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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.

In the majority of incorrect proofs, the respondent will cite PEMDAS or BODMAS, however these solvers we all agree that parentheses or brackets are processed first. However, some respondents incorrectly resolve the [P]arentheses or [B]rackets of PEMDAS - BODMAS, ignoring the Distributive Law. For for some reason, they these solvers believe that they can solve resolve within the parentheses, but ignore 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'' confirms that this is the way we handle on parenthetical expressions:

::&ndash;[http://mathworld.wolfram.com/Parenthesis.html Wolfram Parenthesis]

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

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 =====6÷2(1+2) is not equal to 6÷2*Explicit vs Implicit (1+2implied)multiplication ==<div style="text-align:center">Standing alone, 6 will always equal 6 and 2(1+2) will also always equal six, so dividing these two terms will always equal 1. We would not write 2({{Quote|text='''''Implied multiplication''' has a+b) as (2(a+b))higher priority than explicit multiplication to allow users to enter expressions, nor would we re-write in the given equation same manner as 6÷2*(a+b)they would be written. Granted, the ''<br/> &ndash;[https://wwwepsstore.wyzantti.com/resourcesOA_HTML/blogscsksxvm.jsp?nSetId=103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators] }}</14831/the_obelus_for_division obelus div> Explicit operations have explicit operators (* ÷+ - )] 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 separate terms. ''Implied multiplication is an archaic symbol for divisioneverywhere, 1x = x and x/1=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='''im·plic·it''' ''adjective'' :1. implied though not plainly expressed. :2. essentially or very closely connected with; always to be found in. '''im·plied''' ''adjective'' :1. suggested but this is not a postgraduate level equationdirectly expressed; implicit. '''ex·plic·it''' ''adjective'' :1. stated clearly and in detail, this is middle-school level mathleaving no room for confusion or doubt.

{{Quote|text=''Implied multiplication 6÷2(1+2) has a higher priority than explicit two EXPLICIT operators, division and addition. However, IMPLIED multiplication tells us to allow users to enter expressionsIMPLICITLY multiply, [2(2+1)], in the same manner as they would be writtenP step of PEMDAS. In the division step, we must apply division to the (entire) implicitly connected parenthetical expression. We cannot apply division to the coefficient (2) and then not apply division to the factor (2+1).''<br/>

&ndash'''Incorrect:''' (''because division is not EXPLICITLY applied to the entire parenthetical expression'')<div style="color:red;"><div style="width:60px; text-align:center; float:left;[https">6 ---- 2 </div><div style="width:60px; text-align:center; float:left;"><br/><nowiki>*</nowiki><br/></div> <div style="width:60px; text-align:center; float:left;"><br/>(2+1)<br/epsstore.ti.com></OA_HTMLdiv> <div style="width:60px; text-align:center; float:left;"><br/csksxvm.jsp?nSetId>=103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators] }}<br/></div> <div style="width:60px; text-align:center; float:left;"><br/>3 * (1+2) <br/> </div></div></div><div style="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?]