Apparently simple math: 6÷2(1+2) = ?

I am your college professor that you requested, with a doctorate in Mathematics. I will break this down as simply as possible and end this debate as approx. 10 students have already asked me this today.

The problem as it is written is 6÷2(1+2) , the ÷ cannot be substituted with a fraction bar because they have different ranks on the order of operations. It is an illegal math move to do this. The bar ranks with parentheses, ÷ is interchangeable with *. therefore the problem must be solved as 6÷2(1+2) NOT 6 (over) 2(1+2) we do the parentheses first, so 6÷2(3), the parentheses are now no longer relevant, because the number inside is in it's simplest form. Every single number has implied parentheses around it.

6÷2(3)

(6) ÷(2)(3)

6÷2*3,

or even converting the division to multiplication by a reciprocal (a legal math move)

(6)(1 (over) 2)(3)

are all correct ways to write this problem and mean exactly the same thing. Using pemdas, where md and as are interchangeable, we work from left to right, so (3)(3) or

3*3= 9

Just because something is implied rather than written does not give it any special rank in the order of operations.

The problem in it's simplest form, with nothing implied would look like this:

(1+1+1+1+1+1 (over) 1) ÷ (1+1 (over) 1) * ((1(over) 1) + (1+1 (over) 1))

From here, nothing is implied, This again, works out to 9.

If the symbol '/' was used this whole debate would be ambiguous since that symbol can mean "to divide by" or it could mean a fraction bar.

HOWEVER, because the ÷ symbol is used, it can not be changed to mean a fraction bar because that would change the order of operations and thus the whole problem, you can't change a symbol to mean something because you want to, in doing so you are changing the problem.

Once and for all, the answer is 9.

Hopefully some of my students see this so I can stop answering this question.

End of debate... hopefully.

The answer can be nothing but 1. If you get 9, it means you are reading 6÷2 as a fraction all by itself 6/2, which will give the 3 that when multiplied by the 1+2 in brackets gives the nine, but if you keep the 6÷2 as what it is without making it a fraction, the logical answer has to be 1, as you would add 1+2, always learned that you work out brackets first, so it would be 6÷6, since 2(1+2) = 6. Only 1 can ever make sense to me. Sorry, I am with the Engineer on this one

6 Divided By 2 1 2

6÷2(1+2)

(3) 3 = 9

6÷2(1+2) =

6÷2 (3) =

3 (3) =

9

Wow. We had the same ideas. The very best explanation for it is to convert all the operations into Multiplication.

It can be done into :

6 x (1/2) x (3)

Any thing divided into 2 can be converted as multiplied into 1/2

Thus, getting the answer 9 whatever way.

18 x 1/2

3 x 3

1.5 x 6

Done.

The answer is one and I can solve this using algebra. If you sub any number in the equation with a letter to sub for when like this

6÷2(1+2)=1 as one of fallowing

a÷2(1+2)=1 or 6÷b(1+2)=1 or

6÷2(c+2)=1 or 6÷2(1+d)=1

Every number we took out will become the number it was.

First I will do with the answer being 1 for

a÷2(1+2)=1 multiple both sides by a to isolate.

a×a÷2(1+2)=1×a

2(1+2)=a. Now solve rest for a

2(3)=a

6=a

Now

6÷b(1+2)=1 multiple by 6 both sides

6×6÷b(1+2)=1×6

b(1+2)=6

b(3)=6.

3b=6. Now isolate b divid both side by 3

3÷3b=6÷3

b=2

Now

6÷2(c+2)=1. Multiple both sides by 6

6×6÷2(c+2)=1×6

2(c+2)=6 multiple inside bracket both numbers by 2

2c+4=6 subtract 4 from both sides

2c+4-4=6-4

2c=2. Isolate c divide both sides by 2

2÷2c=2÷2

c=1

Now

6÷2(1+d)=1 multiple 6 on both sides

6×6÷2(1+d)=1×6

2(1+d)=6. multiple inside both brackets numbers by 2

2+2d=6

2-2+2d=6-2

2d=4. Isolate d divid both side by 2

2÷2d=4÷2

d=2

You can not find any number in 6÷2(1+2) if you do this with 9.

6/2(1+2)=9

6/[2(1+2)]=1

Now I understand why our economy is failing. It is because so many people DO NOT understand simple math. BEDMAS, PEMDAS or BODMAS are all the same and it says Brackets and Exponents are done first in order left to right inthe order they appear. Then it is Divide and Multiply or Multiply and Divide in any order from left to right and it ends with Add and Subtract in any order left to right.

The KEY to remember is LEFT to RIGHT.

So the question 6÷2(1+2)is the same as 6÷2x(1+2) the lack of a symbol between the 2 and the bracket infers a muliplication symbol.

6÷2(3) is the same as 6÷2x(3) and now the brackets are not needed because you can't do any more to the 3 so now it is 6÷2x3 and I sure hope everybody can figure out that one now.

We only us / instead of ÷ because many computers do not have ÷ keys because no type writers had a key for ÷. The / does not make it a fraction although that still works. Just remember if you do not see a symbol where you need one in an equation then you must do it as multiplication.

The answer is only 9 and any teacher that says otherwise is doing a huge disservice to the future of our children.

In a typed format, 6/2(3) is equal to (6/2)(3). Thats it. For it to be 6/(2(3)), it MUST be stated WITH the parenthesis. The fraction bar also does not make everything after it the denominator unless there is a parenthesis on it. If there isn't one, you follow left to right. When using any acronym like PEMDAS, you have to remember that its NOT Multiply before Divide, Those two are really placed at an equal level. It's 9.

You can also compare it to an easier equation involving addition and subtraction:

2 - 3 + 4 = 3

What many people who consider the original equation's answer as 1 would do is make the 3+4 into 7, and then do 2 -7 = -5.

This is EXACTLY what people are doing with 6/2(1+2) but at a level of multiplication and division.

Using the same example, you can use algebraic properties to move 2 - 3 + 4 = 3

2 - 3 + 4 = -3 + 4 +2

the (3) is negative because there is a minus sign before it, obviously.

Using the same principles, you would do

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

In this case, there is a DIVISION sign before 2, so if you move 2 to the front, you have to move the DIVISION sign before it. Just like how you move the minus sign with the 3.

The answer is 9 6/2(1+2)=9 you always do parentheses first so 1+2=3 then you divide 6/2because it came before 2*3 then you multiply 3*3 and get 9