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Back to the content 'Idiot'
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Okay there's a lot of confusion being caused here by the bad syntax in the question and by the looks of things, people not being properly taught the application of PEMDAS. I'll try and break it down as much as possible and explain why we use these forms and orders.
Firstly, the important thing to note is that the posed question: 6/2(1+2) does not have an operator between the '2' and '(1+2)'. This very simple little thing is what's causing your application of PEMDAS to go awry. Consider for a moment the similar problem: 8/(6x+2y) which we can simplify by taking out the common factor as 8/2(3x+y). Now if you apply PEMDAS assuming that the 2(3x+y) is the same as 2*(3x+y) you'll get an incorrect answer.
It's easier to think of this same equation as 6*(1/2)*(1+2). Since something divided by two is the same as multiplying by 1/2 it would be the same thing. This would conclude that 6*(1/2)*3 would equal 9
This is because when you have 2(f(x)) is is shorthand for (2.(f(x))) - why the extra brackets? Simply because that 2 'belongs' to the bracket. In high school you will probably get away with using 'classroom shorthand' meaning 2*(1+2) which is not strictly correct. If the f(x) were a vector then it would be equivalent to 2.(V) which gives a scalar and so must go first (you cannot divide by a vector.)
This is why you're getting a 9. Cont below.
It's probably simpler for you to think of the '2' of (1+2) as 'belonging' to the bracket and so it gets resolved during out first parentheses stage. Hence
You might also like to think of 2(1+2) as being equivalent to (2*(1+2)) rather than simply 2*(1+2) - which would have to have been written as such for it to apply in the same way as the division sign - so that your PEMDAS application can be rigidly applied. This is also the cause of error when putting it into a calculator because it relies on you spotting this.
Wrong wrong wrong
2(3) = 2*3 and 6/2 and 2*3 follow the order of who comes first.
therefore 6/2 = 3 and then you do 3*3 = 9
You've made the same error that I was talking about. 2(3) here is not 2*(3) it works out as (2*(3))
The 2 is still coupled with the bracket in exactly the same way as if you simplify 6/(2+4) to 6/2(1+2) by taking out a factor of 2 from the denominator.
I hope this clears things up for those of you who thought PEMDAS was broken. I'm happy to try and explain it further and take any questions.
Look, it's much more simple than that. This problem was purposely made this way so that someone could feel superior when people began to get various answers. The simple explanation is that whoever made it didn't put the other set of parentheses in. The question should be: " 6/ (2(1+2)) = ? " , which is what you eventually said. This is just like those problems that teachers gave when you first learn PEMDAS - the "insert the parentheses" problems.
or you can skip those paragraphs and realize that because of the parentheses the quantity 2(2+1) is being multiplied so it comes be for the dividing. PEMDAS
If you would like to simplify it like that then I'm sure it will work out fine however multiplication and division are on the same 'level' and so can be done in either order - usually from left to right as they appear. In this case doing it that way is confusing because the 2(1+2) actually all counts in the 'P' stage.
Division is just multiplication by 1/the function which is why they are on equal footing. :)
indeed i should have said by the distributive property its it simplified because 6/2(1+3) should be looked at as 6/(2*1+2*2)
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