Bedlam with BODMAS and BEDMAS

Suppose you have this calculation:

One third of 3^2 + 1

Depending on the order in which you do the operations, you could get either 2 or 4 as answer. My intuition tells me you should get 4, i.e. if we combine the BODMAS and BEDMAS rules, we should get BEODMAS, not BOEDMAS.

Any comments from our fine mathematical minds here?

As it stands, the result is 10/3. That’s because the wordy part of the question enjoys the lowest priority, and the question is not (32)/3+1 = 4, but instead (32+1)/3 = (9+1)/3 = 10/3, or 3.333…

I don’t see how you can get 2. Even 9 is more likely: (32+1)/3 = 27/3 = 9.


And if I rephrase:

1/3 of 3^2 +1 ?

What do we get now?

In short, BODMAS and BEDMAS seem to be in potential conflict. How do we resolve the conflict? What gets precedence, “of” or the exponent? Is it BEODMAS or BOEDMAS? Or neither, and if so, what then?

The exponent. “Of” is the same as “multiply with what follows”.

1/3 of 3^2 +1 = ⅓ x (32 +1) = 10/3


The mathematical/arithmetical formulations always take precedence over the worded parts. That’s the way I’ve both seen it treated and treated it myself. Thus, “1/3 of 3^2 +1” is still “1/3×(3^2 +1)”, as before.

Any mathematician worth their salt would in any case carefully avoid such potential ambiguities.


Maybe thus

1/3 of 3^2 +1 = (⅓(3))2 +1 = 2

Yes, that’ll get you 2 as an answer. With a real stretch, I can get the answers 2/3 and 4/3. These results hinge on a different interpretation of the “^” operator, one with which C programmers will be familiar.


I like
1/3 x 3^2 + 1 = 4

Well this is just me as an ordinary layman, I like this answer best, because when I read 1/3x3^2 +1 I don’t see why I must assume that the operation (3^2+1) would be meant to be in brackets the way the question was put, but I’m certainly no math guru, just to me that makes the most sense, but if you would’ve asked

What is one third of the sum of 3^2 and 1 I would then think to use brackets

And this is indeed what I was taught in grade 8 (then standard 6) when I first learned it - that “of” takes precedence. Hence BODMAS, not BDMASO. Apparently they taught me wrong, but that BODMAS rule is still taught in schools all over the world. Under that rule, the calculation 1/2 of 4 + 2 would yield 4, not 3.

Except in some schools they teach BEDMAS, and pretend that “of” does not exist as operator.

Question is now what the heck do I teach a grade 7 class? Seems to me BEDMAS is probably better, and then to word problems in such a way that ambiguity will not arise when “of” makes its appearance? When at all in any doubt, always use brackets?

Precisely. You can never use too many brackets. (It’s by far the most useful thing (short of the author himself pointing out what he meant originally (when he first wrote the text (whether at his home or office (whilst slightly drunk)))) for clarifying the meaning of an expression.)


I must ask: Did this question appear in a maths test? And at what level? If it did, shame on the teacher who set up the test. The question is meant to trick pupils, not test any understanding of BODMAS. That teacher no doubt meant that (1/3)×(3²)+1 = (1/3)×9+1 = 3+1 = 4 is the correct answer. It is this sort of question that validates the complaint about pupils being trained to pass exams, not understand the subject matter.

If one were to encounter such an expression in a book or a journal paper, that would be a very rare occurrence, and the context would make clear what is meant, for example, “… and so the final answer becomes one-third of the previous expression 3²+1, or 10/3.”

As a side note, (and I think I may have pointed this out in another thread) it is beyond me why BODMAS has been made so overly complicated. There is no reason to distinguish between multiplication and division, or between addition and subtraction. Division is just multiplication by the divisor’s inverse, and subtraction is just addition by the addend’s negative. Thus, BEMA or BOMA would concisely capture all of the precedence rules of arithmetical operations.


With roots presumably lumped in with exponents, since nx = x1/n ?

Yup, just as division is multiplication in drag and subtraction is addition in a clown suit, extracting roots is exponentiation in a tuxedo.


I still don’t see why the rule is BODMAS, or BOMA, if “of” does not in fact take precedence…

But I agree that it overly complicates the issue. On the other hand, if I teach the kids BEMA, and then they go to another school, they may end up confused.

Had this quiz on Facebook the other day: 4x4-4x4+4-4x4=? 20 in my book despite the lack of brackets…many said 320.

It’s the introduction of the “of” construct in mathematical/arithmetical expressions that is the problem. It is completely artificial and exists only in the strangely contorted OCD minds of postmodern maths teachers and educational authorities whose megalomania vastly exceeds their numeracy. In the substantially more real world inhabited by mathematicians and mathematically literate professionals, such forms are NEVER used. (Is that sufficient emphasis?) And they are NEVER used precisely because they are ambiguous, and ambiguity goes with maths the way cheese goes with barbed wire.

–12. (4×4 – 4×4 + 4 – 4×4 = 16 – 16 + 4 – 16 = 0 + 4 – 16 = –12.)


sorry my bad: the original quiz was: 4x4+4x4+4-4x4=?? 20 or 320 according to some. I was taught that multiplying and dividing take predence over +/-…in other words the order is key.

Yes that is what I was thought as well , how the hell do they get to 320?

  1. (4×4 + 4×4 + 4 – 4×4 = 16 + 16 + 4 – 16 = 20.)

In naïve left-to-right reading order, the result obtained is wrong:
4×4 + 4×4 + 4 – 4×4
= 16 + 4×4 + 4 – 4×4
= 20×4 + 4 – 4×4
= 80 + 4 – 4×4
= 84 – 4×4
= 80×4
= 320.


Yes, it is kind of difficult to imagine many real world problems where the “of” construct would be used, except perhaps in things like “only 12% of the 300 math educators could do elementary multiplication.” :slight_smile:

Now I am lucky enough to work for an independent school, and we do not absolutely have to do whatever the education department wants, but we nevertheless try to remain fairly close to the official syllabus because the students will eventually have to pass a government exam. Also, as I have pointed out before, if one of our students go to another school, or we get students from other schools, it can create difficulties if the two schools followed too radically different syllabuses.

Hmm, reading around on the web, I now see a new twist in the tale. When I was in school, they told us the O in BODMAS stands for “of”. But according to some pages, such as this one:

it actually stands for orders - also known as exponents. I.e. BODMAS and BEDMAS are not two conflicting rules. They are the same thing. I.e. my grade 8 teacher was just an uneducated moron. As if the fact that he was also a sadistic psychopath wasn’t bad enough.

One more data point in my theory that we need teachers who are educated in the subjects they teach, rather than in education, and that closing down the teachers’ colleges might well have been the one rational thing the ANC did in its quest for better education. :slight_smile:

PS: I also get -12 on that thing with the long string of multiplied, subtracted and added 4s… :slight_smile: