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'''6÷2(1+2)=?''' is the pre-algebra question that a majority of respondents seem to get wrong. If you are in the camp that knows the correct answer is 1, read no further. If you believe the answer is 9, you should probably review this document -- {{highlight|there is only one number that 6 can be divided by to result in 9, and that number is 2/3.}}
==Processing rules==
In the majority of incorrect proofs, the respondent will cite PEMDAS or BODMAS, however these solvers incorrectly resolve the [P]arentheses or [B]rackets of PEMDAS - BODMAS, ignoring the Distributive Law. For some reason, they believe that they can solve within the parentheses, but ignore the adjacent and dependent coefficient.
<blockquote>
::'''''Distributive law''''', ''in mathematics, the law relating the operations of multiplication and addition, stated symbolically, a(b + c) = ab + ac; that is, the monomial factor a is distributed, or separately applied, to each term of the binomial factor b + c, resulting in the product ab + ac.''
::–[https://www.britannica.com/science/distributive-law Encyclopedia Britticana]
</blockquote>
Therefore, in 6÷2(1+2)=?, we start with the Distributive Law in the Parenthesis step of the Order of Operations.
''Wolfram Mathworld'' also confirms that this is the way we handle parenthetical expressions:
<blockquote>
:: 1. ''Parentheses are used in mathematical expressions to denote modifications to normal order of operations (precedence rules)...''
::[...]
::3. ''Parentheses are used to enclose the variables of a function in the form f(x), which means that values of the function f are dependent upon the values of x.''
::–[http://mathworld.wolfram.com/Parenthesis.html Parenthesis]
</blockquote>
So the fatal error with solvers who emphasize the left to right rule, is in incorrectly handling the Parentheses step, disregarding the Distributive Law.
=== The Fundamentals of Algebra (1983) ===
<blockquote>
{{Quote|text=
''Parenthetical Expression. The parenthesis was described in Chapter 1 as a grouping symbol. When an algebraic expression is enclosed by a parenthesis it is known as a parenthetical expression. '''When a parenthetical expression is immediately preceded by coefficient, the parenthetical expression is a factor and must be multiplied by the coefficient.''' This is done in the following manner.''
–[https://books.google.com/books?id=BabtEFxgZ2AC "Technical Shop Mathematics / Edition 2"], by John G. Anderson, ISBN-13:9780831111458, Industrial Press, Inc., 02/28/1983, Page:138
}}
</blockquote>
==The equation is not ambiguous ==
Our given equation is two terms, 6 is always equal to 6 and 2(1+2) is also always equal to 6, dividing these two numbers must always equal 1. The equation is not ambiguous, it challenges the theory that multiplication and division are equal in terms of the Order of Precedence. First, let's review fundamental fact that x÷1 = x and that 1*x = x. When we enclose a term within parentheses, implicit multiplication assumed, if the multiplier is not given, the multiplier is the number 1 because 0 times any number is 0.
<div style="background-color:#E8E8E8; padding-left:30px;">
Example that ignores the Distributive Law:
'''3÷3=1'''
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'''Therefore,''' if the calculator module has already processed the division 6÷2 by the time it reaches the parenthesis, the coefficient and the accompanying implicit multiplication has been eliminated. The division must be sustained to apply to the remainder of the function, in left to right processing, it cannot be presumed that an implicit/implied operator always infers multiplication in computer mathematics modules. In this case, the processing must branch.