How to Perform PEMDAS Correctly

Disclaimer: This is rehashed from an article I wrote in June 2016 for GineersNow. Some parts are added or edited for clarity.

There are lots of math problems going around Facebook that can essentially be answered by the concept of PEMDAS. This should be easy for engineers who experienced a considerable amount of math in college; but some may have banged a wall that caused amnesia and totally forgot about this elementary math concept.

There is only one definite answer to every arithmetic problem that requires the concept called order of operations.

Others are already familiar with PEMDAS, or Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction, which indicates the sequence. Others even use the mnemonic “Please Excuse My Dear Aunt Sally,” but they may be missing something: it’s not like reading English that is always done from left to right. The arithmetic of order of operations has special rules.

Solving such problems start by performing first the operations inside the parentheses or the brackets. This is followed by exponents. Then, the multiplication AND/OR division. Finally, the addition AND/OR subtraction.

What is usually missed is the rule of AND/OR. Once the exponents are addressed, the next thing to do is to assess whichever between multiplication and division comes first, then that is to be performed. This is also true to addition and subtraction.

So the concept of PEMDAS is not actually absolute. It can either be PEDMAS, PEDMSA, or PEMDSA.

To explain further, let’s take for example the thumbnail photo of this article. Or we can use the photo below with different calculators used.

 The equation is: 6÷2(1+2), but it showed two different answers, 1 and 9. Which one is correct?

Following the PEMDAS rule, the answer is 9.

First the equation inside the parenthesis is performed, leaving us with:

6÷2(1+2)
= 6÷2(3)

But this is where it gets tricky. Other may interpret that the 2(3) is a parenthetical operation so it should be done first, but actually it is a multiplication. Since according to the PEMDAS rule, division and multiplication have the same precedence, the correct order is to evaluate from left to right.

= 6÷2×3
= 3×3
= 9

Now a question is raised, “Why would the other calculator show that the answer is 1?”

One Quora user from 2012 (yes, this PEMDAS problem has plagued the community that long) said, “The calculator on the left is interpreting everything after the division sign as a group. [Handwritten], it would be 6/(2*(2+1)).”

So in this interpretation, this will yield the answer to be 1.

15 thoughts on “How to Perform PEMDAS Correctly”

  1. You missed one rule the distributive property. When you do that operation the answer is 1. Definitely the correct answer is 1. And one more thing try to search the implicit multiplication for that.

  2. Oh my… You see, I know that there’s this left-to-right thing going on, but I sometimes forget to apply it because of the parenthesis. But now I know why. Thank you! This might help me for my future career. I still don’t know if I should take STEM (Engineering) or just pursue my music career. Hopefully, I’ll come up with a fixed answer someday. Thanks for reading my ted talk.

    – Future Engineer/Musician ¿

  3. I disagree! While 2×3 and 2(3) yield the same result, as both of them are multiplication, one takes precedence over the other.

    f(x)=6÷2x let x=3
    f(3)=6÷2(3)

    It would be wrong to divide it first.
    f(3) was always and always will be equal to 1.

    One may argue that the P in PEMDAS means to evaluate first whatever is inside the parenthesis, WRONG!
    It means that we have to eliminate the grouping symbol (e.g. parenthesis, brackets, etc.) by any means. We may evaluate what is inside first or we may apply the distributive property. The goal is to eliminate the grouping symbol.

    Ganern!

  4. Thats why people with faked math college degrees tend to push 9 is the correct answer. 2(3) is the same with 2 x 3 in simple terms yet when put on complex equation, a special rule arises regarding 2(3). 2 x 3 is the same as (2)(3) but less on 2(3)

  5. hmm yeah we’ve already eliminated the grouping by adding 2 & 1 so we got 3 or (3). pranethisis also means multiplication right ? so it could also be written as 6÷2×3 after we added 2 & 1. and here as what the article said (the tricky part) coz pemdas could also be pedmas or pedmsa ir pemdsa.. so we got 9.

  6. to put in fractional form. (for better understanding of this glitch in the matrix :>.
    6
    __(2+1)… we get an answer of 1 coz
    2

    we put (2+1) besides 2. 6
    ______
    2(2+1)

  7. This why buildings and other structures didn’t last long because of this confusion, when in fact Mathematics must be absolute !

  8. hay…. there is a topic that says in simplifying a group of expression you always free it of parentheses and exponents before proceeding.. and you might forget the use if a parentheses is to group a set of terms and obviously what precedes a parentheses is its coefficient.. an expression group into a parentheses is a singular term..

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