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Technical Order of Operation Question

  1. Aug 12, 2015 #1
    The problem is presented like so:

    1) 8+20/10-3

    = 28/7

    But what if I had rewrote the problem as:

    2) 8+20 ÷ 10-3 ?

    8+2-3 = 7

    Order of operation would divide 20 by 10 and then continue solving left to right.

    What am I missing here? The first problem was presented and solved in a textbook.

    With this type of problem, does it come down to how the problem is presented?
  2. jcsd
  3. Aug 12, 2015 #2


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    The book has it wrong. Division always takes precedence over addition/subtraction regardless of what symbol you use to represent the operation. It's just a convention and can only legitimately be overcome by parentheses.

    EDIT: UNLESS the point of the problem is to divine the non-standard precedence required to give the stated answer.
  4. Aug 12, 2015 #3


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    No grouping symbols are shown.
    Do multiplications and divisions first, from left to right.
    Next do additions and subtractions, from left to right.

    Your #1 and #2 mean exactly the same thing.

    That is the value of the expression given.
  5. Aug 12, 2015 #4


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    If the problem is printed so that the 8+20 is raised above the 10-3, then it's usually taken as a fraction and the 8+20 and 10-3 are grouped together.

    If it's just written straight across with a slash, then it's the same as your case 2.
  6. Aug 12, 2015 #5


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    Right. Grouping symbols are necessary when one needs to change some meanings.
  7. Aug 12, 2015 #6


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    Was #1 written as?

    ----------- --> equivalent to --> (8+20) / (10-3) = 28 / 7

    then the book answer of 28/7 is correct.

    The operator precedence rules are:

    ( ) expressions are evaluated before x or /
    x and / are evaluated left to right
    x and / are evaluated before + and -
    + and - are evaluated left to right

    In truth though we will tend to use parentheses when we feel there may be confusion as in your #1 example.

    8+20/10-3 = 8+2-3 = 7 vs 8+(20/10)-3 = 8+2-3 = 7
  8. Aug 13, 2015 #7

    The book considers a faction bar a grouping symbol... I didn't know this!

    Some people also consider the fraction bar to be a vinculum, which is
    how it originated, since it also groups the numerator and denominator
    together without needing parentheses around them:

    21+2 " (from Dr. Math I believe)
    Last edited: Aug 13, 2015
  9. Aug 13, 2015 #8


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    I would spend as little time as possible on this. All this "order of operations" is, IMHO, a load of nonsense and, although it may be relevant in computer programming, it's not mathematics. Something like:


    is, to me, meaningless, ambiguous and not worthy of attention. Use brackets and move on to some "proper" mathematics.

    (Although the posters above are correct in how this should be interpreted, if you meet something as awful as this, there's no way to be sure what it means. Anyone who could write such an expression might mean anything by it!)
  10. Aug 13, 2015 #9

    Many high schools in the US echo with the mnemonic device PEMDAS, Please Excuse My Dear Aunt Sally, the letters of which stand for Parentheses, Exponentiation, Multiplication and Division, Addition and Subtraction. I prefer to tell students that the order is better remembered as Grouping, Exponentiation and taking Roots, Multiplication and Division, Addition and Subtraction GERMDAS, but I doubt my personal flavor has the inertia to overcome. Either way, you're right in that the fraction bar implies grouping the numerator and the denominator. Another place where grouping is implicit is under radicals. Part of the order of operations is convention, and arithmetic convention as taught has changed in the US from the 1950's when the rules for operations without explicit grouping underwent a slight shift. I always tell students, just use parentheses when in doubt, because calculators are stupid and need the guidance from the user, and that usually helps. One particular case always comes up working with students, and that is the difference between the two ways students calculate ##-3^2##. Many students enter this in the calculator and amazed to find that the answer is ##-9## and not ##9## as they expect. This is because ##-3^2 = -(3^2) = -9## and not ##(-3)^2 = 9##.
    Last edited: Aug 13, 2015
  11. Aug 13, 2015 #10
    Good responses. I am going to incorporate GEMDAS (G=grouping=brackets, bracers, fraction bar, radicals, and absolute value). I won't spend too much time on it, but I am a physics major and am doing an in depth review of math to fill in some gaps of knowledge.
  12. Aug 13, 2015 #11

    @starrynight108, if you're interested in investing in your education, I'd recommend http://www.ixl.com. You can go to the site and do problems for free. Best site I've found thus far.
  13. Aug 13, 2015 #12


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    When there is any possibility of ambiguity, use parentheses. (8+20)/(10-3)
  14. Aug 13, 2015 #13
    Do you work for this company? I've seen you link the same site and it seems like you're a salesperson. Not that it's a bad thing, but does it really belong on this type of forum?
  15. Aug 13, 2015 #14
    Yes, but that's not getting to the root of the issue. That's just a shortcut way of thinking, which will suffice for a lot of people, but we're interested in finding the core answer.

    See above on why a fraction bar is considered a grouping symbol.
  16. Aug 13, 2015 #15


    Staff: Mentor

    What is missing? It seems to me that your question has been answered.

    If you see an expression like this
    $$\frac{8 + 20}{10 - 3}$$
    it's pretty clear that the addition and subtraction operations are to be performed before the division, so in simplified form, the above results in 28/7 = 4.

    On the other hand, if you see 8 + 20/10 - 3, written on a single line, the division should be performed before the addition and subraction, yielding 8 + (20/10) - 3 = 8 + 2 - 3 = 7.

    The precendence rules that have been mentioned in this thread came out of computer programming languages, I believe, which don't ordinarily have the option of writing fractions on two lines, like the example at the top here, and the way such problems typically appear in textbooks. If you want a subexpression to be evaluated before the precedence rules say this would be done, put parentheses around the subexpression to force it to a higher precedence.

    What are you still uncertain about?
  17. Aug 13, 2015 #16


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    The core of the answer is that math expressions are a way to communicate ideas. It's more important to get the correct idea across (or receive it) than to argue that your personal preference or convention is the correct one. At the same time, given that different conventions exist, it's useful to be aware of them so that you can interpret (in context) what an author means.
  18. Aug 14, 2015 #17
    "it's pretty clear that the addition and subtraction operations are to be performed before the division" It's not clear if you write the problem with a division sign. The question was already answered and my original reply to you was to point out that your response is common, but it didn't address the question. That's why I said "see above." The precedence for this operation actually came during the Middle Ages. The 'fraction bar' was called a vinculum aka 'a grouping symbol.'

    You all meant well and I thank you for your responses.
  19. Aug 14, 2015 #18
    I am not arguing about personal preference. I was trying to understand a mathematical concept and the reasoning behind it. The answers that followed the convention of: "when in doubt use parentheses" isn't very mathematical. The idea was to understand the different conventions and a solution was found.

    Seriously, I really think only a few people actually read and understood the post. I won't post these types of questions in the future.
  20. Aug 14, 2015 #19


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    It isn't about your personal preference, it's about the preference (or carelessness) of whoever is communicating the problem or whatever.

    You were talking about finding a mathematical "core" or "root" behind it all, but it's really about convention.
  21. Aug 14, 2015 #20


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    Thread is closed.
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