}}
'''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.}}
==Proof incorrect solution ==
We substitute x for 2(1+2) from the given equationand backtest the equation with the result of 9:
If
:6÷9 = x
there is a Common Denominator common divisor of 3 so we reduce the fraction...
:2÷3 = x
Now we compare the value given in the equation to the value we find found (above) for x.
:2÷3 '''!=''' 2(1+2)
:6÷2(1+2) '''!=''' 9
<p style="text-align:center;"> {{highlight|Without fail, 2(1+2) will always equal (2*1 + 2*2), this is the '''Distributive Law.'''}}</p>
== Proof of correct solution ==
''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.''
==Processing rules==
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. For , for some reason, they these solvers believe that they can resolve within the parentheses, but ignore neglect to multiply the adjacent and dependent coefficient.
=== PEMDAS ===
</blockquote>
=== Parenthetical expressions Parentheses ===
''Wolfram Mathworld'' on parenthetical expressions:
<blockquote>
::<s>'''<div style="color:red;">3*(3)</div>'''</s>
-->
=== The Fundamentals of Algebra (1983) Parenthetical expressions ===
<blockquote>
}}
6÷2(1+2) has two EXPLICIT operators, division and addition. However, IMPLIED multiplication tells us to IMPLICITLY multiply (, [2 * (2+1))], normally in the P step of PEMDAS. Then in In the division step, we must apply division to the (entire) implicitly connected termparenthetical 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 (1+2) the entire parenthetical expression'')<div style="color:red;"><div style="width:200px60px; text-align:center; float:left; color:red;">::6 :--------- * (1+2) </div><div style="width:60px; text-align:center; float:2 left;"><br/><nowiki>*</nowiki><br/>
</div>
<div style="clearwidth:both60px; text-align:center; float:left;"><br/>(2+1)<br/></div>
'''Correct:''' (''when EXPLICIT division is applied, it is applied to the entire term.'')<div style="width:200px60px; text-align:center; float:left; color:green;">::6 (1+2)<br/>:--------- ÷ ---------=::2 1<br/>
</div>
<div style="clear:both;"></div>
'''Correct:''' (''For multiplication, inverting the denominator will maintain its value.'') <div style="width:200px60px; text-align:center; float:left; color:green;">::6 1<br/>:--------- 3 * ---------::2 (1+2) <br/> </div></div>
</div>
<div style="clear:both;"></div>
As we can see'''Correct:''' (''when EXPLICIT division is applied, we cannot add an EXPLICIT operator it is applied to 6÷2(1+2) without maintaining the IMPLICIT relationship that was stated with IMPLIED multiplicationentire term. 6÷2(1+2'') <div style="color:green;"><div style= 6÷(2*("width:60px; text-align:center; float:left;">6 ---- 1+2)). </div><div style="width:60px; text-align:center; float:left;"><br/>÷<br/></div>
<div style==The equation is not ambiguous =="width:60px; text-align:center; float:left;">===6÷2(1+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. </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 full equation to include an implicit multiplier if we take care to maintain the grouping with additional bracketing 6÷(2*(a+b)). align:center; float:left;"><br/>=<br/></div>
Granted, the [https<div style="width:60px; text-align://web.archive.org/web/20210417145938/httpscenter; float://www.wyzant.com/resources/blogs/14831/the_obelus_for_division obelus (÷)] is an archaic symbol for division, it represents a fraction with one dot being the numerator and the other being the denominator. Computers use the forward slash for division 6/(left;">2*(a+b)) because the keyboard does not have an obelus key. * 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)=? 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÷26 *(1+2). </div></div></div><div style==Correctly solving left to right=="clear:both;"></div>
If we insist on a left to right '''Traditional solution, ignoring both PEMDAS and the Distributive Law, we can do this with the Least Common Denominator rule:'''
::''6÷2(1+2)<div style=?''"color::''3÷1(1+2)=?'' ::''3÷(1+2)=?''::''3÷(3)=?''::''3÷3=1'' When we reduce by the common denominator, this does not complete the division operation, division must be applied to both the factor and the coefficient. Division is sustained.* In the notation above, a multiplier of one is always implied for any number. For example, x == 1x. ==Correct solution written as a fraction ==In the denominator, both the coefficient and the factor must be divided in the following manner. green;"><div style="width:100px60px; text-align:center; float:left;">
6
----
2(1+2)
</div>
<div style="width:100px60px; text-align:center; float:left;">
<br/>
=
</div>
<div style="width:100px60px; text-align:center; float:left;">
3
----
1*(1+2)
</div>
<div style="width:100px60px; text-align:center; float:left;">
<br/>
=
</div>
<div style="width:100px60px; text-align:center; float:left;">
3
----
</div>
<div style="width:100px60px; text-align:center; float:left;">
<br/>
= 1
<br/>
</div>
</div>
<div style="clear:both;"></div>
<!--
As we can see, we cannot add an EXPLICIT operator to 6÷2(1+2) without maintaining the IMPLICIT relationship that was stated with IMPLIED multiplication. 6÷2(1+2) == 6÷(2*(1+2)).
-->
==Correctly solving left to right==
If we insist on a left to right solution, ignoring both PEMDAS and the Distributive Law, we can do this by expressing the common divisor first. However, division is sustained and must be applied to everything to the right of the explicit division operator.
::''6÷2(1+2)=?''
::''3÷1(1+2)=?''
::''3÷(1+2)=?''
::''3÷(3)=?''
::''3÷3=1''
When we reduce by the common divisor, this does not complete the division operation, it simply reduces the coefficient to one. Division must be applied to both the factor and the coefficient of the parenthetical expression.
==The equation is not ambiguous ==
===6÷2(1+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.
We would not write 2(a+b) as (2(a+b)), nor can we re-write the given equation as 6÷2*(a+b) and maintain the same value. We can however, rewrite the full equation to include an explicit multiplier if we take care to maintain the parenthetical expression with additional bracketing 6÷(2*(a+b)).
=== Obelus ===
Granted, the [https://web.archive.org/web/20210417145938/https://www.wyzant.com/resources/blogs/14831/the_obelus_for_division obelus (÷)] is an archaic symbol for division, it visually represents a fraction with one dot being the numerator and the other being the denominator. Computers use the forward slash for division because the standard keyboard does not have an obelus key.
For those who claim that the obelus is outdated, why then is it present on virtually every modern calculator?
== The calculator problem ==
The [https://www.desmos.com/scientific Desmos Scientific Calculator] handles our given equation correctly. The error checking appears to be built into the division key.
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.
'''6÷2(1+2) == 6÷(2*(1+2))'''
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?]