# Math problem

I’ve seen this on facebook, and I’m afraid I don’t understand why there is confusion. 6÷2(1+2)=? Is it 1 or 9?

Long answer: This is why I tell my student to forget PEMDAS that was taught to them in elementary school. The problem is 6 / 2 x 3. Students that memorized PEMDAS invariably interpret it as saying multiply then divide. What you are supposed to do is if you have both multiplication and division, do whichever is to the left so the proper solution is
6 / 2 x 3
3 x 3
9

Math life would be easier if instead of PEMDAS (mini rant: what about roots? at least teach PERMDAS or better yet stay away from the mnemonic), elementary teachers taught
()
^ r
X /

I think it’s ambiguous and that’s the point. Personally, I think everything after the divide sign should be treated as a single entity simplified to 6/6 = 1. If there was an actual multiplication sign between the 2 and the parentheses I would do the parentheses first then go left to right per standard math rules to get 6/2 = 3 then 3*3 = 9. I think it’s a crappy way to write the equation, but once again I think that’s the point.

6÷2(1+2)=?
do brackets first: 6÷2(3)=6÷2x3
The next rule is to do operations left to right;
First do all additions subtractions; there are none.
Now do multiply and divide.
6÷2 = 3 and 3x3 =9

The trick is to see that 2(3) means 2x3, but is not a 2x3 that takes precedence of order over 6÷2

And you would be incorrect.

It is not ambiguous (see my post) and to claim 6÷2(1+2) should be read as 6÷[2(1+2)] is simply erroneous.

PEDMAS.

P. Terms inside parentheses.
E. Exponents and roots.
M. Multiplication.
D. Division).
S. Subtraction.

So I would do:

2 + 1 = 3
2 x 3 = 6
6 ÷ 6 = 1

Well, the Ancient Society of the No Homers has decreed it is actually 4 and a squiggle.

But seriously, it’s both.

Sorry about the ambiguous answer, but that’s the best we can do. The confusion comes from some textbooks and teachers teaching that if you see an implied multiplication, like 2(1+2), you should do that first. If that’s the case, the answer is 1.

Other textbooks and teachers teach that you should do the division before the implied multiplication, as you would under a normal order of operations order. Then, the answer is 9.

Of course, if there’s no agreement even among different teachers, that means there’s no hard and fast rule.

The problem is the the order of operations is itself a convention. It’s a way of codifying how most people approach arithmetic. In real life (even for mathematicians), we’d probably ask for clarification or curse the problem poser for giving us an ambiguous expression.

Most of us would use an extra set of parentheses to clarify the expression or use a rational expression with a numerator and denominator, instead.

We’ve been seeing this question submitted a lot on the “Ask Dr Math” site, and it’s the 2nd round of it. A similar question was posed a couple weeks ago and also spread like wildfire. These things come in waves.

The way we’ve been explaining it is to use the phrase “American history teacher”. When heard, is that a teacher of American history? Or a history teacher from America? It’s an ambiguous expression with no established rule to resolve the ambiguity.

Actually, thinking on it some more, the “real” answer is whatever your teacher says it is. Doesn’t change the ambiguity in the real world, but it does mean the difference between good and bad marks in class.

I haven’t seen this specific problem, but there was another, almost identical problem that stormed the 'net earlier this month. As far as I’m concerned, the only wrong answer is “I don’t understand why there is confusion.” The long answer is “This is why you don’t write equations this way, and your algebra teacher should have harped on this.” The short answer is 9.

ETA: As a programmer, I can say there’s no ambiguity when it comes to how a programming language will perform the order of operations. If there’s still a debate amongst teachers like Great Antibob suggests, then they all need to take an intro to C++ course and realize that the debate has been settled by the nerds.

Distracted by South Park. Didn’t read it before posting.

Suppose you put the 6 above a division line, and had 2(2+1) below the line?

Then it is a different problem. Absolute value and division bars act as grouping operators and so
NUMERATOR
DENOMENATOR
is (NUMERATOR)/(DENOMENATOR)

And I will also point out that your thought process in an earlier post as to how PEMDAS work is exactly why it should NOT be taught.

I understand this is a bit a snark, but I wanted to address this point directly. There’s no implicit multiplication in C++. You need to use an explicit ‘*’ for multiplication. That’s the crux of the debate. I suppose we should have engineered spoken and written language to avoid ambiguous statements altogether.

Also as a programmer, even though the compiler knows & enforces an order of operations, it’s important for programmers (and future maintenance programmers) to understand it clearly. So when doing program coding review walkthrus, whenever this came up, we just demanded that the coder rewrite this, using parenthesis to make the order explicitly clear. Saved lots of confusion & errors in future years.

I’m not entirely convinced, now let me preface it that my gut would be 9, because of the way I was taught order of operations, but let’s do a simple substitution.

let (1+2) = x

6/2(1+2) = 6/2x

Now it seems a lot more ambiguous to me, is it (6/2) * x, or 6/(2x)? The general tendency in algebra is to treat a coefficient/variable pair as one entity, not in need of parens to separate it, but at the same time, (6/2) * x can’t be ruled out, if they expect you to use the order of operations without question, or even if they considered “6/2” to be the coefficient of x. In fact, with the variable substitution I’d probably be more likely to treat it as 6/(2x), whereas with the literal statement I’d be more likely to treat it as (6/2)(1+2). Interestingly, making a simple addition to 6/2x -> 6/2 * x makes it seem to me that the grouping is (6/2) * x, even though technically no ambiguity should have been altered. It’s funny how syntax psychology works sometimes.

Actually, it’s not so clear cut, here’s a picture of several calculators, some of which are from the same company, solving a similar problem. Even the calculators don’t agree with how to interpret it.

It occurs to me that PEMDAS really needs to be PEMA. Roots (and logs) are already rolled into Exponents, so why not have the other inverses rolled together? Subtraction is addition by a negative and division is multiplication by a reciprocal.

I suppose then we’d have to teach kids some extra facts when they learn the order of operations, but I don’t think it’s insurmountable.

This is why I use parentheses when I code.

FWIW, my Ti-30 says the answer is 9. I haven’t tried it on my Ti-84 or any of the HPs. (Too lazy to dig them out.)

Well, enter 6÷2(1+2) into Google, and they agree that the answer is 9.

This is completely incorrect. Saint Cad already laid out the order of evaluation including the fact that division and multiplication are evaluated from left-to-right. This is the rule; there is no ambiguity.

You don’t get to add a set of parenthesis around 2x just because it looks like they belong together. If they were meant to go together, then they should have had a set of parenthesis explicitly added or they should have both been written as the denomenator underneath the 6.

It’s not ambiguous. The order of operations in math is absolute. The answer is 9.

Wolfram Alpha actually agrees with my interpretation.

Now, Wolfram isn’t God, but it is often pretty good about laying out generalities like that. Now, should it be unambiguous? Yes, it would make the world a much easier place, but just because the order of operations should be followed like law doesn’t mean it IS. The fact is, depending on who is using what, the correct answer will change. “Math” is universally true and correct if you follow the premises, mathematical syntax is a lot closer to natural language in ambiguity sometimes.

As Al Gore would say, “There is no controlling legal authority” that determines operator precedence. The ‘square root of yellow’ is just as valid an answer to the question as any other the way it is expressed.