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I saw this photo on my social network.

The ambiguous expression $6\div2\times3$ yielded 2 different answers.

The Picture

The difference is the order of operations. If the division's done first then the answer is 9. If the multiplication was first then the answer would be 1.

What is correct?

There are rules for the order of operations like "BODMAS" for example (I use it) but it doesn't say what is correct in such situation.

Is it possible that both answers are correct unless "brackets" are specified? Or is there one correct answer? Is there a rule for the "direction" in which those must be done? "left to right" for example?

edit: It looks like it's known that it should be done from left to right but the question now is whether:

  • It's just a convention.

  • It's a definite rule that makes us say that one of the answers is wrong.

tag should be "operator-precedence" for example. Help me with the tag please :)

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marked as duplicate by Najib Idrissi, Jyrki Lahtonen, MJD, GEdgar, Daniel Fischer Mar 4 '14 at 19:03

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

It's a matter of convention. – user2345215 Mar 4 '14 at 13:19
Odd that they have the same manufacturer. – David Mitra Mar 4 '14 at 13:22
@DavidMitra odd indeed. could be photoshop for all we know. but aside the fact it is the same manufacturer, this is a legit question. I don't like that someone downvoted, if non math people want to know, we should make them feel welcome. – Sabyasachi Mar 4 '14 at 13:24
@MinaMichael In a quiz, I advise taking the questioner to court. – Sabyasachi Mar 4 '14 at 13:59
@Warren - That suggest this calculator is interpreting the multiplication-by-juxtaposition as a higher-precedence operator than regular multiplication (see That's not entirely crazy, given how we'd interpret $5x/30x^2$. – Govert Mar 4 '14 at 14:22

There is no contradiction here. The first calculator interprets it as $$\frac{6}{2(1+2)} = 1$$

The second interprets it as,

$$\frac{6}{2}(1+2) = 9$$

It is a matter of how the calculators are designed. In general, it is bad practice to write ambigous equations like that, since your intention is not always clear.

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Yes I know and agree with all that but I'm asking if there's a specific rule to govern which way it should be interpreted. – Mina Michael Mar 4 '14 at 13:24
@MinaMichael no there isn't. it is a matter of convention, and a wise person always sticks in enough parenthesis to make it unambiguous. – Sabyasachi Mar 4 '14 at 13:25
Hmmmm alright that's good enough for my mind! Thanks a lot :D – Mina Michael Mar 4 '14 at 13:26
@MinaMichael Sure. Anytime. Happy math. – Sabyasachi Mar 4 '14 at 13:27
I'd say there is a very clear rule of my comment above and the other answer. – DonAntonio Mar 4 '14 at 13:37

There is a a speciefic rule for this: the order of operations. Multiplication and division apear on the same place in this rule so they must be interpreted from left to right so 9 is the correct answer. The other calculator is just wrong.

This probably has something to do with the fact that most people never write division inline like on a calulator, but rather as a fraction


where both the numerator and the denominator need to be evaluated first, as if the where in parenthesis. This is however not the case so it is wrong.

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" Add to this the fact that it makes no sense at all to write a/bc if you wish to convey the meaning ac/b, and one can almost safely say that anyone who writes a/bc means a/(bc)." – Sabyasachi Mar 4 '14 at 13:57
I agree with @jenders: the operations preceedence is a rule and, anyway, I think the downvote is uncalled for. – DonAntonio Mar 4 '14 at 14:25
No,no. There is a rule not in the sense "that I'd do this or that", but a mathematical rule, period. One could argue this rule is not an international one, not everywhere it is applied, etc., but kids in several parts of the world are taught to apply it. – DonAntonio Mar 4 '14 at 14:32
@DonAntonio I would argue that some primary school technical teaching is flawed. Just because people teach it, shouldn't make it right. Being right, should make something right. – Sabyasachi Mar 4 '14 at 14:35
According to your "rule" all those math texts that use “$1/2\pi$” to mean $1/(2\pi)$ are "just wrong". Sorry, but that is just wrong. – MJD Mar 4 '14 at 16:23

protected by Willie Wong Mar 4 '14 at 15:42

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