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What is the correct answer of 6/2(2+1) =...?

___________________________________________

Answer
    1
Explanation

The above question is looking very simple but there are many things to understand it. Mostly student just try to give the answer quickly and forget many mathematical things. So I suggest don't answer blindly. Here because my answer is 1 so I give the proof why answer should be 1.

1-Brackets

Parentheses (brackets) are just like a king of BODMAS because in BODMAS we have to solve the brackets first. _____ vinculum, () parentheses, {} braces, [] square brackets are all different type of paretheses (brackets).
We should also remember this that brackets have their own rules first solve the _____ vinculum, after this solve the ( ) parentheses, after this solve the { } braces, after this solve the [ ] square brackets.

    _____
          Vinculum was actually used before parentheses
      ___
    a b+c  meaning  a(b+c)
        ___
    [{(     )}]

In question 6/2(2+1), brackets are exist so we have to solve the brackets first.

2-Common factor

Second important point is factoring, simplifying equations or distributive property.

Let we have a number 6.

6 = (4+2). We know 6 is total of 4 and 2, 
    So we can write it as (4+2)

  = 2(2+1) Take 2 as a common factor here. 

The common factor 2 will remains a part of the (2+1) or the common factor 2 will remains inner part terms at all times. 2 cannot be used in another operation by itself because 2(2+1) show that 2 is came out form (4+2).

In given question 6/2(2+1), 2 is common factor which come from (2+1). Also note it that 6÷2(3), 6÷(2)3, (6) ÷(2)(3) are same thing.
If you are still confused then I have some mathematicians proof.

Introduction to Real Analysis by William F. Trench 

Page # 123: 
        k(ε/2k) = ε/2

http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF

To see more proof let us check more than 20 refrences from different mathematicians books.

3-Fraction

Sign '/' use for horizontal fraction bar. This fraction bar is also exists in above question. Let us discuss on it too.

Let we have  14/7n.
    Here 7n has one and only one meaning
    That is 7n = n+n+n+n+n+n+n 

  14 / 7n 
= 14 / (n+n+n+n+n+n+n) 
= 2/n

The 7n is a group (or product) of variables and coefficients is bellow the horizontal fraction bar.
All factors of a product are the coefficients of the remaining factors. e.g.; f7abc = 7bac = a7bc = abc7 etc etc.

In question 6/2(2+1) this is clear that 2(2+1) is bellow the horizontal fraction bar.

4-Denominator
Let we have 1/2x. It clearly show that 2x is denominator. There is no need any parentheses around 2x. It is understood that 2x is denominator
i.e. 1/2x = 1/(2x) = (2x)^-1

This sort of notation is used especially with pi, ln, or e. We have never had to say 1/(2pi). We use to simply written as 1/2pi or 1/2e^2 without any parentheses.
Similarly in ab/cd, ab is nominator and cd is denominator. Without any parentheses and its mean (ab)/(cd) and parentheses no need to use here.

Similarly in given question 2(1+2) is denominator. No need to write it as {2(1+2)}

5-Coefficient
Let we have a = 1a or a= 1(a) then

We know there is always an invisible 1 as a ceefficient of a variable and a/1a =1 is true. But some people blindly using BODMAS/PEMDAS, they move toward wrong way and try solving it as...

1 = a/a

  = a/1a

  = a/1×a (they apply BODMAS/PEMDAS)
          (Solve from left to right) 

  = a×a 

1 = a^2 

They claimed that math is false because we have prove a^2 = 1. That's why I always say being careful to apply bodmas/ pedmas.
In above example

Keep in mind that in above question 6/2(2+1), here 2 is came out from (2+1). People are trying to solve this by calculator, but that is the wrong way. Solve it using algebra techniques with a pen & paper.

6-Don't change parentheses to multiply

Let xy=z
Here it is understood that xy=x×y or x(y) or (x)y
writing xy=z is better than writing x×y=z because it is unambiguous and convenient.

Now if a/z then a/xy. We do not write it as a/x×z 

because
        a
a/xy = ----
        xy

and 
a/x×y = ay/x 
    
are two different things

So when we write 6/2(1+2), we could not write as 6/2×1+2), because they are two different things. 6/2(1+2)=1 but 6/2×(1+2)=9

7-Is 6/2(2+1) = ambiguous?

I disagree, the statement 6/2(2+1) is absolutely not ambiguous. It is just written linearly (in one line)

And we know
               6
 6/2(2+1) = ------- = 1  
            2(2+1)

but if (6/2)(2+1)=(3)(3) = 9  

The people whose answer is 9 they should write the question as (6/2)(2+1)

8-Answer will be 9 if and only if...
If you still want to get answer 9 then you would have to written the multiplication sign as
6÷2×(1+2)   

Or use parentheses as
(6÷2)(1+2)

or 
 6
--- (1+2) = 9
 2

The question 6÷2×(1+2) or (6÷2)(1+2) is not the same as 6/2(1+2).  
9-Machines like Google 

Most of the people who give their answer 9 and supporting their answer with machine. They put 6÷2(3) into calculator, Google or Wolfram Alpha. All of these first convert the parentheses into multiplication sign (X). i.e. the expression 6/2(2+1) becomes 6/2×(2+1) then they get answer 9.

Here’s the thing. When you put this question (6/2)(1+2) in Google. Google give you the answer 9. Actually Google reads it as – first term (6/2) multiplied by the second term (1+2). That is why Google gets you 9. By the way is incorrect. This equation in fact has two terms 6 which is divided by the other term 2(1+2).

Some calculators do not follow the Order of Operation. For example, Casio Fx570 give the result 6/2(2+1) = 1 while Casio Fx50fw show the result 6/2(2+1) = 9


Why we forget that any procedure or a machines solution needs to be verified by human.



10-Mathematicians

This brings us to factors, such as 5ab = b5a = ba5=ab5. Factors can be rearranged in any order because their product result remains the same. (Means the quantity doesn’t change).
Because the quantity is the same therefore there is no need of parentheses. These rules was very clear in past because people were actually taught basics of mathematics but these problems appears in today, when more and more people are relying on calculators, most of the programmings languages does not follow the above rules, they always tell you that you should use parentheses when entering the data.

For the student, teachers and all the lovers of math should also study that How mathematicians explained the product, factors, division and coefficients in their books. "How famous math experts explain 6/2(1+2), 1/2a, 1/2pi or ab/cd in their books and some web link in support of my article." Because I have study on it in detail and collect much material on it so I have put it in a separate article.

11-A step by step solution

Let us solve the parentheses step by step

  6/2(2+1)
= 6/2(3)

Here the parentheses still exist so solve it first and don't try to convert it into multiplication sign 'x'. The problem is that people do not finish the parentheses step before jumping to the multiplication/division step and move toward wrong answer.

= 6/6
= 1

Some other weblink which support answer.

  • http://www.purplemath.com/modules/orderops2.htm
  • http://www.askamathematician.com/2011/04/q-how-do-you-calculate-6212-or-48293-whats-the-deal-with-this-orders-of-operation-business/


  • 12-Second Solution
    By reverse solution
    
    Let   1
               (Multiply and divide by 6 
        = 6/6    
               (Take 2 factor from denominator) 
        = 6/2(3)
               (Because 3 = 2 + 1 so...)
        = 6/2(2+1)
    


    13-Third Solution

    There is another easy way to prove it. Take X = 2+1 = 3 and put it in 6/2(2+1)=6/2(3) for check the answer 1 or 9.

    When X = 3 for answer 9
        6 / 2 (2+1)     
        6 / 2 X = 9   
        6 = 9 × 2 X      
        6 = 18 X        
        6 / 18 = X
        1 / 3 = X
    

    In first case we get X=1/3 or 0.33333 which is not correct therefore X=3 value is not satisfied 6/2(2+1)=9.

    When X = 3 for answer 1
        6 / 2 (2+1)
        6 / 2 X = 1
        6 = 2 X 
        3 = X
    

    In this case we get X=3 which is correct X=3 value is satisfied for 6/2(2+1)=1.

    Therefore answer of 6/2(2+1) should be only and only 1.



    All the above is clearly proof that answer should be 9. But some still think that Why answer of 6/2(2+1) should be 9.

    ___________________________________________

    ORDER OF OPERATION ([{(-)}], ()^2 √ , ÷, ×, +, -)

  • What is the correct answer of 6/2(2+1)=? (1 or 9)NEW
  • How MATHEMATICIANS explain answer of 6/2(2+1)=? (1 or 9)NEW
  • Why the answer of 6/2(2+1) should be 9NEW
  • What is the denominator in 6/2(2+1)=? (2, (2+1), 2(2+1), Any other)NEW


  • What is BODMAS? Why we use it?NEW
  • What is PEMDAS? What is different between PEMDAS and BODMAS?NEW
  • What is difference between BODMAS, BEDMAS, BIDMAS, PEDMAS and PEMDAS?NEW

  • Who invented the BODMAS? When BODMAS was introduced?NEW
  • Where we should solve the factorial(!) in BODMAS ? NEW
  • When percentage(%), logarithms and trigonometric involve in BODAMS. How we solve it? NEW

  • Why PEMDAS create confusions?NEW
  • BODMAS Short Questions

  • What is BODMAS?
  • ______________________

  • What is order of operation?
  • ______________________

  • Who invented BODMAS?
  • ______________________

  • Solve 3 - 3 × 2 ÷ 6 = ?
  • ______________________

    BODMAS Questions

  • What is square root of 25?
  • ______________________

  • Solve 10 ÷ 4 of 5 =...?
  • ______________________

  • Solve 2 / 2 - 2 + 2 × 2 =...?
  • ______________________

  • Solve 10-4×0+3÷3 =..?
  • ______________________

  • Solve 7+7÷7+7×7-7 =...?
  • ______________________

  • Solve 3 + 5% =...?
  • ______________________

  • Solve 2e × 100% =...?
  • ______________________

  • If √(0.04)(0.4)a = (0.4)(0.04)√b then a/b ... ?
  • ______________________



  • Solve 5 × (1/5) ÷ (1/5) ÷ (1/5) × 5 =...?
  • ______________________



  • 3 ÷ (5 ÷ 7) = (3 ÷ 5) ÷ 7
  • ______________________

  • Solve 4 - 2 ÷ (1/2) + 3 =...?
  • ______________________

  • Solve 2 - 2 × 2 - 2 ÷ 2 =...?
  • ______________________

  • Solve 3 - 3 × 3 - 3 ÷ 3 =...?
  • ______________________

  • Solve 4 + 4 × 4 - 4 ÷ 4 =...?
  • ______________________

  • Solve -1 + 2 × 0! =...?
  • ______________________

  • Solve -2 + 3! - 1 =...?
  • ______________________

  • Which step is wrong?
  • ______________________



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