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6÷2(2+1)


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The problem:
6÷2(2+1)

The solution:
1

The disagreement:
People are arguing if the correct solution is 9 or 1.

The online disagreement over this has been incredible.  In my attempt to enlighten both sides of this disagreement, I have been called all kinds of unsavory things, so I am bracing myself for a large number of downvotes here.  In my assertion that the answer is 1, I have done a great deal of background reading and research, and am more firm than ever in my commitment that there is a singular correct answer, and that answer is indeed 1.  However, I am open to having my mind changed.  Just please, if you disagree with me, explain why I am wrong.  Show me where I made a mistake.  For the love of all that is good in this world, please don't just launch into personal attacks, telling me how wrong I am, without actually giving any contrary argument.

The (wrong) argument for the answer being 9:
For the most part, this argument focuses on the order of operations, primary relying on the mnemonic device of PEMDAS or similar acronyms.  Following the order of operations, you would do Parentheses first, followed by Exponents, then Multiplication and Division in order of left to right, finally followed by Addition and Subtraction, also calculated left to right.

= 6÷2(2+1)
= 6÷2(3)         Performing the operation in parentheses first.
= 3(3)             Performing multiplication and division from order of left to right, thus dividing 6 by 2 first getting 3.
= 9                  Multiplying 3 by 3 gives you 9.

The (wrong) argument for the answer being 1:
Many people are getting the correct answer, 1, but arriving at it incorrectly.  They are also attempting to apply PEMDAS, but never learned, or forgot, that multiplication and division take equal priority and are performed in order of left to right.  Because multiplication comes first in this acronym, they believe it must be performed first.

= 6÷2(2+1)
= 6÷2(3)        Performing the operation in parentheses first, according to the order of operations.
= 6÷6             Incorrectly performing multiplication first, multiplying 2 by 3 to get 6.
= 1                 Six divided by six equals 1.

 

Misconceptions leading to the wrong ways of solving this problem
The mistakes people are making seem to stem from a couple of misconceptions.  It seems there are widespread lapses in how math is being taught, leading to many having these misconceptions.  This includes people who really should know better, like math teachers and engineers.

 

Misconceptions about PEMDAS
The order of operations is hammered into us when we learn math.  Not addressed, however, is why we learn the order of operations or why it is applied the way it is.  It is not just a made up way of doing things that we decided is correct.  Rather, it is a helpful tool derived as a shortcut to make solving algebraic problems simpler.

The order of operations is not technically part of algebra.  The modern order of operations has been around for less than a century, yet algebra itself is much older than this.  Theorems proved using algebra hundreds of years ago are still just as valid today, even though they were proved without the benefit of the modern order of operations or PEMDAS.  How could this be?

Algebra is simply a set of properties, known appropriately enough as the algebraic properties.  I believe there are some 15 common algebraic properties, and some other less common ones, though I admit I don't know all the algebraic properties off the top of my head so I could be wrong on this.  However, algebraic problems must be consistent under all the algebraic properties.  Otherwise, you could end up with an equation being solved two different ways with two different answers.  The algebraic properties themselves are not simple conventions either but are mathematically provable. 

The order of operations is a shortcut.  When followed properly, the result should be that equations are solved in such a way that it is in full accordance with all the algebraic properties.  This way you can do algebra, without sitting there doing the same problem a dozen different ways, testing every single way against all the algebraic properties for consistency.  But that is all the order of operations is, a shortcut to achieve consistency with the algebraic properties. 

In arguing with people about the correct answer to this problem, it seems people now are under the impression that the order of operations is some deep mathematical law.  It is not.  It is only a modern convention to make things a bit easier.  It is not mathematically provable the way the algebraic properties are.  Yet I have had math teachers try to tell me that the algebraic properties don't apply in this case because they run contrary to the order of operations.  The algebraic properties always apply, and if you are running into a conflict with them, something is wrong.

 

Misconceptions about obelus (÷)
The division symbol, properly called an obelus, is not typically used in algebra today for exactly the kind of reason we see here;  It opens the door to ambiguity about what is actually being divided. People are reading this problem 6÷2(2+1) as being equivalent to:

Capture.JPG

This is not the case though.  Though today we have computers that help us write mathematical formulas, if we travel back in time we will find an era in which typewriters reigned supreme.   It was much more common in this era to use the obelus in algebraic equations that were typed, but this could lead to a bit of ambiguity about what exactly was being divided.  The established convention was that everything to the left of the obelus is the numerator and everything to the right the denominator.  If this was not the case, the convention held that you should use parentheses to denote this.  Since no such parentheses exist, the correct way to interpret the division here is as follows:

Capture1.JPG

 

Solving the correct way

One of the algebraic properties relevant here is the distributive property.  Remember, algebraic properties are not optional.  They are mathematically provable statements and must be applied consistently.  Therefore, if we make sure to properly apply the distributive property we get:

= 6÷2(2+1)
= 6÷(4+2)         Applying the distributive property.
= 6÷6
= 1

Now, this might seem like we are running contrary to the order of operations, but this is not true.  If we follow the proper conventions for using an obelus for division, the order of operations holds true.

= 6÷2(2+1)

=Capture1.JPGApplying the obelus symbol for division follwing established conventions.

=Capture2.JPGPerforming the operation in parentheses first, according to the order of operations.

=Capture3.JPGSimplfying the numerator and the denominator, in accordance with the order of operations.

=1       Six divided by six equals one.

 

As you can see, if we follow proper conventions for using the obelus for division, there is no conflict between the order of operations or the algebraic properties, as it should be.  A major source of confusion here, I believe, is that the obelus is not typically used in algebra today, so the conventions surrounding its use would not typically be taught.  Another issue is that the order of operations are being taught as an absolute mathematical law that takes precedent overall, not what it actually is, a simple shortcut that has only been around in its current form for the last century.

I hope someone finds this helpful.  If you think I'm wrong, tell me why 🙂

 

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To me it's 9. I was taught in school that in the order of operations multiplication and division are of the same rank and they are done fron left to right in the order in which they appear. Even my calculator gave me a result of 9. This type of equation, at least to me, is not an algebraic one.

Edited by johnnyc82
Added more info (see edit history)
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2 minutes ago, johnnyc82 said:

To me it's 9. I was taught in school that in the order of operations multiplication and division are of the same rank and they are done fron left to right in the order in which they appear. Even my calculator gave me a result of 9

I'm not sure you actually read my entire post.  If you did, you will see that the order of operations should yield the correct answer, 1, if you follow the proper conventions regarding using an obelus to represent division in algebra.

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9 hours ago, Imouto Kitten said:

So, would 9 be the correct answer if it was written

6/2(2+1)

As such expressions are usually typed on account of the divided by symbol not being a common symbol on modern keyboards(and was it really a common symbol back in the days of typewriters? If so, it begs the question of why it was dropped considering the qwerty layout is a relic from the typewriter days)?

If so, I think part of the problem might be that most aren't taught, or have forgotten that the divided by symbol is supposed to be one-line shorthand for a divider bar. At the very least, I can't recall hearing this rule before, and even the name is unfamiliar.

That said, I do think it a common failing of common mnemonics(the one I was taught was please excuse my dear aunt sally) that they almost never communicate that multiplication and division are supposed to have equal priority(I myself made this mistake for many years, and I was considered good at math for most of my k-12 education). Also probably doesn't help that middle and high school algebra classes typically teach very little algebra and focus almost entirely on solving polynomial equations while failing to demonstrate the usefulness of solving equations, and the history of math is almost never touched upon.

Though in all fairness, even when taking the higher maths required for my BS in Computer Science, my mistakes were usually in the arithmetic and not whichever branch of higher maths was at hand, and every person who makes heavy use of math I've talked to has had the same complaint(in particular, off-by-one errors seem to be the bane of everyone from physicists to biologists, pure mathematicians to computer scientists, and pretty much any other field that deals with numbers and their arithmetic manipulation).

All that said, if I was typing this expression into a computer calculator or trying to communicate it to another person, I'd probably throw in another set of parentheses to remove any ambiguity and in general, I'd argue "when in doubt, add parentheses" is a good rule of thumb to ensure you and the reader are parsing an expression in the same way. Hell, I wouldn't be surprised if this expression was written as such specifically to sow discord and cause arguments over the correct way to interpret it or so someone could feel smug for getting it right when so many get it wrong.

Also, I figured out both answers before entering the thread, but I'm not sure which I thought to be correct and which I thought to be the common error, and was expecting a debate over nitpicks in parsing maths expressions, but I wasn't expecting a right for the wrong reasons explanation to show up.

I wrote it on google search with both ÷ and / and gave me 9 both times. I've always wrote it like this:

6÷2(2+1)

6÷2(3)

3(3)

9.

 

I don't think the distributive property applies here.  Or should it apply in every mathematical equation?

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This feels to me like any number of ambiguous sentences you might find on the internet.  If I said "I saw a man on a hill with a telescope," did I have the telescope or did the man?  Neither answer is wrong, and if you're trying to communicate effectively you'll probably try to avoid expressions like this.  Mathematics is, in a way, a language.  And how you write it down is just a matter of convention.


Under your interpretation, where everything to the left of the obelus is the numerator and everything to the right is the denominator, it is perhaps appealing that the distributive property works in the way you find natural.  And that could argue for the superiority of your proposed convention.


But in the competing interpretation, in which only the things immediately to the left and right of the obelus are the numerator and denominator, it's a faulty application of the distributive property to distribute 2(3) into (6).  Those two things aren't being multiplied by each other.  The two is interpreted as the denominator and would have to be represented as one-half if you wanted to distribute it.


Even the Wikipedia article on order of operations has a section on "exceptions".  


I think the predominant convention would give you the answer of 9 here.  Mathematica evaluates 6/2(2+1) as 9.  MATLAB evaluates 6/2*(2+1) as 9.  And, to produce a simpler example, if someone wrote 1+1/2, I would think they probably meant 1.5, not 2.  But I wouldn't call either answer wrong, exactly.  Maybe I was looking through the telescope and saw the man, and maybe I simply used my unaided eye to spot a man who had a telescope.  It's a clumsy attempt at communication.

Edited by jb2ja
Formatting and typo (see edit history)
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4 hours ago, johnnyc82 said:

I wrote it on google search with both ÷ and / and gave me 9 both times. I've always wrote it like this:

6÷2(2+1)

6÷2(3)

3(3)

9.

 

I don't think the distributive property applies here.  Or should it apply in every mathematical equation?

The distributive property should always work when the situation is presented.  The algebraic properties are not things you can just pick and choose when to apply- For mathematics to be consistent the basic properties of math must be consistent.  These properties are mathematical truths that are derived from pure mathematics and provable to be true, not just made up by someone who issued a decree that it is so.  The Pythagorean theorem still works today because it is based on fundamental mathematical principles. If algebraic properties were only applied sometimes, then mathematical theorems based on them would only work sometimes, and that isn't the way the world works.

The real question is the use of the obelus for division and the ambiguity it introduces.  In this equation, are we distributing six halves by the quantity of (2+1)?  Or are is 6 the numerator with 2(2+1) the denominator?  The obelus is not used to represent division in modern algebra, so to answer this question I had to look up historical conventions regarding its usage from when its usage was more common.  According to the accepted conventions for the use of the obelus, everything to the left is part of the numerator and everything to the right is part of the denominator unless parentheses indicate otherwise.

3 hours ago, jb2ja said:

This feels to me like any number of ambiguous sentences you might find on the internet.  If I said "I saw a man on a hill with a telescope," did I have the telescope or did the man?  Neither answer is wrong, and if you're trying to communicate effectively you'll probably try to avoid expressions like this.  Mathematics is, in a way, a language.  And how you write it down is just a matter of convention.


Under your interpretation, where everything to the left of the obelus is the numerator and everything to the right is the denominator, it is perhaps appealing that the distributive property works in the way you find natural.  And that could argue for the superiority of your proposed convention.


But in the competing interpretation, in which only the things immediately to the left and right of the obelus are the numerator and denominator, it's a faulty application of the distributive property to distribute 2(3) into (6).  Those two things aren't being multiplied by each other.  The two is interpreted as the denominator and would have to be represented as one-half if you wanted to distribute it.


Even the Wikipedia article on order of operations has a section on "exceptions".  


I think the predominant convention would give you the answer of 9 here.  Mathematica evaluates 6/2(2+1) as 9.  MATLAB evaluates 6/2*(2+1) as 9.  And, to produce a simpler example, if someone wrote 1+1/2, I would think they probably meant 1.5, not 2.  But I wouldn't call either answer wrong, exactly.  Maybe I was looking through the telescope and saw the man, and maybe I simply used my unaided eye to spot a man who had a telescope.  It's a clumsy attempt at communication.

 

I just want to quickly point out that this isn't my interpretation.  Rather, these were the recommended and agreed upon conventions regarding the usage of an obelus to represent division in an algebraic equation as adopted by both Oxford and Harvard Universities and has stood as the standard convention for over 100 years.

For the alternative interpretation, where 6 is the numerator and two is the denominator, that would, of course, change things.  You would then be distributing six halves to what is in the parentheses, which would, of course, give you nine, but this would involve having to interpret the meaning of the obelus in a non-standard way that doesn't follow historical conventions.

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Years and years the word PEMDAS comes to play, you were always required to do that, it was established in math for a specific reason otherwise you would come up with multiple answers, Parentheses, Exponent, Multiplication, Division, Addition, and Subtraction

6/2( 1+2) do parentheses first, doesnt matter that addition is in there you HAVE to do whatever is in there first then move on in that order, 1+2=3

now its 6/2(3) because the 2 is next to the parentheses, its multiplication, because of the basic algebra equation of any number next to a number in parentheses is multiplication ( ex. 4(5) is 20) 

so 2x3 is 6

now you divide and the answer is 1

The reason people fail this is because the human generally is taught to read left to right, so we solve whats on the left first then go our way over to the rest, when that isnt the case for each problem.

Edited by Brittanybunny (see edit history)
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To clarify, are you contending that the obelus specifically is to be interpreted differently from the solidus?  Is 6÷2(2+1) different from 6/2(2+1)?  Or do you believe both are 1?

I am also curious as to where you would land on the interpretation of log(10).  In my experience, most Americans would evaluate that as 1.  If an American intends to use base e instead of base 10, they'll typically write it as ln(10) = 2.302...  But in many other countries, the prevailing interpretation is that "log"refers to the natural log.  Is one side wrong here?  Does whether they're wrong depend on whether they can cite a 100-year-old style manual consistent with their cause?

If, as you say, the obelus is not used to represent division in modern algebra, then it seems a little faulty to pick a fight as to what it means with a modern audience.  ISO 80000-2 apparently discourages the use of the obelus for division, which is perhaps a further statement that, whatever its historical meaning, the symbol is incompletely defined in the context of modern discourse.  Does that standard supersede your Harvard and Oxford references?  Conventions do change sometimes.  The word toilet used to refer to the cloth cover of a lady's dressing table, but it would be counterproductive to good communication to try to use the word that way today.

Again, I'm not declaring either answer to be wrong.  Just exploring what the question means.

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Imouto: Yes, that's the symbol we're talking about.

To TVGuy's point, the following 1917 paper observes that it was (at the time) the predominant interpretation that an entire product to the left or right of a division sign would be regarded as the numerator or denominator.  And that paper points out that it was inconsistent with the stated rules of order of operations, but that it was (at the time) what such expressions were widely understood to mean.

Lennes, N. J. “Discussions: Relating to the Order of Operations in Algebra.” The American Mathematical Monthly 24.2 (1917): 93-95. Web. http://www.jstor.org/stable/2972726?seq=1#page_scan_tab_contents

It seems to me that this was a sloppy exception to the rules that came about as a result of trying to present things cleanly while using the limited typesetting technology of the day.  We no longer have the technology limitations that originally motivated this untidy exception, and so we as a society seem to have largely forgotten it ever existed.  And that's probably for the best, if it was motivated by the typesetting technology of the time and not by an actual mathematical consideration.

I recall my algebra teacher telling us that the PEMDAS convention came about after many heated arguments.  Duels, even, I think she said.  I don't know if that's true or not, but it seems that 6÷2(2+1) has been a topic of similarly heated debate in more modern times.  There's no fundamental right or wrong here.  It's like driving on the left side of the road versus the right side.  It doesn't really matter, as long as everyone driving on the same road at the same time does it the same way.  A hundred years ago, people drove on the other side of this mathematical road.  Today, we don't.  Either way, we get where we're going.

(And as an aside, I wonder if there are similar debates as to whether names starting with Mc still get alphabetized as though it were Mac...  Given that most things are no longer looked up by humans rifling through filing cabinets, I suspect that may be another antiquated "rule".)

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11 hours ago, Brittanybunny said:

Years and years the word PEMDAS comes to play, you were always required to do that, it was established in math for a specific reason otherwise you would come up with multiple answers, Parentheses, Exponent, Multiplication, Division, Addition, and Subtraction

6/2( 1+2) do parentheses first, doesnt matter that addition is in there you HAVE to do whatever is in there first then move on in that order, 1+2=3

now its 6/2(3) because the 2 is next to the parentheses, its multiplication, because of the basic algebra equation of any number next to a number in parentheses is multiplication ( ex. 4(5) is 20) 

so 2x3 is 6

now you divide and the answer is 1

The reason people fail this is because the human generally is taught to read left to right, so we solve whats on the left first then go our way over to the rest, when that isnt the case for each problem.

PEMDAS should be written like this

P (Parenthesis)

E (Exponents)

MD (Multiplication and Division)

AS (Adding and Subtracting)

Multiplication and division, as well as adding and subtracting,  are done from left to right in the order in which they appear (if the division comes first it should e done first and so on).  If multiplication is done first you get the result of one but if it's done like I stated above you get 9.  This is how PEMDAS was taught to me and I've searched online and all sites I've seen value multiplication and division as being of the same rank, meaning they're supposed to done from left to right.

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7 hours ago, johnnyc82 said:

PEMDAS should be written like this

P (Parenthesis)

E (Exponents)

MD (Multiplication and Division)

AS (Adding and Subtracting)

Multiplication and division, as well as adding and subtracting,  are done from left to right in the order in which they appear (if the division comes first it should e done first and so on).  If multiplication is done first you get the result of one but if it's done like I stated above you get 9.  This is how PEMDAS was taught to me and I've searched online and all sites I've seen value multiplication and division as being of the same rank, meaning they're supposed to done from left to right.

Except based on the history of Pemdas, its not suppose to be left to right, it was stated through all my algebra courses even in college that you go in order, doesnt matter, its not always left to right, we read left to right so we solve that way but it isnt correct, and math isnt something you can just google, you have to do the work yourself, relying on the internet for math questions is automatically a fail, because you will get multiple answers because people dont do it correctly. 

the reason people get 9 is they do the division first but pemdas is still doing whats in parentheses first no matter what is in them, then once that cancels out you do exponents if there are any, then multiplication, then division, addition and subtraction, i remember getting marked wrong so many times for having different answers if i didnt do it this way. 

Even if you had a random equation that only had multiplication, addition then division, you still go in the order of pemdas or its wrong

if the equation is something like 6/3x5 its still multiplication first, otherwise you get it wrong based on what ive been taught for several years in math

Edited by Brittanybunny (see edit history)
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46 minutes ago, Brittanybunny said:

Except based on the history of Pemdas, its not suppose to be left to right, it was stated through all my algebra courses even in college that you go in order, doesnt matter, its not always left to right, we read left to right so we solve that way but it isnt correct, and math isnt something you can just google, you have to do the work yourself, relying on the internet for math questions is automatically a fail, because you will get multiple answers because people dont do it correctly. 

the reason people get 9 is they do the division first but pemdas is still doing whats in parentheses first no matter what is in them, then once that cancels out you do exponents if there are any, then multiplication, then division, addition and subtraction, i remember getting marked wrong so many times for having different answers if i didnt do it this way. 

Even if you had a random equation that only had multiplication, addition then division, you still go in the order of pemdas or its wrong

if the equation is something like 6/3x5 its still multiplication first, otherwise you get it wrong based on what ive been taught for several years in math

Using 6/3x5 as an example.  I put this in a calculator that lets you write equations and it gave me 10.  Maybe I put it wrong in the calculator but it did the division first, instead of showing 6 divided by 15.

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  • 3 weeks later...
Guest Blurple

If we’re going by what was hammered into us in school then it’s 9. Now I’m completely unsure what the answer would be with the common core teaching, but a lot of people feel that the problem should be solved just like they’re reading from left to right. Any answer they get from that is probably going to be wrong if it’s mixed operations. Then you have the people who completely ignore the parentheses and get 1.

 

I think for your logic to work the problem would have to be written as 6/2 x (2+1)/1 which would still equal 9.

Edited by Blurple (see edit history)
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  • 1 year later...

Anyone believing the answer is 9 isn't thinking about this with the experience of higher level math classes such as algebra and calculus.

Multiplication by juxtaposition such as 3x or 3(5) clearly takes precedence over all other multiplication or division.

So the fact that we need to multiply first is not because of the kind of division sign--that's irrelevant.  It's because the multiplication is implied by juxtaposition rather than with a x or * sign.  If the problem were written 6/2*(2+1) then we would proceed left to right and answer 9.  But with 6/2(2+1), we do parentheses, then juxtaposition, then the division last, to get 1, because multiplication by juxtaposition binds tighter than explicit multiplication or division.

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It can only be one.

 

The first division depends on the multiplication, and half of that depends on what's in the parenthesis.

Then like David said the "Multiplication by juxtaposition such as 3x or 3(5) clearly takes precedence over all other multiplication or division."

 

(My text again, font got stuck )

so after you collapse the parentheses you'll see that you're first division is dividing by a juxyapose of two numbers you have to multiply the juxtapose numbers first. If there is no symbol in between the values they have a higher priority like parenthesis. 

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Ok it's simple. 

 

Machine logic: 9

Algebraic Convention: 1

 

Algebra convention has implicit multiplication precedence. Something machines don't always or have to account for. 

 

Neither trumps the other in this case because there is no basis to decide which to use. 

 

I'm a drafting engineer so is 1, someone who works with machines might say 9.

 

There is no wrong answer because there is no basis to determine which system to use.  

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  • 1 month later...

I’ve not heard of PEMDAS before. When I was at school (too many years ago, and in the UK) we used BODMAS:

Brackets (Parentheses)

Orders (I.e. powers)

Division

Multiplication

Addition

Subtraction.

It was understood that multiplication and division were of equal rank and likewise for addition and subtraction.

However I always strive to avoid any ambiguity in the first place. If adding a redundant pair of brackets can avoid the ambiguity then you should do so. If this had been done with the expression above then much argument could have been avoided. Whoever wrote the expression in the form given should be shot. Imagine if you’d done it like that in a safety-critical engineering application. 

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  • 6 months later...
On 2/21/2019 at 5:35 PM, TVGuy said:

The (wrong) argument for the answer being 1:
Many people are getting the correct answer, 1, but arriving at it incorrectly.  They are also attempting to apply PEMDAS, but never learned, or forgot, that multiplication and division take equal priority and are performed in order of left to right.  Because multiplication comes first in this acronym, they believe it must be performed first.

= 6÷2(2+1)
= 6÷2(3)        Performing the operation in parentheses first, according to the order of operations.
= 6÷6             Incorrectly performing multiplication first, multiplying 2 by 3 to get 6.
= 1                 Six divided by six equals 1.

This first argument is actually not wrong.

In a way, when you write it as a fraction (which I agree it’s the most logical and natural way to interpret it), you are automatically putting a parenthesis like this around the denominator:

6÷[2(2+1)]

So, it doesn’t matter if you add first to get 6÷[2(3)] and then multiply to get 6/6, or if you distribute 6÷[(4+2)] and then add to get 6/6.

You don’t necessarily have to apply the distributive property of multiplication, but if you do, you get the same result because you consider the denominator as a whole, solve the denominator first, and then perform the division.

Edited by blitzomo (see edit history)
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In Dutch we have "Meneer Van Dalen Wacht Op Antwoord", and the first letters corrospond with the manner of calculating:

Meneer (mister) = Machtigen (to the power of...)

Van (of) = Vermeerderen (multiply)

Dalen (it's a surname) = Delen (devide)

Wacht ([is] waiting) = Wortel (root)

Op (on) = Optellen (addition)

Antwoord ([an] answer) = Aftrekken (subtraction)

So, going in this order it's first everything within the (   ), so 2 + 1 = 3.

Now that everything is on the same 'priority' we go down the line, starting with multiple, so 2 * 3 = 6

After that we're left with 6 / 6 = 1

E-Z

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