# These PEMDAS problems drive me crazy.

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I keep seeing these problems, and apparently there are 2 ways to do it.  The article talks about the old way and the new way?  What?  Math is math, right?  Is the problem just that they should have used more parenthesis to clarify what to do?  Thinking it over, if I had an equation like that to figure out, it would be because I had written it as a step in a larger problem, and would know which came first.  Right?  Ugh.

Thoughts?

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I think the problem is that there is no multiplication sign. If it'd said 6/2*(1+2), then yeah, 9. But the omission of a x, *, or dot for multiplication to me implies that the 2(1+2) should be considered as one unit, so the answer is 1.

And yes, in reality, you typically have an actual problem to solve, so you'd know what you mean. Or there'd be more parentheses. Or w/e.

ETA: I think technical correctness is less important than good communication. Don't write problems that are going to be misinterpreted, even if technically they're correct.

ETA2: I also don't know that it has anything to do with old or new math or w/e.

Edited by luuknam
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Better notation people who object to multiplication signs:

6

---------  = 1

2(1+2)

Or:

6

-  (1+2) = 9

2

ETA: That 6/2 should just look like a fraction in the latter... unfortunately it's spaced kind of far apart vertically.

Edited by luuknam
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IIRC, a fraction line is a grouping symbol while a simple division sign is not.

So, the way the problem is written, which is how you should do it, the answer is 9.

If they had written it so that 6 was clearly a numerator and the rest was clearly a denominator, it would be different. I don't think the presence or not of a distinct multiplication symbol makes a difference.

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Well, it sounds like the "old way" is actually not the way any of us learned math, but the way they were teaching math 100 years ago.

Would you have gotten the correct answer if you'd been taught using the mnemonic "BEDMAS", I wonder? (Where D comes before M?)

There is an interesting explanation below about the differences in the treatment of the division sign in problems like this between math today and math in the early 1900s which boils down to: "Historically the symbol Ã· was used to mean you should divide by the entire product on the right of the symbol."

But...my sense is that there may be a less fascinating explanation for arriving at different answers to this problem, which is that many people use the mnemonic longer than they remember that multiplication and division actually take equal precedence in the order of operations, if not in the mnemonics.

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I have a math degree and the answer is 9. First you solve in the parens they then go away and you have

6 divide 2 * 3

Sorry don't have a divide sign on my iPad. This is solved left to right to get 9. I don't know about old vs new. I'm old. Lol. But both my kids got this right at 13 and 16 and I am not a hands on math teacher. There would need to be a 2nd pair of parens to get 1.

In pemdas, divide and multiply are equivalent and add subtract. They are solved left to right.

http://www.purplemath.com/modules/orderops.htm

Edited by WoolySocks
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It is simply that people who get it wrong don't understand the order of operations.

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I hate these too, because the purpose of language, including mathematical language, is to communicate. If I say or wrote something that is widely misunderstood, even if I'm technically fully correct, I've done a poor job communicating and should clarify or revise. Given that there are other, clearer, ways to write this problem, it is poorly written. Also, the proliferation of this type problem on Facebook gives the impression that math is some "gotcha" thing that only a persnickity elite can solve.

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This is not really an "order of operations" problem, aside from knowing to execute the parenthesis first. After that, you go left to right and perform operations that have the same rank in consecutive sequence. 6/2*3=9.

It is a problem that illustrates how an imprecise notation confuses people. If you want to have 6/[2(1+2)], say so. If you want to have (6/2)(1+2), say so, because the implied multiplication in does not automatically imply parentheses with the last character. Best, use an equation editor and make it clear by using horizontal lines as denominators.

ETA: It is also am problem that illustrates how horrid the PEMDAS mnemonic is, because it creates the incorrect impression that multiplication ranks before division.

Edited by regentrude
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On  a related note: this is also an issue that occurs frequently in trigonometry. Which is why in writing complex expressionss with numerical factors, the trig function is usually placed last to make absolutely clear what the argument of the function is.

ab sin theta is unambiguous.

sin theta ab would benefit from parenthesis.

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I don't know that PEMDAS is so horrible.  Maybe it should be written out as PE(MD)(AS)?  It doesn't make sense to me logically that multiply/divde and add/subtract wouldn't be equal priority in operation?  Honestly, I think there are just a bunch of crappy math teachers out in the world that don't emphasize the actual important concepts and marginally know what they're doing.

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100 years ago my algebra teacher taught us PEMA (and not PEMDAS).  M stood for multiplication or division from left to right.  In other words, it is not necessarily that multiplication comes first, but rather they are step 3 from left to right.  To me PEMDAS makes it seem like multiplication always comes before division.

It's one of the few things I remember from algebra that long ago. LOL

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I don't know that PEMDAS is so horrible.  Maybe it should be written out as PE(MD)(AS)?  It doesn't make sense to me logically that multiply/divde and add/subtract wouldn't be equal priority in operation?  Honestly, I think there are just a bunch of crappy math teachers out in the world that don't emphasize the actual important concepts and marginally know what they're doing.

Logically it makes no sense - but students who memorize the mnemonic frequently have the misconception that M goes before D.

And many math teachers do not actually understand that division is just multiplication: with the inverse.

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The German mnemonic is SO much nicer: "Dot calculations before line calculations". It refers to the signs for the operation: dot for multiplication, double dot (colon) for division; plus and minus for addition/subtraction. It completely eliminates the misconception PEMDAS creates.

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##### Share on other sites 100 years ago my algebra teacher taught us PEMA (and not PEMDAS). M stood for multiplication or division from left to right. In other words, it is not necessarily that multiplication comes first, but rather they are step 3 from left to right. To me PEMDAS makes it seem like multiplication always comes before division.

It's one of the few things I remember from algebra that long ago. LOL

That's essentially what we were taught too. I didn't even know what PEMDAS stood for until college!

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I was out of college before I ever heard of PEMDAS.  In fact, I can't ever remember the acronym and I find that students who try to do math based on the acronym generally don't know how to set up a basic algebra problems. It impacts their ability to understand how they need to add or subtract an entire term from both sides of an equation.

I remember spending a lot of time in math in late elementary/middle school talking about terms and the logic of how a math equation was set up.  We did this about the same time that we were learning a lot of grammar (distinguishing compound subjects vs. compounded sentences and making sure that series were parallel).  I don't know if it is the way my mind works or because of the way I was taught but I think of the parts of an equation much as I do the parts of a sentence.

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I have a math degree and I never used PEMDAS to remember either.  I just knew after doing a bunch of problems.

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"Please Excuse My Dear Aunt Sally" is how I was taught to remember PEMDAS.  I guess my teachers explained about M/D and A/S being the same though because I was able to do it correctly.

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The problem is that the fraction bar and the division symbol can mean the same thing. In this case that leads to two different answers. As others have said, it is a communication problem, rather than anything to do with old math/new math PEMDAS/BEDMAS. In my amateur opinion.

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I have thought about this problem a lot as well.  I don't have a math degree, but I do have an engineering degree.

The expression, as it is written, is ambiguous.  It is not clear whether the denominator is 2, or 2(1+2).  Brackets are needed for clarity.

That brings me to an analysis of what might lead some people to one solution, and others to another.  I think there are at least two factors involved.  One is the way math is taught (in the US, at least), and being over 60yo has little to do with it.

For one thing, PEMDAS Order of Operations is a middle-school skill that is covered once a year for a couple weeks, then set aside for the most part.  Once a kid moves on to Algebra, they move into thinking of groups of variables and constants as Terms.  They learn to simplify each term before combining the terms together as much as possible to simplify expressions.  Also, once a kid moves on to Algebra, multiplication signs (or even dots, for the most part) disappear.  I was one who, on initial quick calculation of this problem in my head, came immediately to "Oh, 6 divided by 6 is 1".  Only after I was challenged did I go back and rework the problem, mentally jumping through the hoops of placing the multiplication sign back into the expression so that I could think of it as an Order of Operations problem.

Then, I tried to figure out why so many people (myself included) worked it the "wrong" way (algebraically, by terms, rather than by PEMDAS).  I decided to come up with a real life problem for each scenario.  I came up with these:

6/2(1+2) could be derived from a problem like this:

Suppose you had two tables. Sitting at each table were 1 boy and 2 girls. You have 6 apples to divide between them. How many apples would each child get?

6 apples / 2 tables (1 boy + 2 girls at each table).

That could be rewritten as 6 apples / 2 tables (3 children at each table), which could then be considered 6 apples / 6 children.

6 apples divided amongst 6 children gives each child 1 apple.

Therefore, in this scenario, the answer is 1.

But it could also be derived from a problem like this:

Suppose you had 6 pies on a table, and half of each pie was gone. And, suppose there were two more tables, and each of them had 6 pies that were each half eaten. How many pies would you have?

6 pies / 2 x (1 table+2 tables).
This can be rewritten to say 6 pies/2 x 3 tables.
6/2 = 3, so there are 3 pies on each of the 3 tables.
3 pies X 3 tables = 3 X 3 = 9

There are 9 whole pies altogether.

This exercise only brought me back to the position that the problem is ambiguous.  It needs brackets for clarity, or a multiplication sign, or some indication that it is an Order of Operations problem, not an Algebra problem.   :001_unsure: :tongue_smilie:   Oh, and perhaps we need to bring Order of Operations back into Algebra instruction.  After all my years of high school and engineering mathematics, the only reason I remember Order of Operations (and PEMDAS) is because I homeschooled my kids through middle school.   :001_tt2:

Edited by Suzanne in ABQ
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The problem that I see, especially in students being able to set up simple problems in business, is that there is a difference in being able to use order of operation skills and simply answering a textbook order of operations problem.  An example in business might be:

You are making 100 widgets.  Your machine costs $20. You must buy$1 worth of materials for each widget you make.  What is the per unit cost to make a widget?  So a student starts writing:

20 + 1 *100/100  rather than (20 + 1*100)/100.

Then they put the formula into Excel and don't know why they aren't getting the correct answer.  Learning PE MD AS may help with an equation that is written but students don't know where to put the parenthesis when they develop the equation if they don't understand what is behind PE MD AS

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I think the problem is that there is no multiplication sign. If it'd said 6/2*(1+2), then yeah, 9. But the omission of a x, *, or dot for multiplication to me implies that the 2(1+2) should be considered as one unit, so the answer is 1.

And yes, in reality, you typically have an actual problem to solve, so you'd know what you mean. Or there'd be more parentheses. Or w/e.

ETA: I think technical correctness is less important than good communication. Don't write problems that are going to be misinterpreted, even if technically they're correct.

ETA2: I also don't know that it has anything to do with old or new math or w/e.

Agreed. Especially because when instead of (1+2) you have a letter (algebra)  This would be 6/2x  and no one argues THAT should be 3x.  it is 6 divided by 2x.

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Agreed. Especially because when instead of (1+2) you have a letter (algebra)  This would be 6/2x  and no one argues THAT should be 3x.  it is 6 divided by 2x.

But it would be imprecise notation if a student did not put parentheses or used a horizontal fraction line. That is exactly one of the reason why I require students NOT to use the / for fractions - because there are too many implied assumptions. It is sloppy notation.

We assume the 2x is supposed to be grouped because we see it often like that - but as soon as it's all variables, it is unclear. Put parentheses that indicate whether a / b c is supposed to mean a/(bc) or (a/b)c - or use a decent equation editor and not this one line sloppiness.

The implied multiplication with the omitted dot does NOT imply parentheses with the previous symbol.

Edited by regentrude
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It is simply that people who get it wrong don't understand the order of operations.

Not necessarily. Think of it as a statistics problem:

We assume the 2x is supposed to be grouped because we see it often like that

So, statistically speaking, the person who wrote that equation is probably more likely to mean that they should be grouped together, so odds are that the answer they were going for is 1 (if we're going back to the original equation). So, even if you know the order of operations, if you also know how people work, you could figure the answer is 1 or 9, and guess the wrong answer.

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I have a math degree and the answer is 9. First you solve in the parens they then go away and you have

6 divide 2 * 3

Sorry don't have a divide sign on my iPad. This is solved left to right to get 9. I don't know about old vs new. I'm old. Lol. But both my kids got this right at 13 and 16 and I am not a hands on math teacher. There would need to be a 2nd pair of parens to get 1.

In pemdas, divide and multiply are equivalent and add subtract. They are solved left to right.

http://www.purplemath.com/modules/orderops.htm

Yeah, I'm a dummy--I thought one could distribute the number outside the parenthesis first, like if you did not have two numbers and one was an "x" or something, so that would read:  6 divided by 2(1 + x) = 6 divided by 2 +2x and if the divided by symbol means something different than "/" and it is a grouping thing, then what is the answer if x=2?

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The problem should not be written in that manner. Like regentrude, I don't let my students use / for division.

Edited by Caroline
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I think problems like this are easier to do handwritten than in digital format because then you can write it as a fraction under a fraction line or otherwise for clarity whereas when it has to be converted to a single line of text more care needs to be taken with notation.

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I'm going to be obnoxious and say the problem is perfectly clear, and there is a correct, and perfectly understandable order to do it in. But I guess my teachers drilled order of operations into my head pretty well.

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Agreed. Especially because when instead of (1+2) you have a letter (algebra)  This would be 6/2x  and no one argues THAT should be 3x.  it is 6 divided by 2x.

But actually, if you type 6/2pi into a lot of calculators, it will be interpreted as 3pi. This causes loads of problems with my precalc students who do not understand that their calculator takes order of operations absolutely 100%.

With respect to the original problem, it irritates me. Yes, according to a strict reading of the order of operations, you would perform the division and then the multiplication. But there are plenty of people who casually and informally write 3/2x to mean that 2x is in the denominator when they are typing quickly online. I do not like the way the problem is phrased to deliberately trip people up, and the only reason I'd ever use such a problem as a teaching device is to illustrate as to why when you're typing into a calculator/excel you should always be aware of how completely the computer will follow what you said and not what you thought you said.

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But actually, if you type 6/2pi into a lot of calculators, it will be interpreted as 3pi. This causes loads of problems with my precalc students who do not understand that their calculator takes order of operations absolutely 100%.

I see this too with my physics students. Sigh.

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Agreed. Especially because when instead of (1+2) you have a letter (algebra)  This would be 6/2x  and no one argues THAT should be 3x.  it is 6 divided by 2x.

I would see this as 3x.  I spend so much time with computers that I strictly see PEMDAS.  I would bet anyone that programs computers would see 3x.

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But actually, if you type 6/2pi into a lot of calculators, it will be interpreted as 3pi. This causes loads of problems with my precalc students who do not understand that their calculator takes order of operations absolutely 100%.

With respect to the original problem, it irritates me. Yes, according to a strict reading of the order of operations, you would perform the division and then the multiplication. But there are plenty of people who casually and informally write 3/2x to mean that 2x is in the denominator when they are typing quickly online. I do not like the way the problem is phrased to deliberately trip people up, and the only reason I'd ever use such a problem as a teaching device is to illustrate as to why when you're typing into a calculator/excel you should always be aware of how completely the computer will follow what you said and not what you thought you said.

Absolutely.

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I would see this as 3x.  I spend so much time with computers that I strictly see PEMDAS.  I would bet anyone that programs computers would see 3x.

I do program databases.

I did see it your way originally until someone pointed out the algebra, etc.

And evidently some calculators do it one way and some the other. So for a lot of it -- it depends on how the calculator is programmed.  I can see the argument that because there is no actual times sign, the number before the parentheses is part of the parentheses operation and thus done before you do the MD part of PEMDAS.

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And many math teachers do not actually understand that division is just multiplication: with the inverse.

Which is both frightening and depressing.

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But actually, if you type 6/2pi into a lot of calculators, it will be interpreted as 3pi. This causes loads of problems with my precalc students who do not understand that their calculator takes order of operations absolutely 100%.

I absolutely hate calculators that do the order of operations for you.  I realize that I'm a dinosaur in that regard.

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I'm going to be obnoxious and say the problem is perfectly clear, and there is a correct, and perfectly understandable order to do it in. But I guess my teachers drilled order of operations into my head pretty well.

Still I have my question: if you say  6 divided by 2(1+x) and you distribute the 2 first, which I thought one could do, it becomes 6 / 2 + 2x and then you find out that x=2 as in the original equation.....one should get the same number whether or not one knows x first?  And, if you do the division first 6/2=3 and then + 2(x)=4, one would get 7 as the answer?  If you take the division bar to mean everything below it is in parenthesis then you have 6 / (2 + 4)= 6/6 so I still don't get 9.....

How does distribution fit into the equation?

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I would see this as 3x.  I spend so much time with computers that I strictly see PEMDAS.  I would bet anyone that programs computers would see 3x.

Me too.

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Still I have my question: if you say  6 divided by 2(1+x) and you distribute the 2 first, which I thought one could do, it becomes 6 / 2 + 2x and then you find out that x=2 as in the original equation.....one should get the same number whether or not one knows x first?  And, if you do the division first 6/2=3 and then + 2(x)=4, one would get 7 as the answer?  If you take the division bar to mean everything below it is in parenthesis then you have 6 / (2 + 4)= 6/6 so I still don't get 9.....

How does distribution fit into the equation?

You cannot "distribute" the 2 because the (1+x) is NOT in the denominator as a factor next to the 2, unless there are additional parentheses. The expression does NOT mean "divide 6 by [2(1+x)]". That is precisely the point of this problem.

6/ 2 (1+x)= 3(1+x)=3+3x=9 if x=2

If you mean the (1+x) to be in the denominator together with the 2, you must have an extra set of parentheses

6/[2(1+x)]

Edited by regentrude
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Still I have my question: if you say  6 divided by 2(1+x) and you distribute the 2 first, which I thought one could do, it becomes 6 / 2 + 2x and then you find out that x=2 as in the original equation.....one should get the same number whether or not one knows x first?  And, if you do the division first 6/2=3 and then + 2(x)=4, one would get 7 as the answer?  If you take the division bar to mean everything below it is in parenthesis then you have 6 / (2 + 4)= 6/6 so I still don't get 9.....

How does distribution fit into the equation?

But it's not 6 divided by 2(1+x). It is 6/2 times (1+x). Or you could say six halves times (1+x)

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It is the same a 6 X (1/2) X (1+ X) .   So you could distribute the 6 across the (1 + x) and then multiply by 0.5.  Or you could distribute the (1/2) across the (1 + x) and then multiply by 6.

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Does this website do LaTex?

It all depends on if you see the problem as $\frac{6}{2}(1+2)$ or $\frac{6}{2(1+2)}$

ETA: no it doesn't do Latex  Edited by Leav97

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