That is a classic. The problem is that there is no universal agreement on how to parse expressions with a mix of multiplications and divisions, or with more than one division. Does 1/2a mean 1/(2b) or (1/2)a? It is a matter of convention. Mathematica treats it as (1/2)a, but you may find books where it is meant as 1/(2a), perhaps because this way you save typing parentheses. Usually the ambiguity can be decided from the context, as only one interpretation makes sense. I teach my students to avoid expressions such as a/b/c and always write parentheses to make sure they are not misunderstood. There is a Wikipedia article on the "Order of operations". The calculation 6 ÷ 2(1+2) = 6 ÷ 2 + 4 = 7 is a gross mistake in my view.

It all depends on what they call the precedence of the operators. For Mathematica (Wolfram Language) the Divide has a higher precedence than Times. So it will first do divisions then multiplications:

Precedence[Divide]
Precedence[Times]
470
400

But in other cases you also have to look at how it groups:

6/2/3

could be 6/(2/3) or (6/2)/3, giving different answers.

have a look here:

https://reference.wolfram.com/language/tutorial/OperatorInputForms.html

To sum up: the author of that equation is just sloppy; you have to assume something in order to solve it, and depending on the conventions... so therefore always add extra parenthesis to rules out those cases...

Yes it comes down to whether a space in front of parenthesis in an equation has any significance when parsing a formula. Mathematica doesn't care and even inserts a space to make the formula appear nice. Of course in Mathematice as in many other notations, the multiplication sign can be replaced by a space like 4 5=45 obviously leaving out the space there would be the number 45. with parenthesis the space might not be necessary 2(1+2) = 2 (1+2)=2(1+2), I have not seen the interpretation 2(1+2)=(2*(1+2)), but it is all a matter of agreed convention.

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

Out[1]= 9

When in doubt, the best way to insure clarity one can write out the formula like this:

In[2]:= 6 / 2*(1 + 2)

Out[2]= 9

In[3]:= 6 / (2*(1 + 2))

Out[3]= 1

Yes, without further hints it is not good notation, you need to know the rules of the game. It is like using parenthesis like

 [3*(2+3]*5)

we don't know how to deal with that without some extra rules...

6÷2(2+1), the answer is actually 9. For the people who used this principle x/2y, 2y is obviously together in a group. Polynomials is a simpliest form of an equation the doesn't need parentheses. for example, if the problem is written as 6÷[2(2+1)], in algebraic expression it can easily written as 6÷2y or 6/2y with y=2+1. For the people who said that we should use the distribution, using distribution in 2(2+1) is actually a multiplication method which means you break the rule of PEMDAS that Multiplication and Division are tied in priority which who comes first from left to right should be solve first. 1st step: 6÷2(2+1), Inside the parentheses, 6÷2(3), the parentheses sign is indicate a multiplication, eq. of multiplication signs [x, *, ()] 2nd Step: 6÷2(3), since () is indicate a multiplication, you can rewrite the problem as 6÷2x3, since Division comes first, used Left to Right rule, solve 6÷2=3 3(3) or 3x3 or 3*3 3rd Step 3(3)=3x3= ****9****.