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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>

''Encyclopedia Britannica'' on parenthetical expressions:

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.

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.<!--::'''Note''' Even more puzzling is that incorrect solvers will cite PEMDAS and then suspend it, doing the addition within the parenthesis before multiplication. Rather than correctly distributing the multiplication across the parenthetical term. I.e., 2(1+2) == (2 * 1 + 2 * 2). (See the Distributive Law below.)-->

=== Parenthetical expressions Parentheses ===

=== The Fundamentals of Algebra (1983) Parenthetical expressions ===

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 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 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1+2)<br/>:--------- ÷ ---------=::2 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1<br/>

'''Correct:''' (''For multiplication, inverting the denominator will maintain its value.'') <div style="width:200px60px; text-align:center; float:left; color:green;">::6 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1<br/>:--------- 3 * ---------::2 &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (1+2) <br/> </div></div>

==The equation '''Correct:''' (''when EXPLICIT division is not ambiguous =====6÷2(1+2) applied, it is not equal applied to 6÷2*(1+2the entire term.'')<div style="color:green;"><div style=="width:60px; text-align:center; float:left;">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><div style="width:60px; text-align:center; float:left;"><br/>÷<br/></div>

We would not write <div style="width:60px; text-align:center; float:left;">2(a1+b) as (2(a+b)), nor can we re ----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)). Granted, the [https://www.wyzant.com/resources/blogs/14831 1 </the_obelus_for_division obelus (÷)] is an archaic symbol for division, computers use the forward slash for division 6/(2*(a+b)).div>

* 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)<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)."width:60px; text-align:center; float:left;"><br/>=<br/></div>

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

<div style="width:100px60px; text-align:center; float:left;">

<div style="width:100px60px; text-align:center; float:left;">

1*(1+2)

<div style="width:100px60px; text-align:center; float:left;">

<div style="width:100px60px; text-align:center; float:left;">

<div style="width:100px60px; text-align:center; float:left;">

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))