-5) = ?

A) The 2 is related to the numbers in the parentheses. Using the
distributive property to show this, the equation would then be    
36 / (2x8 -2x5) 
								
	  = 36 / (16-10)
	  = 36 / 6
	  = 6
or

B)   36/2(8-5) = also written

     36/2x(8-5) = parenthesis first and then they disappear

     36/2x3 =   order of operations says divide and multiply in order 

 so  36/2 = 18  then multiply 18x3 = 54

The answer changes due to the division taking precedence when 
following the right-to-left order for division and multiplication.

Date: 10/05/2001 at 12:14:29
From: Doctor Ian
Subject: Re: Order of operations

Hi Tressie,

If the person who wrote the expression intended for the 2 to be 
related to the numbers in the parentheses, he should have written

  36/(2(8-5))

As it is, the standard order of operations specifies that parentheses 
are to be resolved before anything else; and once parentheses have 
been resolved, multiplications and divisions should be resolved next, 
working from left to right; so

  36/2(8-5) = 36 / 2 * (8 - 5)

            = 36 / 2 * 3

            = 18 * 3

            = 54

The original expression, without the extra parentheses, is a little 
like this sentence: "Bob and Alice were drunk, so we took their keys 
and drove them home."  Most people would 'know' that the speaker drove 
Alice and Bob home; but the rules of English say that the speaker 
drove the keys home, leaving us in the dark about what happened to 
Alice and Bob.

This kind of ambiguity is largely ignored in everyday language, where 
semantic information is sufficient for disambiguation. However, in 
math, numbers (like '2') have no semantic information associated with 
them, so it becomes essential to know what the rules are, and to 
follow them. 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/