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| caption1 = The Casio fx-9860 GII SD returns the correct answer to the equation as given, and . The next row provides the answer that a less sophisticated calculator would return.

'''6÷2(1+2)=?''' is the pre-algebra question that a majority of many Internet 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.}}

We substitute x for 2(1+2) from the given equationand backtest the equation with the result of 9:

there is a Common Denominator common divisor of 3 so we reduce the fraction...

Now we compare the value given in the equation to the value we find found (above) for x.

<p style==Processing rules== In the majority of incorrect proofs"text-align:center;"> {{highlight|Without fail, the respondent 2(1+2) will cite PEMDAS or BODMAS, however these solvers incorrectly resolve the [P]arentheses or [B]rackets of PEMDAS - BODMASalways equal (2*1 + 2*2), ignoring this is the '''Distributive Law. For some reason, they believe that they can solve within the parentheses, but ignore the adjacent and dependent coefficient. '''}}</p> == Proof of correct solution ==''Encyclopedia Britannica'':

Therefore, in 6÷2(1+2)=?, we start with the <span style="color:green;"> Distributive Law </span> in the <span style="color:green;">Parenthesis </span> step of the Order of Operations.

::'''6÷2(1+2)=?'''::'''6÷<span style="color:green;">(2*1 + 2*2)</span>=?'''

==Processing rules== In the majority of incorrect proofs the respondent will cite PEMDAS or BODMAS, we all agree that parentheses or brackets are processed first. However, some respondents incorrectly resolve the [P]arentheses or [B]rackets of PEMDAS - BODMAS, for some reason, these solvers believe that they can resolve within the parentheses, but neglect to multiply the adjacent and dependent coefficient. === PEMDAS ===<blockquote> ::P parenthesis::E exponentiation::M multiplication::D division::A addition::S subtraction ::&ndash;[https://mathworld.wolfram.com/PEMDAS.html Wolfram PEMDAS]</blockquote> === Parentheses ===''Wolfram Mathworld'' also confirms that this is the way we handle on parenthetical expressions:

::&ndash;[http://mathworld.wolfram.com/Parenthesis.html Wolfram Parenthesis]</blockquote> ===Distributive law===''Encyclopedia Britannica'' on parenthetical expressions: <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.''::&ndash;[https://www.britannica.com/science/distributive-law Encyclopedia Britticana]

Correct processing:::<s>'''<div style="color:greenred;">6÷3*(2*1 + 2*23)</div>'''</s>-->=== Parenthetical expressions ===

''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 <u>must be multiplied by the coefficient.</u>''' This is done in the following manner.''

==The equation is not ambiguous Explicit vs Implicit (implied) multiplication ==<div style="text-align:center">{{Quote|text='''''Implied multiplication''' has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written.''<br/> &ndash;[https://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=6÷2103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators] }}</div> Explicit operations have explicit operators (1* ÷ +2- ) which are delimiters that divide equations into separate terms. Implied multiplication is notation that informs us that the value of a variable, bracketed function or exponent are connected and not equal to 6÷2*(separate terms. ''Implied multiplication is everywhere, 1x = x and x/1+2)==x. Any number times 1 is that number and any number divided by 1 is that number.'' Implied multiplication is also implicit multiplication. {{Quote|text=Standing alone, 6 will '''im·plic·it''' ''adjective'' :1. implied though not plainly expressed. :2. essentially or very closely connected with; always equal 6 to be found in. '''im·plied''' ''adjective'' :1. suggested but not directly expressed; implicit. '''ex·plic·it''' ''adjective'' :1. stated clearly and 2in detail, leaving no room for confusion or doubt. &ndash;Oxford}} 6÷2(1+2) will also always equal sixhas two EXPLICIT operators, so dividing these two terms will always equal 1division and addition. We would not write However, IMPLIED multiplication tells us to IMPLICITLY multiply, [2(a+b) as (2(a+b)1)], in the P step of PEMDAS. In the division step, nor would we re-write must apply division to the given equation as 6÷2*(a+bentire)implicitly connected parenthetical expression. Granted, We cannot apply division to the [https://www.wyzant.com/resources/blogs/14831/the_obelus_for_division obelus coefficient (÷2)] is an archaic symbol for and then not apply division, but this is not a postgraduate level equation, this is middle-school level mathto the factor (2+1).

2'''Incorrect:''' (1+2) ''because division is understood to be a function, placing this function anywhere in a larger equation must always resolve not EXPLICITLY applied to the same value. entire parenthetical expression'')<div style="color:red;"><div style="width:60px; text-align:center; float:left;">6 ---- The equation, 6÷2(1+2) </div><div style=? is illustrative to why we cannot substitute explicit multiplication for implied multiplication, in other words, 6÷2(1+2) is not equal to 6÷2"width:60px; text-align:center; float:left;"><br/><nowiki>*(1+2).</nowiki><br/></div>

{{Quote|<div style="width:60px; text=''Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written.''-align:center; float:left;"><br/>(2+1)<br/></div>

&ndash<div style="width:60px; text-align:center; float:left;"><br/>=<br/></div> <div style="width:60px;[httpstext-align:center; float:left;"><br/>3 * (1+2) <br/epsstore.ti.com> </OA_HTMLdiv></div></csksxvm.jsp?nSetIddiv><div style=103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators] }}"clear:both;"></div>

'''Correct:''' (''when EXPLICIT division is applied, it is applied to the entire term.'')<div style="color:green;"><div style=Correctly solving "width:60px; text-align:center; float:left to right=;">6 ---- 1 </div><div style="width:60px; text-align:center; float:left;"><br/>÷<br/></div>

If we insist on a <div style="width:60px; text-align:center; float:left to right solution, ignoring both PEMDAS and the Distributive Law, we can do this with the Least Common Denominator rule;">2(1+2) ---- 1 </div>

<div style="width:60px; text-align:''6÷2(1+2)=?''center; float:left;"><br/>::''3÷(1+2)=?''::''3÷(3)=?''<br/>::''3÷3=1''</div>

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1*(1+2)

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= 1

Entering the equation into a programming language, or low quality calculator , requires the explicit multiplication symbol and outer parenthesisto maintain value of the parenthetical expression.

Other calculators, including Google and Wolfram will simply strip the parenthesis and solve a different equation, 6÷2*3. This is because within the programming, the open parenthesis (bracketing) triggers a different function within the programming, an open parenthesis tells the compiler to find the innermost parenthesized term and work outwards. Thus, to get the correct answer from inferior calculators, the input must be formalized with correct bracketing. I.e. 6÷(2*(1+2)) ==Spreadsheets==Entering the equation as =6/2(1+2) into a cell in a LibreOffice Calc spreadsheet will result in Err:509 (Missing operator) , Google Sheets also returns an error (Formula parse error). This forces the user to format the equation using explicit multiplication.To avoid the left to right problem, the compiler must be instructed in advance to prepare for the function with an extra pair of parenthesis =6/(2*(1+2)). ==See also==* [https://www.nytimes.com/2019/08/05/science/math-equation-pemdas-bodmas.html "That Vexing Math Equation? Here’s an Addition"]* [https://www.youtube.com/watch?v=hsZCtgFcL40 "How to Solve 8÷2(2+2) Using Implied Multiplication"]* [https://slate.com/technology/2013/03/facebook-math-problem-why-pemdas-doesnt-always-give-a-clear-answer.html "What Is the Answer to That Stupid Math Problem on Facebook?"]* [https://www.inc.com/dave-kerpen/this-basic-math-problem-is-breaking-internet.html This Basic Math Problem Is Breaking the Internet: How do you solve this simple arithmetic problem?]