Q: How do you calculate 6/2(1+2) or 48/2(9+3)? What’s the deal with this orders of operation business?

Mathematician: Now that the Physicist and I have answered questions like these ones at least 9 times via email, I figure we should get this horrible topic out of the way once and for all with a short post.

The “order of operations” tells us the sequence in which we should carry out mathematical operations. It is merely a matter of convention (not a matter of right or wrong), giving us a standardized way to decide whether 6/2*3 is (6/2)*3 or 6/(2*3). The convention has been accepted just so that when different people see an equation they can agree on its meaning, even if they hate each other’s guts. You could come up with a different convention, but why bother? Let other people waste their time coming up with arbitrary conventions so that you can merely waste time reading posts about them.

The rules to follow are:

1. Things in parenthesis get computed before things not in parenthesis (this is because of the (little known fact) that parentheses kick ass).

2. Then, exponents get dealt with (including roots). Hence a^b*c = (a^b)*c.

3. Next, multiplication and division get carried out from left to right. So a*b/c*d is ((a*b)/c)*d. Note that division can be thought of as still carrying out a multiplication since a/b = a*(1/b), so it makes a certain sense that neither division nor multiplication takes precedence over one another.

4. Finally, additions and subtractions get carried out from left to right. Subtraction can be thought of as being like addition since a – b = a + (-b) so it makes sense that subtraction is treated on an equal footing with addition.

People sometimes use the expressions PEMDAS, “Please Excuse My Dear Aunt Sally” or “Puke Exhaust Mud Dolls Autumn Sasquatch” to remember these rules. These all stand for “Parenthesis, Exponents, Multiplication and Division, Addition and Subtraction”. It is easy to get confused by these mnemonics though because they don’t capture the fact that multiplication is treated the same way as division (carried out left to right), and addition is treated the same way as subtraction (carried out left to right). That’s why I prefer: “Puke Exhaust Mud And Dolls Autumn And Sasquatch”

One interesting thing to note about the order of operations is that there is nothing interesting about them whatsoever. But that implies there is something interesting about them. This is a contradiction, and hence the order of operations form a paradox. They are also about as fun as getting to watch a fish crawl up the stairs as a reward for answering math questions.

But, that won’t stop me from giving an example. Consider:

a^b*c/d+e-f.

Without a convention, it has many possible interpretations, such as

a^(b*(c/(d+e)))-f

or

(a^b)*((c/(d+e))-f)

which could be horribly confusing for people trying to communicate with each other. But, the correct interpretation according to the rules mentioned above, is:

(((((a^b)*c)/d)+e)-f).

When in doubt about what the orders of operation are for an expression, there is a simple solution which works even better than tattooing PEMDAS to your face. Just paste the expression into the google search bar and hit “search”. For instance, if I paste in:

4^2*3/6+1-5

it gives back

(((4^2) * 3) / 6) + 1 – 5 = 4. 

Not only is this the correct answer, but it shows you explicitely the order of operations that were used, so it’s useful for learning.

Google even gets this one right:

4^2*3/6 + 1 – 5/14*3 + 6/10 – 4*2/14*6^3/18 + 14

which is the kind of question little sisters give you when they are old enough to finally sort of get what a mathematician is.

It’s important to note that not all calculators handle the order of operations in the standard way. Why? Because people who make calculators hate children. The preceding sentence was not intended to be a factual statement.

Also, a lot of programming languages use the standard order of operation rules, but then you have exceptions like Smalltalk.

Oh, and in case you were wondering:

a^b^c  = a^{b^c}.

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212 Responses to Q: How do you calculate 6/2(1+2) or 48/2(9+3)? What’s the deal with this orders of operation business?

  1. Jeffrey says:

    Dear Kate first of all i dont care for what my calculator says, a incorrectly programmed.machine doesn’t have to be the smarter one just because it says so! Its a computer and computers are operated, if you operate them correctly ,they will give you.the correct answer to this equation.

    In order to let your calc see the correlation between the multiplier “2” over the brackets, you need to let it prioritize that part firsts.. If not it will follow pemdas and give you a invitee answer 9!

    Let me explain why this answer is in correct and maybe you’ll understand it afterwards..

    Lets get clear that the concept of multiplication is nothing else than short writing a certain amount of the same you have. For example if I have 2+2+2 I have 3 times the unit “2” , right!?

    So the multiplier becomes 3. To write this in a multiplication, that could be 3*2 in other words and also in literal speaking you pronounce this as 3times 2..so far so good right?

    OK the same concept apply if you replace the value of the unit..for example with an letter “b”.. If I say you have 2b that means literally you have 2 times whatever “b”stands for. Perfectly equal to saying 2b is exactly b+b!

    Ok ..now let’s explain that in this case writing down 2b there is a unwritten multiplication symbol between 2 and “b”, that is OK as it eases things up, but the correlation of 2 over “b” remains!

    Now if we get back to the equation 6÷2(1+2),

    You see the 2 standing directly before the parentheses, even though in many case it would be fine to place the * symbol yourself, now its not! Why…
    Because if you do so, you’ll ignore the fact of why the symbol is left out.. The 6 and the 2 are separate units as the are split by the “÷” symbol, the “2” is not separated and should not be from whatever value is between the parentheses.

    Your order of operations ( pemdas) rules, tell you to solve parentheses first. But as the correct way of dealing with the parentheses would be to see or rewrite it as
    2 units of (1+2), like in 2b or 4a etc..

    That can be rewritten as ((1+2)+(1+2)) or as ((2*(1+2)), on its own you could do 2*(1+2), but not now, since its part in a equation!
    If you do, your pemdas will neglect the correlation and true meaning over what multiplication is! what the multiplier is over which unit..

    So solving the equation as 6÷2* (1+2) would be incorrect as in this case it is absolutely not the same as in doing 6÷((2*(1+2))!

    Hope it explains it a bit.. So to be honest yes the calc is wrong.

  2. Kenya Carter says:

    I asked both my husband and my 15 year old son and both said 1. Thank god. I was about to evict everyone out of this house. The calculator is wrong and so is the internet to come up with 9. Of course if you have a real calculator (scientific or graphing) that can handle an equation, that will spit out 1 all day long. The 2 belongs to the (1+2) portion of the equation. 6/2(3) The parentheses don’t just disappear! You still have to resolve it before you can solve! I’m 41 years old taking pre-calc. Undoing a life time of misunderstanding about math.

  3. Kate says:

    Just so u know the graphing calculator says nine. The answer is nine. You only do what’s inside the parentheses first. Here they are telling you to multiply which u do after you divide left to right.

  4. Kate says:

    Also since many of you seem to forget you can’t have a coefficient without a variable. Let’s check our work just to be sure.
    6/2(x)=9 or 6/2(x)=1 if we solve for x it must equal 3
    6/2(x)=9
    6(x)=18
    X= 3
    6/2(x)=1
    6(x)=1 ITS DEFINITELY NOT ONE PEOPLE. I DONT UNDERSTAND WHY PEOPLE WANT TO PUSH IMPLIED MULTIPLICATION. ITS NOT A THING IT NEVER WAS. THIS ISN’T ALGEBRA ITS FOURTH GRADE ARITHMETIC. YOU DO WHATS INSIDE THE PARENTHESES FIRST AND THEN YOU MOVE ON. ITS 9 .. so says my scientific calculator, a graphing calculator, and Google. Tighten up folks

  5. Kenya Carter says:

    We can beg to differ on this one..but 6÷2(1+2) is not the same as 6/2(1+2) when you enter it into a calculator or into google. The calculator is trying to solve 6 OVER 2 (or 6 HALVES like a fraction.) and handles the problem left to right. It doesn’t understand “multiplication by juxtaposition” rules or that a(b+c) is the same as ab+ac.

    The Casio fx115ES Plus Natural is the best calculator you will ever buy. It allows you to enter an equation exactly as stated, even a fraction with the numerator over the denominator.

  6. Kate says:

    I didn’t need the calculator to solve this.
    I read it as 6 divided by 2 (2+1) also. I do not have a division sign on my phone. Multiplication and division are done from left to right. Even you distributed the answer is 9 people. You forget, seriously
    6/2(2)+6/2(1) = 3(2)+ 3(1) = 6+3=9

    The answer is 9

  7. Kate says:

    None of these things should be needed to solve this. This is grade school math, it isn’t even algebra. You simply follow the order of operations. INSIDE the parentheses first, exponents, multiply or divide left to right and add or subtract left to right. Those have always been the rules of math and still are.

  8. Aksa says:

    “multiplication denoted by juxtaposition is interpreted as having higher precedence than division, so that 1/2x equals 1/(2x), not (1/2)x. For example, the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a slash,[9] and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lifshitz and the Feynman Lectures on Physics.[nb 1]”

  9. mathman says:

    Aksa, any book may announce its convention. The problem here is that some people assume that without convention, the “standard” or “default” is 1/2x = (1/2)x because 2x = 2 * x

    I will post later with all the math texts I found that ALL use 1/2x = 1/(2x) and NONE use 1/2x = 1/2 * x.
    They explain what quantities are and factor groups as well. Basic stuff. Sure 2x = 2 * x, but 2x is a product of factors before it is used in the order of operations. in the order of operations, we need to know first what the operators and operands are. In this case, 2x is a single quantity, that is, it is one operand.

    Does anyone what to argue this fact? If so, provide a reference please.

  10. kate says:

    Yes. 1/2(x) is literally one half multiplied by x . I spoke to multiple professors about this and they say neither of us our correct because the problem is ambiguous how it is written. Type the problem into any scientific calculator, Google or wolphram alpha because they all have the same answer as me.

  11. kate says:

    None of you are making sense. U can’t prove what you are saying
    6/2(3)=9
    Let’s substitute (3) with (x) so I can prove it
    6/2(x)=9 multiply both sides by 2
    6(x)=18
    X=3 . U can’t do it if the answer is 1. The guy way above is correct it’s different when there are variables and constants.

  12. sasha says:

    kate you just proved yourself wrong. when you multiply by 2 on both sides the x is still at the bottom of 6 so you will get 2 = 9.

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