6÷2(1+2)



6÷2(1+2) =
6÷2(1+2) =


Least Common Denominator (LCD) before we divide.

6̸ ÷ 2̸(1+2) == 3 ÷ (1+2)

The parenthetical expression now has a coefficient of 1, i.e., 1(1+2).  (This is because 6 ÷ 2 = 3 AND 2 ÷ 2 = 1)

3 ÷ 1(1+2) =


Therefore, finding the LCD does not "clear" the parenthetical expression, nor does it complete the division. Next, we have to process the parenthesis to obtain an integer. (1*1) + (1*2) == 3.  (The 1 is always present, because 0(1+2) == 0)

3 ÷ 1(1+2) == 3 ÷ 3

Finally, we can express division to the parenthetical factor  -- finishing the division step.

3 ÷ 3 = 1

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Sanity Check

Some people are incorrectly arriving at 6 ÷ 2(1+2) = 9.

There is only one number we can divide 6 by to arrive at 9, and that number is 2/3. However, 0.666666666666666 will give us a pretty close decimal representation.

6 ÷ 0.666666666666666 = 9

But, when we enter 2(1+2) into any calculator (or distribute manually) the answer is always 6.

0.666666666666666  represented as 2/3 below.

   6          2  
  ―   ÷   ―  = x
   1          3      


   6          3  
  ―   *   ―  = x
   1          2

      

  18    
  ―   = 9
   2         

If the denominator in 6 ÷ 2(1+2) does not resolve to 2/3, the answer cannot be 9. Therefore, these solvers have an error in their logic.  The only way this denominator can equal 2/3 is to change the implied multiplication in 2(1+2), to explicit division. 2 ÷ (1+2) == 2/3.

These "left to right" solvers have some bizarre theories, first that the Distributive law does not apply to first order of operations, they solve within the parenthesis (the factor) and then ignore the coefficient.  2(1+2)


Then, left to right solvers believe that they can substitute explicit multiplication where implied multiplication is given.  Effectively by inserting explicit multiplication   6 ÷ 2 * (1+2), they are changing the equation to  6(1+2) ÷ 2 = 9.  This is not the equation as given.

In cases where we are required to insert explicit multiplication (cheap calculators and computer programs), we must protect the parenthetical expression with outer parenthesis,  6 ÷ (2 * (1+2)) = 1.

To error is human, to really screw up, you need a computer.


An example of failure to observe The Distributive Law.

The Desmos online calculator will handle the user input correctly for this equation when using the keypad.

https://www.desmos.com/scientific

Advanced Casio calculators also process correctly.



Google and Wolfram use the older math modules that don't recognize the coefficients (implied multiplication) in some parenthetical groupings.  For these older modules, we must be careful to enforce the groupings with another pair of parenthesis: 6÷(2*(1+2))
The Casio fx-9860GII SD returns the correct answer to 6÷2(1+2)
The Casio fx-9860GII SD

The Casio fx-9860GII SD returns the correct answer to the equation as given and the alternate notation if you were expecting a result of 9. (Source)
Therefore:

 6÷2(1+2) != 6÷2*(1+2)

AND 6÷2(1+2) == 6÷(2*(1+2))


Additional proofs below:
For those who claim that the obelus "÷" does not represent a fraction. See: The Obelus for Division

Whole numbers can be represented as fractions:

   6          2(1+2)    
  ―   ÷   ―――   = x
   1               1      

Is equivalent to multiplying the inverse:

 6             1
―   *   ―――    = x
 1         2(1+2)  

Is equivalent to:

     6
  ――     = x
  2(1+2)  

Is equivalent to:

     6
  ――     = 1
     6 

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.

distributive law | mathematics | Britannica.com

https://www.britannica.com/topic/distributive-law

6 ÷ 2(1+2) =

Parenthesis is the first step in the PEMDAS order of operations, so we apply the Distributive Law.


 6 ÷ 2(1+2) =

Distribute

6 ÷ 2(1+2) =

6 ÷ 2(3) =

6 ÷  6 = 1

Remedial: 6÷2(1+2)

We have 6 people. We also have 2 baskets that contain (1+2) apples in each basket. Divided evenly, how many apples will each person get?

First: we find that there are 3 apples in each basket (1+2) = (3).
Second: we perform the implied multiplication to find that there are 6 apples in 2 baskets 2(3) = 6.
Third: we divide the number of people by the number of apples 6 ÷ 6 = 1.

We don't care about how many baskets, nor are they relevant to the people, we are finding for apples. In a parenthetical expression, the (factor) is the number of apples, the coefficient() is the number of baskets.





Footnotes:
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.

distributive law | mathematics | Britannica.com

https://www.britannica.com/topic/distributive-law

The Fundamentals of Algebra (1983)

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 must be multiplied by the coefficient. This is done in the following manner.
5(a + b) = 5a + 5b
3a(b - c) = 3ab - 3ac

Parenthesis

3. Parentheses are used to enclose the variables of a function in the form f(x), which means that values of the function f are dependent upon the values of x.

 Weisstein, Eric W. "Parenthesis." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/Parenthesis.html
TEXAS INSTRUMENTS:
Does implied multiplication and explicit multiplication have the same precedence on TI graphing calculators?

Implied multiplication has a higher priority than explicit multiplication to allow users to enter expressions, in the same manner as they would be written. For example, the TI-80, TI-81, TI-82, and TI-85 evaluate 1/2X as 1/(2*X), while other products may evaluate the same expression as 1/2*X from left to right. Without this feature, it would be necessary to group 2X in parentheses, something that is typically not done when writing the expression on paper.

Implied Multiplication Versus Explicit Multiplication on TI Graphing Calculators.
Schaum's Outline of Precalculus

By Fred Safier
McGraw Hill Professional, Oct 22, 1997 - Mathematics - 407 pages

DISTRIBUTIVE LAWS:

a(a + b) = ab + ac; also (a + b)c = ab + ac: multiplication is distributive over addition.
(Page 1)


ORDER OF OPERATIONS: In expressions involving combinations of operations, the following order is observed:

1. Perform operation within grouping symbols first, If grouping symbols are nested inside other grouping symbols, proceed to the innermost outward.

2. Apply exponents before performing multiplications and divisions, unless grouping symbols indicate otherwise.

3. Perform multiplications and divisions in order from left to right before performing additions and subtractions (also from left to right), unless operation symbols indicate otherwise.

(Page 4)

https://books.google.com/books?id=EwnGj0GpJq4C

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