6÷2(1+2)
The question is whether the bracket is (1+2) ends with the addition and give 6÷2x3
or the the brackets remain and 2(3) is a separate term giving 6÷[2(3)].
Either way I am assuming either a multiplier or a bracket making this question redundant due to lack of information . :/
How would you write these 2 seperate problems in single line form without having the ability to make a fraction or similar character from a standard keyboard device ??
6
------
2(1+2)
6
--- (1+2)
2
I would say that the first one is: 6 / [2(1+2)]
And the second is: (6/2)(1+2)
In both instances, I had to include extra brackets to the original equation to remove any possible source of ambiguity that might come into play with differences of opinions. If you look at the 2 ways I wrote the equations, there is no chance of misinterpretation which therefore tells me the original question was written poorly and leaves room for error. Hence, this drawn out forum over such a simple issue.
My final stance, there is a possibility of 2 correct answers depending on clarification of the original equation for which expressions go where.
Somebody brought a dead thread back to life... lol
Anyway... I think it depends on what you were taught in school. I was taught that it was 1 simply because in terms of order, multiplication trumps division. But for some, multiplication and division are in the same order and thus you go from left to right, which gives you 9.
So, the fault(s) lies in the school education system and the damn bloody lousy equation that is ambiguous.
Either answer is RIGHT depending on what you are taught.
AND WRONG because what you want is a definitive answer since you can do plenty damage with equations like those. I am a structural engineer. Getting those wrong can cause a building to collapse. Put in the parentheses and remove the ambiguity. That's the CORRECT answer.
Either answer is RIGHT depending on what you are taught.
.
LOL this is the fact....while we are taught the answer is 9 (which it is) the actual number nine is a representation of a value created by man which is why the answer can also be 1 or 5 or 33 all depending on how you were taught...lol
Math is man made and while nature gives us examples of formulas and representations of (proper) math, math is in fact an idea....
Someone please feel free to correct me if I'm remembering this wrong--after all, it was back in 1982 that I took College Algebra.
There shouldn't be any ambiguity about this equation because you're supposed to solve the contents of parentheses first, hence we do the (1+2) to get 3. Then, since the remainder of the problem involves division and multiplication, we simply go from left to right to solve it.
Someone please feel free to correct me if I'm remembering this wrong--after all, it was back in 1982 that I took College Algebra.
There shouldn't be any ambiguity about this equation because you're supposed to solve the contents of parentheses first, hence we do the (1+2) to get 3. Then, since the remainder of the problem involves division and multiplication, we simply go from left to right to solve it.
We have:
6 / 2 * (1 + 2) or
(6 / 2) * (1+2) or
(6 / 2) * (3) or
(3) * (3) for an answer of 9.
Right...?
Yep, you are 100 percent correct. Under standard decimal arithmetic, that is the procedure to use. Many people learn the acronym PEMDAS, or the phrase "Please Excuse My Dear Aunt Sally", and incorrectly learn or assume that multiplication has higher precedence than division. Under standard rules of arithmetic, multiplication and division have the exact same precedence and must be computed from left to right. Addition and subtraction also have the same precedence and also should be completed from left to right.
This is why when I teach this topic, I prefer to use the phrase "Please Exclude Mom or Dad As Sane". It gets a chuckle out of people and also inserts the idea that it's M or D, not M then D.
Yep, you are 100 percent correct. Under standard decimal arithmetic, that is the procedure to use. Many people learn the acronym PEMDAS, or the phrase "Please Excuse My Dear Aunt Sally", and incorrectly learn or assume that multiplication has higher precedence than division. Under standard rules of arithmetic, multiplication and division have the exact same precedence and must be computed from left to right. Addition and subtraction also have the same precedence and also should be completed from left to right.
Yes, that's exactly how I do it.
Quote:
This is why when I teach this topic, I prefer to use the phrase "Please Exclude Mom or Dad As Sane". It gets a chuckle out of people and also inserts the idea that it's M or D, not M then D.
I like that.
I wish my dad (actually my father-in-law) was still around. He died two years ago. He was a college math professor and I'd LOVE to ask him about this.
At any rate, when I solved it, I just pictured myself back in class taking an exam, and it seemed clear as day how to solve it. But, as I said, it was actually 1982 when I was in that particular class, so maybe I was just remembering wrong.
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You know what got me? I was under the impression in PEMDAS "Multiplication or division, which ever comes first" meant what I had to use first to get rid of the parenthesis. Was never taught left to right. Maybe they meant that when they said which ever comes first. And the parenthesis as still there....so I multiplied to get rid of it.
6÷2(1+2) =
6÷2(3) =
3*3 = 9
6÷2(1+2) =
6÷2(3) =
6÷6 = 1
I was hung up on getting rid of the parenthesis. Its the first rule of PEMDAS!!!! lol.
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In order to get 9, you need to add the brackets {parentheses} in the latter equation which were not in the original question. If you start implying {} in different parts of the equations, you will get different answers.
FYI, up in Canada, we use the acronym BEDMAS (not that that makes any difference)
Last edited by funpig; January 6th, 2013 at 03:03 PM.
I used the {} to signify a fraction, not as an extra pair of parenthesis. By mathematic definition, an operation is a function that takes 2 numbers in a specific order and yields one result. An operation can be delayed by the of parenthesis BEFORE the second number. Since the 2 is not wrapped in parenthesis, it is part of the division and since division and multiplication have equal precedence according to the order of operations, it would be done before the multiplication. there's no denying that the first operation done is the addition because of the parenthesis. Since there is no exponents, that brings us to multiplication AND division. They have EQUAL precedence, therefore it is computed from left to right. 6/2 =3, then 3*3 is 9.
Another thing that might help people see that the answer is 9, is that the definition of division is the first number times the multiplicative inverse(aka reciprocal) of the second number. If using this definition, then after the parenthesis, the expression would change to 6*{1/2} * 3 or 6*.5*3.
Intuitively (and incorrectly), I interpreted the "÷" so the question appears as 6 over (2 times 3).
6
2(3)
I sourced an 1898 math reference over the internet which specifically says a ÷ b x c should be interpreted as (a ÷ b) x c. So the answer would be 9.
I am too old for this to make any difference other than looking dumb on the internet, but I am going to have to go over this with my kids. I could see this as some admissions test, and judging from poll results, almost half of us get it wrong.
It's a common thing that I see, I even saw this question pop up in a picture on facebook with the claim that 93% get it wrong. I don't know how true that is, but I know that order of operations questions are quite popular. I personally feel that mathematics is poorly taught to students because a large number of those that are really good with math can make bigger money elsewhere, leaving numerous secondary and primary math teachers that aren't on the same level. Here in NY, you need to have a masters degree to teach, and the starting pay is horrible. I don't know many people that would be thrilled with going to college for 6 years to start with a salary of 37,500
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seriously? I am by no means a math whiz.. in fact I consider myself somewhat mathematically challenged....In fact I almost doubted myself because it seemed too easy.....
the answer is 9
1+2=3
*always do the equation in parenthesis first
6 divided by 2 =3
*if there is nothing between the two numbers always multiply
3x3=9
so the answer is 9 pieces of bacon!!!LOL
Last edited by huh; January 7th, 2013 at 02:57 PM.
"There are three words in the english language".....
The English Language = three words
But you've left out the rest of the sentence, and the rest of the paragraph, therefore your solution doesn't make sense.
Here's what's said:
Quote:
There are three words in the English language that end in "gry." "Angry" and "hungry" are two. What's the third?
So the statement is that there are "three words in the English language that end in 'gry.'" Not "there are three words in 'the English language.'" And the question being asked refers to the three words that end in "gry"--"Angry" and "hungry" are two. What's the third?
Unless I'm having a major brain fart (and that's ENTIRELY possible!), it just doesn't make sense.
I believe it has something to do with the original math question allegedly having ambiguity, which is what I think the cartoon makes a reference to. I could be totally wrong as well.
Hey jhawkkw, is answer still same if question was written 6/2(3)?
Yes, the answer is still 9. The parenthesis part of PEMDAS means an operation within parenthesis. Since only the 3 is inside the () and no operation, that step is done. No exponents, so that is skipped. The multiplication is outside the (), and since multiplication and division have equal precedence, you do them in the order they occur from left to right.
Thanks. I just have to accept the mathematical convention, but it sure looks funny in my mind's eye, like an optical illusion. I can still see my self mixing up 1/2n as 1/(2n) as opposed to (1/2)n. Good thing I am not an engineer or bridges would be falling down all over the place.
What this section states (or implies) is that if you try to interpret 6÷2(1+2) strictly by the HUMAN-MADE order of operations, it is ambiguous. The system is not perfect. There are gaps where additional parens are needed to remove ambiguity. But as long as ambiguity exists, it can go multiple ways.
Of course the problem is, most people don't believe that it's ambiguous; they think there must be a correct answer and that their interpretation of PEDMAS is absolutely correct. And in both directions, it's a correct interpretation. That is the definition of ambiguous. That's the reason people are split 50/50 no matter where this question is posed.
It doesn't matter how many PhDs you have and in what field. I'm sure there are super-intellects on both camps. The right answer is: don't expect a consensus until you add parentheses to the original problem to make it unambiguous. The problem was designed to exploit the gap in the order of operations.
Quote:
Originally Posted by XplosiV
It seems to me that the answer is somewhat split between the simple folk and the math geeks. Ask anyone in the street and they will probably solve the question using left to right. where as ask anyone with some kind of higher education in math, and they will probably use the math parenthesis. Theres not a lot of point 'simplifying' the problem, the problem 'as is' done by a mathematician, using the common laws of math, will simply answer it as 1.
Who can tell me the highest number, that when written in the queens English, consists of only three words?
(IE 110 = one hundred and ten, 4 words)
I voted for 1.
Although I follow and appreciate all the 9 explanations, I think the above posts nicely explain why we are ALL RIGHT.
For me in particular I would be more inclined to accept 9 if the problem had been set out as 6÷2x(2+1) instead of 6÷2(2+1)
But because the 2 is adjacent to the parenthesis it looks like they are connected in a more intimate and therefore immediate way than are the 6 and 2.
As another post pointed out, 6(2+1)÷2 would be more obviously 9 than would 6÷2(2+1), because 2(2+1) appears to be (2*2) + (2*1)
In a democracy we might say that the answer with the most votes is the correct one, but we might be wrong... it is only the accepted one
Without the additional parenthesis to clear up the ambiguity it is as clear as saying "I helped my uncle jack off his horse" without adding the necessary capitals
And my guess at XplosiV's puzzle (after his clarification that he meant the smallest number) is... one half dozen
Although I follow and appreciate all the 9 explanations, I think the above posts nicely explain why we are ALL RIGHT.
Sorry, but math doesn't work that way.
Quote:
For me in particular I would be more inclined to accept 9 if the problem had been set out as 6÷2x(2+1) instead of 6÷2(2+1)
How it LOOKS has no bearing on how it's supposed to be solved according to mathematical rules. Those rules have been stated in previous posts, including mine, so I won't bother repeating them.
Quote:
But because the 2 is adjacent to the parenthesis it looks like they are connected in a more intimate and therefore immediate way than are the 6 and 2.
As another post pointed out, 6(2+1)÷2 would be more obviously 9 than would 6÷2(2+1), because 2(2+1) appears to be (2*2) + (2*1)
Again, none of that matters. How it looks doesn't determine how it's solved. That's based on rules of math.
Quote:
Without the additional parenthesis to clear up the ambiguity it is as clear as saying "I helped my uncle jack off his horse" without adding the necessary capitals
That's a funny analogy...but not an accurate one. Unlike written English, where things like capitalization matter in how a sentence is interpreted, math doesn't work that way.
Although I follow and appreciate all the 9 explanations, I think the above posts nicely explain why we are ALL RIGHT.
For me in particular I would be more inclined to accept 9 if the problem had been set out as 6÷2x(2+1) instead of 6÷2(2+1)
But because the 2 is adjacent to the parenthesis it looks like they are connected in a more intimate and therefore immediate way than are the 6 and 2.
As another post pointed out, 6(2+1)÷2 would be more obviously 9 than would 6÷2(2+1), because 2(2+1) appears to be (2*2) + (2*1)
In a democracy we might say that the answer with the most votes is the correct one, but we might be wrong... it is only the accepted one
Without the additional parenthesis to clear up the ambiguity it is as clear as saying "I helped my uncle jack off his horse" without adding the necessary capitals
And my guess at XplosiV's puzzle (after his clarification that he meant the smallest number) is... one half dozen
Well, I teach math at the college level so I guess that contradicts XplosiV's idea that mathematicians would vote 1. These math question threads that pop up continue to convey the idea that mathematics is something you can opinionate, and is not absolute fact. If I add 3 apples to 5 apples, you get 8. Not 2, not 10, but 8. The laws of mathematics are absolute and breaking them results in contradictions. An example is one of the many division by zero fallacies. I'll show this one as an example:
let a=b,
by multiplying both sides by a, you would get a^2 = ab.
subtract b^2 from both sides yields a^2-b^2 = ab - b^2.
factoring using the difference of squares on the left side, and the common factor of b on the right side yields (a+b)(a-b)=b(a-b).
dividing (a-b) on both sides yields a+b = b.
since a=b, making the substitution yields b+b or 2b = b.
finally, dividing both sides by b yields a final result of 2=1, which is clearly not correct. This is because of the division of (a-b) because that is equal to zero. Division by 0 isn't allowed under standard mathematical arithmetic, which causes the contradiction of 2=1.
6÷2x(2+1) is identical to 6÷2(2+1) under the standard rules of arithmetic. One of the topics one would encounter in the 4th year of a bachelor's in mathematics called modern/abstract alegrba, is a course where you investigate operations at it's absolute core. Addition and multiplication form an abelian ring, otherwise known as a field, on the set of real numbers. Subtraction and division is nothing more than addition and multiplication respectively by use of it's corresponding inverse. For example, 3-5 = 3 + -5; as well as 6 ÷2 = 6 * (1/2) (<--one half). Using this idea, you can change the original problem: 6÷2(1+2) to 6*(1/2)*(1+2). Using PEMDAS normally, you would get 6*(1/2)*3. Performing the multiplication, 6*(1/2) gives you 3, and then 3*3 gives you 9.
Well, I teach math at the college level so I guess that contradicts XplosiV's idea that mathematicians would vote 1. These math question threads that pop up continue to convey the idea that mathematics is something you can opinionate, and is not absolute fact. If I add 3 apples to 5 apples, you get 8. Not 2, not 10, but 8. The laws of mathematics are absolute and breaking them results in contradictions. An example is one of the many division by zero fallacies. I'll show this one as an example:
let a=b,
by multiplying both sides by a, you would get a^2 = ab.
subtract b^2 from both sides yields a^2-b^2 = ab - b^2.
factoring using the difference of squares on the left side, and the common factor of b on the right side yields (a+b)(a-b)=b(a-b).
dividing (a-b) on both sides yields a+b = b.
since a=b, making the substitution yields b+b or 2b = b.
finally, dividing both sides by b yields a final result of 2=1, which is clearly not correct. This is because of the division of (a-b) because that is equal to zero. Division by 0 isn't allowed under standard mathematical arithmetic, which causes the contradiction of 2=1.
6÷2x(2+1) is identical to 6÷2(2+1) under the standard rules of arithmetic. One of the topics one would encounter in the 4th year of a bachelor's in mathematics called modern/abstract alegrba, is a course where you investigate operations at it's absolute core. Addition and multiplication form an abelian ring, otherwise known as a field, on the set of real numbers. Subtraction and division is nothing more than addition and multiplication respectively by use of it's corresponding inverse. For example, 3-5 = 3 + -5; as well as 6 ÷2 = 6 * (1/2) (<--one half). Using this idea, you can change the original problem: 6÷2(1+2) to 6*(1/2)*(1+2). Using PEMDAS normally, you would get 6*(1/2)*3. Performing the multiplication, 6*(1/2) gives you 3, and then 3*3 gives you 9.
I do follow that but I guess I agree with XplosiV because I studied electronic engineering at uni (a long time ago) and my maths is ok but I did see the 2(2+1) as having operational priority, placing the product in the denominator. I guess I didn't read it as a computer algorithm would interpret it but tackled the parenthesis and its immediate product first. I'm not claiming I'm right, just that the presentation of the problem invites alternative interpretations due to ambiguity.
I do follow that but I guess I agree with XplosiV because I studied electronic engineering at uni (a long time ago) and my maths is ok but I did see the 2(2+1) as having operational priority, placing the product in the denominator. I guess I didn't read it as a computer algorithm would interpret it but tackled the parenthesis and its immediate product first. I'm not claiming I'm right, just that the presentation of the problem invites alternative interpretations due to ambiguity.
I like this thread.
Indeed. There is no doubt that because of the way one types(all on one line), versus hand writes, there is inherent ambiguity. Maybe the author of this thread intended it to be that way to cause the debate, maybe not. It certainly has spurred some.
Indeed. There is no doubt that because of the way one types(all on one line), versus hand writes, there is inherent ambiguity. Maybe the author of this thread intended it to be that way to cause the debate, maybe not. It certainly has spurred some.
It certainly has.
I read through the whole thread and I don't know how it got reprized but I'm glad it did and I can see it continuing for a good while.
It took me a long while to reconcile myself to the probability that I might have chosen the wrong answer. Lots of hedging there; I still can't accept it completely
Last edited by davoid; January 10th, 2013 at 07:25 PM.
I do follow that but I guess I agree with XplosiV because I studied electronic engineering at uni (a long time ago) and my maths is ok but I did see the 2(2+1) as having operational priority, placing the product in the denominator. I guess I didn't read it as a computer algorithm would interpret it but tackled the parenthesis and its immediate product first. I'm not claiming I'm right, just that the presentation of the problem invites alternative interpretations due to ambiguity.
But, as I see it, there is no ambiguity. There's only PERCEIVED ambiguity by people who don't understand how math problems are supposed to be solved!
But, as I see it, there is no ambiguity. There's only PERCEIVED ambiguity by people who don't understand how math problems are supposed to be solved!
Or there's only perceived ambiguity by people who do understand how maths problems are supposed to be written?
As jhawkkw, who teaches maths at college level, says in his post above:
Quote:
Originally Posted by jhawkkw
There is no doubt that because of the way one types(all on one line), versus hand writes, there is inherent ambiguity.
The ambiguity is caused because maths problems are not usually written in one typed line as the OP provided. That is more akin to a line of computer code, and the way a computer interprets a line of code is not the same as the way a hand written maths equation is usually solved. But to complicate matters, a line of code would not usually have the 2 immediately in front of the parenthesis without an asterisk * to indicate multiplication. The problem 6÷2(2+1) = ? is not a syntax you would come across either in maths or programming. In maths, the 6 would be drawn over the 2, and in code there would be an asterisk present.
Last edited by davoid; January 10th, 2013 at 09:09 PM.
there is no relativity in math. it is constant and predictable..the answer is 9
there is no other correct answer or interpretation
it is exactly the same as 1+1=2
2 is ..and always will be... the only outcome...
the original equation is not written ambiguously it is the standard in mathematics
I think the point of the whole thing is to draw attention to the fact that a simple middle school math problem is difficult and confusing for many and it shouldn't be..so it's a fail on the state of math and education.... ...those dern kids...get off my lawn !
.
Last edited by huh; January 10th, 2013 at 09:10 PM.
the original equation is not written ambiguously it is the standard in mathematics.
I don't believe it's the standard in mathematics to use a "÷" symbol
6
--- x (2+1)
2
That would be a standard equation in mathematics, not a linear formula using a ÷ symbol. And that would not have been ambiguous.
And if the linear formula is to be calculated following a linear order similar to the interpretation of computer code, it shouldn't have the form 2(2+1) from mathematics, but the form 2*(2+1) from programming.
I believe the original problem was designed to be ambiguous
Last edited by davoid; January 10th, 2013 at 09:35 PM.
yikes!! I was just coming back on to say I think thatI may have sounded all stiff and snooty and maybe even a little b I didnt mean it that way.. in fact I don't even like math!!
I don't believe it's the standard in mathematics to use a "÷" symbol
Then why does it exist?
Quote:
6
--- x (2+1)
2
That would be a standard equation in mathematics, not a linear formula using a ÷ symbol.
The 'order of operations' handles linear formulas just fine.
Quote:
And that would not have been ambiguous.
Neither is the way it was written in this thread.
Quote:
And if the linear formula is to be calculated following a linear order similar to the interpretation of computer code, it shouldn't have the form 2(2+1) from mathematics, but the form 2*(2+1) from programming.
But then it wouldn't be a MATH problem. MATH problems are solved according to MATHEMATIC rules. I've been both a math whiz and a programming whiz, and I'm FINE with this!
Quote:
I believe the original problem was designed to be ambiguous
Only to people who don't understand math's rules.
Quote:
The ambiguity is caused because maths problems are not usually written in one typed line as the OP provided.
Again, it doesn't matter HOW it's written, solving it is based purely on the rules of math.
Quote:
The problem 6÷2(2+1) = ? is not a syntax you would come across either in maths
Sure it is. Even I--some 30 years out from College Algebra--knew exactly how to solve it. It's not that unusual to see something written like that.
Quote:
Or there's only perceived ambiguity by people who do understand how maths problems are supposed to be written?
27Likes
. 
to show it.
He was a college math professor and I'd LOVE to ask him about this.
