Hi!

Vi Hart—a rockstar Youtuber and former Khan Academy employee—has a great video reply to a similar problem (skip to 2:32 in the video below):

https://www.youtube.com/watch?v=a-e8fzqv3CE 

I'll paraphrase her argument:

We’ve been trained to care about notation way too much. The answer is that this is a badly formed sentence. It’s like reading, "I would like some juice or water with ice," and trying to figure out the right answer to what that person really meant. You could make claims about conventions and what’s right and wrong, but really the burden is on the author of the sentence to put in some commas and make things clear.

Mathematicians do this by adding parentheses and avoiding this ‘divided by’ sign. 

Hi Sanima, the short answer is: the expression is not written as clearly as it could be, and both results are reasonable. 

The long answer is, 6÷2(1+2) can mean either (6÷2)*(1+2) = 9 or 6÷(2(1+2)) = 1. The extra parenthesis make it clear which order the writer intends.

Computers and calculators will have to choose one way or another to interpret it, google chooses 9. But a Casio model chooses 1. We humans should probably say, "hey that's not written clearly, can you re-write with parentheses?"

It is also true that multiplication does not necessarily come before division. And addition doesn't necessarily come before subtraction. The order of operations "PEMDAS" can also be PEDMSA ... or PEDMAS... or PEMDSA. 

The order of operations is an agreed upon way to decide how we write math expressions. Without it, we might have to describe our steps with many more words: "first add one plus two, then..." The parenthesis are the main tool we use to show people what we mean.

 I would say that the # 9 is correct because you do start from left to right if you have 2 of the some side by side operations so 9 is correct.