Originally Posted by davoid
I voted for 1.
Although I follow and appreciate all the 9 explanations, I think the above posts nicely explain why we are ALL RIGHT.
For me in particular I would be more inclined to accept 9 if the problem had been set out as 6÷2x(2+1) instead of 6÷2(2+1)
But because the 2 is adjacent to the parenthesis it looks like they are connected in a more intimate and therefore immediate way than are the 6 and 2.
As another post pointed out, 6(2+1)÷2 would be more obviously 9 than would 6÷2(2+1), because 2(2+1) appears to be (2*2) + (2*1)
In a democracy we might say that the answer with the most votes is the correct one, but we might be wrong... it is only the accepted one
Without the additional parenthesis to clear up the ambiguity it is as clear as saying "I helped my uncle jack off his horse" without adding the necessary capitals
And my guess at XplosiV's puzzle (after his clarification that he meant the smallest number) is... one half dozen
Well, I teach math at the college level so I guess that contradicts XplosiV's idea that mathematicians would vote 1. These math question threads that pop up continue to convey the idea that mathematics is something you can opinionate, and is not absolute fact. If I add 3 apples to 5 apples, you get 8. Not 2, not 10, but 8. The laws of mathematics are absolute and breaking them results in contradictions. An example is one of the many division by zero fallacies. I'll show this one as an example:
by multiplying both sides by a, you would get a^2 = ab.
subtract b^2 from both sides yields a^2-b^2 = ab - b^2.
factoring using the difference of squares on the left side, and the common factor of b on the right side yields (a+b)(a-b)=b(a-b).
dividing (a-b) on both sides yields a+b = b.
since a=b, making the substitution yields b+b or 2b = b.
finally, dividing both sides by b yields a final result of 2=1, which is clearly not correct. This is because of the division of (a-b) because that is equal to zero. Division by 0 isn't allowed under standard mathematical arithmetic, which causes the contradiction of 2=1.
6÷2x(2+1) is identical to 6÷2(2+1) under the standard rules of arithmetic. One of the topics one would encounter in the 4th year of a bachelor's in mathematics called modern/abstract alegrba, is a course where you investigate operations at it's absolute core. Addition and multiplication form an abelian ring, otherwise known as a field, on the set of real numbers. Subtraction and division is nothing more than addition and multiplication respectively by use of it's corresponding inverse. For example, 3-5 = 3 + -5; as well as 6 ÷2 = 6 * (1/2) (<--one half). Using this idea, you can change the original problem: 6÷2(1+2) to 6*(1/2)*(1+2). Using PEMDAS normally, you would get 6*(1/2)*3. Performing the multiplication, 6*(1/2) gives you 3, and then 3*3 gives you 9.