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'''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="text-align:center;"> {{highlight|Without fail, 2(1+2) will always equal (2*1 + 2*2), this is the '''Distributive Law.'''}}</p>

== The Fundamentals Proof of Algebra correct solution ==''Encyclopedia Britannica'': <blockquote>::'''''Distributive law''''', ''in mathematics, the law relating the operations of multiplication and addition, stated symbolically, a(1983b + 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]</blockquote>

<blockquote>{{Quote|text=''Parenthetical Expression. The parenthesis was described Therefore, in Chapter 6÷2(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+2)=?, we start with the parenthetical expression is a factor and <uspan style="color:green;">must be multiplied by the coefficient.Distributive Law</uspan>''' This is done in the following manner.''<span style="color::''5(a + b) = 5a + 5b'' green;">Parenthesis<br/span>::''3a(b - c) = 3ab - 3ac''step of the Order of Operations.

&ndash;[https://books.google.com/books:'''6÷2(1+2)=?id'''::'''6÷<span style=BabtEFxgZ2AC "Technical Shop Mathematics color:green;">(2*1 + 2*2)</ Edition span>=?'''::'''6÷(2"], by John G. Anderson, ISBN-13+ 4)=?''':9780831111458, Industrial Press, Inc., 02/28/1983, Page:138'''6÷(6)=?'''}}</blockquote>::'''6÷6=1'''

In the majority of incorrect proofs the respondent will cite PEMDAS or BODMAS, we all agree that parenthesis parentheses or brackets are processed first. However, these solvers 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

::'''Note''' Even more puzzling is that incorrect solvers will cite PEMDAS and then suspend it, doing the addition within the parenthesis before multiplication&ndash;[https://mathworld. Rather than correctly distributing the multiplication across the parenthetical termwolfram. Icom/PEMDAS.e., 2(1+2) == (2 * 1 + 2 * 2). (See the Distributive Law below.)html Wolfram PEMDAS]</blockquote>

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

===Distributive law===''Encyclopedia BritticanaBritannica'' on parenthetical expressions:

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

<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 <u>must be multiplied by the coefficient.</u>''' This is done in the following manner.''::''5(a + b) = Implicit vs implied multiplication =5a + 5b'' <br/>::''3a(b - c) =3ab - 3ac''

Implicit operations have explicit operators (* ÷ + - ) which are delimiters that divide equations into separate terms&ndash;[https://books.google. Implied multiplication is notation that informs us that the value of a variablecom/books?id=BabtEFxgZ2AC "Technical Shop Mathematics / Edition 2"], bracketed function or exponent are connected and not separate termsby John G. ApparentlyAnderson, ISBN-13:9780831111458, this rule has been forgotten with the advent of modern calculatorsIndustrial Press, Inc., 02/28/1983, Page:138}}</blockquote>

6÷2== Explicit vs Implicit (1+2implied) multiplication ==<div style="text-align:center">{{Quote|text='''''Implied multiplication''' has one implicit operator outside of a higher priority than explicit multiplication to allow users to enter expressions, in the parenthesis and therefore has two termssame manner as they would be written.''<br/>

<div style="width:100px&ndash; float[https:left;">::6 //epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators]:--------- :2(1+2) }}

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 everywhere, 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 not directly expressed; implicit. '''ex·plic·it''' ''adjective'' :1. stated clearly and in detail, leaving no room for confusion or doubt. &ndash;Oxford}} 6÷2(1+2) has two EXPLICIT operators, division and addition. However, IMPLIED multiplication tells us to IMPLICITLY multiply, [2(2+1)], in the P 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). '''Incorrect:''' (''because division is not EXPLICITLY applied to the entire parenthetical expression'')<div style="color:red;"><div style="width:200px60px; text-align:center; float:left; color:red;">::6 :--------- * (1+2)::2

<div style="clearwidth:both60px;"></div>Where a calculator or programming language requires implicit operators, we maintain the value of the equation by adding an additional grouping pair of parenthesis 6÷(2*(1+2)) for inline notation.<div style="widthtext-align:100pxcenter; float:left;">::6 <br/>:--------- <nowiki>*</nowiki>:(2*(1+2)) <br/>

<div style==The equation is not ambiguous =="width:60px; text-align:center; float:left;"><br/>===6÷2(12+2) is not equal to 6÷2*(1+2)===Standing alone, 6 will always equal 6 and 2(1+2) will also always equal 6, so dividing these two terms will always equal 1. <br/></div>

We would not write 2(a+b) as (2(a+b)), nor can we re<div style="width:60px; text-write the given equation as 6÷2*(a+b). We can however, rewrite the expression to include an implicit multiplier if we take care to maintain the grouping with additional bracketing for use with calculators and programming languages 6÷(2*(a+b)). Granted, the [httpsalign:center; float:left;"><br/>=<br/www.wyzant.com/resources/blogs/14831/the_obelus_for_division obelus (÷)] is an archaic symbol for division, computers use the forward slash for division 6></(2*(a+b)).div>

<div style="width:60px; text-align:center; float:left;"><br/>3 * 2(1+2) is understood to be a function, placing this function anywhere in a larger equation must always resolve to the same value. The equation, 6÷2(1+2) <br/> </div></div></div><div style=? is illustrative to why we cannot substitute explicit multiplication for implied multiplication without additional bracketing, in other words, 6÷2(1+2) is not equal to 6÷2*(1+2)."clear:both;"></div>

{{Quote|text='''Correct:''Implied multiplication'('' has a higher priority than explicit multiplication when EXPLICIT division is applied, it is applied to allow users to enter expressions, in the same manner as they would be writtenentire term.'')<div style="color:green;"><div style="width:60px; text-align:center; float:left;">6 ---- 1 </div><div style="width:60px; text-align:center; float:left;"><br/>÷<br/></div>

&ndash<div style="width:60px; text-align:center;[httpsfloat://epsstore.ti.com/OA_HTML/csksxvm.jsp?nSetId=103110 Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators]left;">2(1+2) ---- }}1 </div>

<div style==Correctly solving "width:60px; text-align:center; float:left to right=;"><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;"> ::''6÷22 * (1+2)=?'' ::''3÷1(1+2)=?'' ---- ::''3÷(6 * 1+2)=?'' </div></div></div>::''3÷(3)<div style=?''"clear::''3÷3=1''both;"></div>

<div style==Solution written as a fraction ==In the denominator, both the coefficient and the factor must be divided in the following manner. "color:green;"><div style="width:100px60px; text-align:center; float:left;">

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1*(1+2)

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Entering the equation into a programming language, or low quality calculator, requires the explicit multiplication symbol and outer parenthesis to maintain value of the groupingparenthetical 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 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 in programming languages, 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?]