6÷2(1+2)=?

May 2nd, 2011  |  Published in Uncategorized  |  19 Comments

6÷2(1+2)=?

It's a question that comes around in Facebook recently (I've also read it somewhere in the past). There are two major answers: "1" and "9".

For "1", (Assuming “multiplication by juxtaposition” has higher precedence than regular division. Whether the assumption is true, is depending on which literature is being referred to. If you don't agree on it, the answer is simply 9)

6÷(2×(1+2))
=6÷(2×3)
=6÷6
=1

For "9",

6/2*(1+2)
=6/2*3
=3*3
=9

Notice the question is interpreted differently, you can tell it from looking at the symbols. One is mathematical notation, another is program operator notation.

The difference between mathematics and programming shown above is that, they use different kind of symbols(operators), so they have different order of operation.

Google thinks that is 9.
Check it yourself.

WolframAlpha thinks that is 9.
Check it yourself.

My Casio calculator thinks that is 1.

One interesting thing is, even in programming, different programming languages may have different order of operation, ie. they have different operator precedence (or operator associativity). The difference is mostly on bitwise operations(eg. << & |), and it has been a nightmare for programmers who want to port algorithms between languages. And luckily haXe, the language I'm in love with, that outputs C++/JS/PHP and others, already abstracted the different by inserting the necessary brackets in the output automatically(see here). So I'm happily writing codes in haXe and share the same result in different platforms ;)

  • Keaton

    I’d have to say I don’t agree. Just because the calculator produced a result does not indicate a correct result. Science and maths work off repeatable principles, and accepted standards need to be accepted, such as common orders of operations, which i would say here, is not being followed correctly by the calculator. The differences being, if I am not misttaken, the calculator rates the multiplication as a higher rated order, and process it before the division, and with a few little calculations such as changing the above to be 6*2 rather the 6/2, you can see the results become the same on paper & on the calculator

    I think the flaw here is the same issues the plague any standard, my html pages render slightly different in all the browsers, yet I code to the same standard. I would hardly blame my web coding, when it’s browser specific. In this situation I don’t think the math is incorrect, I think the device has decided to simplify some of the standards for their simpler applications (calculators don’t got the brains our PC’s do)

    I think if you allow your calculator to produce this result, you are simply allowing yourself to trust in a device that has not held to understood standards, which using our previous analogy, you would blame the browser, and not the standard.

    I guess it sums up to: Just because it came out of a calculator, it is correct? IMHO I don’t agree

  • Andy Li

    Hi Keaton,

    That’s exactly the debate of which is the “correct” standard, whether “multiplication by juxtaposition” has higher precedence (in mathematics). There are reference from the web and from the books that support the two side. So it is kind of “depends”. There should be an authority to publish a standard for the acadamia to follow, but I haven’t found such thing.

    For the calculator, some newspaper simply said that Casio fx-3650P has a bug on order of operation. But I think that, to put higher precedence on “multiplication by juxtaposition” requires introducing an extra operator/token. I mean converting it to simple multiplication is easier to program. So I believe it is done by design.

    And I don’t think the calculator’s answer must be right, just like I don’t think Google’s or WolframAlpha’s is right. They’re just examples to illustrate there are supports on both view.

    To conclude, we need clarification from an authority, that is accepted by everybody. Before that, we should avoid writing/coding in such embarrassing way.

  • Alan

    6÷2(1+2)=z
    6÷2(3)=z

    At this point is where the debate states. However, we still have parentheses and they can only be removed through distribution. Where you distribute the multiplying variable to the items inside the parenthesis in order to finally “remove” them, correctly. This is still part of the order of operation and first as it deals with the parentheses.

    6÷6=z
    1=z

    Now this would be different if you were written 6/2(1+2) or 3(1+2) as the multiplying factor is six halves.

    Hope this helps.

  • Robin

    Good comments Andy. I agree there should be an accepted standard set. I also remember my math teacher emphasizing the importance of writing out problems in a clear manner. The simple addition of a multiplication symbol before the parentheses removes any doubt that the division should be completed first while an extra set of parentheses around the 2(1+2) makes clear that operation should be completed first.

    It’s like asking if 2(y)is equal to 2y or 2 * y. In most cases it wouldn’t matter, but stick 6/ in front of it, and that is where the problem arises.

    Let’s just hope if the solution to a problem like this one is crucial to someone or something that the person solving it will be darn sure what was intended.

  • http://www.yahoo.com.hk RICK LEUNG

    DEAR ALL,

    I’AM FROM HK ( CHINESE – ASIA ).
    IF SOME ENGLISH OR AMERICAN MATHEMATIS PROFESSOR CAN PROVIDE SOME IDEA, MUCH APPRECIATED.
    MANY THANKS!
    RICK

  • viaria

    i think it is 1

  • lion

    use BoDMAS law (Bracket, Division, Multiplication, Addition, Substraction)
    solve the one in bracket first..so the answer is 1..

  • daixtr

    People. If two symbols have equal precedence, then THEY ARE READ or SCANNED from LEFT to RIGHT. That is the subtle trick that you all missed to consider.

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  • gamerktc12

    BIDMAS. it’s simple.

  • gamerktc12

    @daixtr: there is no need here. one is a bracket and the other is divide. bracket always come before divide.

  • kenneth

    isn’t it 6:2(1+2)=z and then 6:2*3=z ??? because you solve the addition inside the parentheses these things must dissapear. And with 6:2*3=z we must accept that division and multiplication are equal to one another so we must read them from left to right.

  • kenneth

    no daixtr is right. When you solve the 1+2 inside the brackets you have to make the brackets dissapear. So then it does become 6:2*3 and then you have to do them from left to right resulting in 9.

    Source : every single math professor I’ve ever had including in university

  • http://www.facebook.com/people/Johannes-Nokkala/1049160457 Johannes Nokkala

    Multiplication and division hold equal precedence. If there are two or more operations with equal precedence those operations should be done from left to right. Thus, the correct answer is indeed 9.

  • lalit

    guys answer is 7..

  • Eagle

    My first thought was

    6/2 (2+1)

    6/2*3

    3*3

    9

    but then I remebered something from doing algebra, where you multiply everything in the brackets from what is outside of the brackets, eg if the equation was

    x=a/b(c+d)

    x=a/(bc+bd)
    then substitute a=6,b=2, c=2, d=1

    x = 6 / (2 x2 + 2 x 1)

    x = 6/ (4+2)

    x=6/6

    x=1

    Just an alternative…

  • pg

    Let y = 2(1+2).

    Then y = 6.

    Question is 6 / y = ? which is 6 / 6 = 1.

    Anything wrong in above approach?

  • Ricky

    I look at it this way. 6 ÷ 2(1+2) = X. You multiply both sides by the 2(1+2). Then it looks like 6 = X * 2(1+2). Then it is 6 = 6X. X = 1. So simple. Or you can make it look like a fraction. 6 / 2(1+2). 6/6. 1.

  • wrot

    Yes, another example with your idea:

    3/1 + 2 = 1; Isn’t it?

    Let y=1+2;

    Then y=3; Got it?

    The question is 3 / y = ? which is 3 / 3 = 1;

    What is wrong? 3/1 + 2 = 5
    :(