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How do you figure out what $$6\div 2(2+1)$$ is equal?

I get $9$, but some people say $7$ or even $1$ and I don't know how they get that?

What does it really equal?

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    $\begingroup$ how does someone get $7$? $\endgroup$ – Jean-Sébastien Apr 10 '14 at 21:27
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    $\begingroup$ If you want to figure out what that equals to, start by using parenthesis properly like everyone else. =) $\endgroup$ – Pedro Tamaroff Apr 10 '14 at 21:27
  • $\begingroup$ It's an awful way to write it, because it's ambiguous. The point of mathematical notation is to clearly communicate an idea, not to make a puzzle. That said, according to the usual conventions of interpretation, it's $9$. $\endgroup$ – user61527 Apr 10 '14 at 21:27
  • $\begingroup$ 7 because 6/2=3 and 3*2=6 and 6+1=7? $\endgroup$ – user142299 Apr 10 '14 at 21:28
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    $\begingroup$ Order of operations is convention. If there is ambiguity, one should usually write more parentheses (or write something or other as a fraction) to make only one interpretation possible. Because multiple interpretations are possible here, I'd say the correct answer is "Who the hell cares?" $\endgroup$ – user98602 Apr 10 '14 at 21:55
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As T.Bongers says in the comments above, mathematical notation is used to communicate an idea. We have our order of operations by convention; if we want to say "Two, added to the product of two and three", we would probably write $2+2\cdot 3$. Our convention says that this is evaluated as $2+2\cdot 3 = 2+6=8$. To us, this is perfectly clear, because we've decided that's how things are read.

However, not everybody agrees on order of operations. Some say PEMDAS: Evaluate inside parentheses, THEN exponentiate, THEN multiply, THEN divide, THEN add, THEN subtract. Some group the latter set together: PE(MD)(AS), and say one evaluates from right to left when operations have the same priority. These are, essentially, the two most common agreed-upon orders. That would give this problem the following solutions respectively: $$6 \div 2 \cdot(2+1) = 6 \div 2 \cdot 3 = 6 \div 6 = 1$$ $$6 \div 2 \cdot (2+1) = 6 \div 2 \cdot 3 = 3 \cdot 3 = 9$$

But these are completely different things. As T.Bongers says in the comments above, mathematical notation is used to communicate an idea. If we have two expressions that give a different answer, we use more parentheses to make it clear with we mean, or use some other common notation; for this problem, perhaps I would write $$\frac{6}{2(2+1)}$$ because this makes it completely clear what I mean. The point of our accepted order of operations is to save space while making it clear what's meant, but the latter is far more important than the former.

Because multiple interpretations are possible here, I'd say the correct answer is "Who the hell cares?"

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The answer is 9. According to the order of operations, you do what's in the parentheses first (2 + 1) which equals 3. From there, the equation looks like 6 / 2 x 3. Again, by order of operations, you go from left to right, 6 / 2 equals 3 and 3 x 3 equals 9. People get confused if they write it like 6 / 2(3) and do the right side first, which is how some get the 1.

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  • $\begingroup$ Who said "left to right"? Yes, that is the order your favorite programming language uses, but it is not defined to be that way. $\endgroup$ – vonbrand Apr 10 '14 at 21:36
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    $\begingroup$ PEMDAS or the Order of Operations says that when dealing with multiplication and division together, you go from left to right. $\endgroup$ – Shell Apr 10 '14 at 21:38
  • $\begingroup$ @Shell But who made that up? How do people decide what the right way is... $\endgroup$ – user142299 Apr 10 '14 at 21:54
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    $\begingroup$ It's not entirely clear who 'made it up'. But the reason we call that the right way is because for an extremely long time, that's the way it has been. Long ago people decided an order and stuck to it only to make communication easier. I'm guessing 'they' chose from left-to-right because that's how we read, but it's not really clear. There really isn't an answer to your question... not one I'm aware of, anyway. The reason is basically just because that's the way it's always been... $\endgroup$ – Shell Apr 10 '14 at 22:05