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
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 = 
You can look to solve it as 6/2(1+2)=x
Do the (1+2) to get 6/2(3)=x
You may then transfer the multiplied 3 under x. 6/2=x/3
6/2=3
3= x/3. Bring the divided 3 back over, 3*3=x
Therefore resulting in 9=x
My issue with this is the distributive factor that nobody addresses. 6/2(1+2) is six divided by 2(1+2). from everything I’ve been taught you have to distribute it to breakdown the parenthesis and that takes precedence. After you distribute the 2 then you can add it up and divide it. There also is no need for 2 to have a parenthesis around it as well. So then (2*1+2*2) is (2+4)=6 then you divide 6/6=1 PEMDAS is still in play but you HAVE to distribute first to properly figure the problem. I am frustrated because this is such a simple concept and I just can’t understand why smart people don’t understand this. I know smart people who do but when you try to show others this they try to use every reason why it can’t be 1. This is a legitimate equation. I can see how 9 is figured but the biggest reason why it can’t be that is the division symbol separating 2(1+3) making the 2 a part of the parenthesis and the direct number for multiplying. It’s so simple had it been 6/2*(1+3) or (6/2)(1+3) otherwise everything directly to the right of the division symbol is it’s own part. You can even plug in an x if you need to in order to understand the distribution part. x=2 so x(1+2) is (x+2x) is (3x)=6 or to write it as a number you would do 2(1+2) is (2+4)=6. There are no rules it is breaking it’s how you distribute numbers and like I originally stated because of the division symbol and nothing indicating that as a whole problem to multiply to (1+3) everything then to the right is one problem.
When I see a number immediately adjacent to parentheses as is the case with the “2″ in this case, everyone knows it is implicit multiplication, but I consider that to be a tighter bond, if you will, than an explicit multiplication symbol. That’s why I’d put the parentheses in different places than the normal order of operations rule would have them. But ultimately, the problem is the ambiguity and there shouldn’t be ambiguity.
Hey Cassie,
The problem you are having with the distributive property is that you are not doing it right. You are confusing this problem with 6/[2(1+2)] which would be done how you described. If you want to distribute 6/2(1+2) you would do it like this (6/2)(1)+ (6/2)(2) this gives you (3)(1)+(3)(2) which gives you 3+6 which gives you the answer 9. If you use the distributive property correctly you still get the same correct answer. Parenthesis next to each other are the same as multiplication 6/2*(1+2) is the same problem as 6/2(2+1). Maybe it would help you if you could see it written out with symbols that are hard to type. Don’t picture it as a fraction with 6 on top of all of the expression 2(1+2) (that would be incorrect) picture it as multiplying two fractions together 6/2 multiplied by the fraction (1+2)/1
Hope this helps you understand where you are going wrong with distribution.
Happy New Year everyone!
I believe this problem started with the division symbol, not / .
@ Oliver, I agree with you, there should be no ambiguity whatsoever in math, and that’s why this annoys me to no end!! I know why it’s 9 to them. This is a terrible mathematical situation and I don’t know why or how it started, but it appears to be a weeding tactic of the worst kind. And it starts with the parentheses…
When students first begin to do multiplication and division, many are told to assume parentheses represent multiplication. If you accept that, without knowing the distributive rule, you will probably get 9 because of that PEMDAS rule that says go from left to right when you have just multiplication and division (don’t get me started on that creepy rule!).
Later, students are introduced to the distributive rule and this problem is then 1 because now parentheses are parentheses and they must be ‘resolved’ within PEMDAS. That would make any student say ‘huh’? How can you possibly know when it’s just multiplication or when it should be resolved? Not right, I call foul!!
Fact of the matter is, because of PEMDAS, this is 1 in my book. Because of the parentheses! When you get to this point
6 / 2(3) You still have parentheses. It is not 6 / 2*3, it’s parentheses. They must go away, and the only way to make them go away is to multiply them by the 2.
For more, check out one of the best rated math resources online – go to the last problem on the page. They are well aware of this argument as you will see…
It also explains why certain software delivers different answers.
http://www.purplemath.com/modules/orderops2.htm
@PSU thats not the same
with dealing with the () you still follow PEDMAS and E comes before the M so the () is still there you do the E then the M and the () goes away then
when you rewrite it the way you say it you are making an error
@PSUWayne
Not so fast PSUWayne…
Attempting to prove your point using exponents is comparing apples to oranges. Of course you have to do the exponents second because the whole point is to get rid of the parentheses. You can’t multiply before applying exponents.
I don’t need to apply the distributive rule to show that is 1, simply realizing the parentheses are still there after doing the addition inside and applying PEMDAS is enough
6 / 2(3) still parentheses. this is not 6 / 2*3 that would be 9.
the only way to lose the parentheses is to multiply it by the 2
6 / 6 = 1
And read my message to Oliver. I really don’t like the fact this is an argument, I’m calling foul on the mathematicians!!
@Edward – Thanks! I actually don’t fault anyone thinking it’s either answer, I’m blaming the mathematicians. Math is supposed to be an exact science. Then why are we having this argument? Because of how we are being taught to deal with parentheses at different stages in our math education – it’s inconsistent!
Step One:
Using algebra rules regarding parenthesis: 2(a+b) = 2a +2b
This equation must be true at ALL times.
(If, at any time, you’re getting a different number on the left hand side than on the right hand side, you are saying that this equation is not true.)
Using a=1, b=2 there are several ways to keep this equation true:
2(1+2) = 2(1) + 2(2) = 2 + 4 = 6 = 2(3)
But, in order for this basic algebra equation to be true in ALL cases, 2(1+2) must equal 6.
Note for those getting a final answer of 9:
Yes, 2(3) is a possible answer, but if you separate the 2 and 3, the equation 2(a+b) will not equal 2a +2b in ALL cases.
Step Two: Do multiplication/division
6 6 6 6 6 6 6
——– = ——– where ——– = ———— = —— = ——– = —— = 1
2(a+b) 2a+2b 2(1+2) 2(1)+2(2) 2+4 2(3) 6
Sue’s last comment of January 2 is the best place to end this disagreement. I still believe the lack of additional context (i.e., parentheses!) does not allow this expression to be evaluated. My view is the most old school, probably; but unfortunately so am I.
Hi Sue, Happy New Year to you, too. After giving this a bit more thought, I’m going to shoot myself down using the following argument. Multiplication is really just repeated addition, so:
6 / 2(1 + 2) =
6 / (1+2)+(1+2) =
6 / (3) + (3) =
2 + 3 =
5
FIVE! JUST KIDDING. I intentionally left out the extra grouping.
6 / [(1 + 2) + (1 + 2)] =
6 / [(3) + (3)] =
6 / 6 =
1
At the end of the day, it’s a syntactically-weak problem that should be more carefully written to avoid such ambiguity. That, I’m sure we can all agree upon. But it was a fun exercise, nonetheless.
An explanation of 6÷2(2+1)
PEMDAS is a mnemonic for children. What you want to do is study Order Of Operations. Also look into the “Distributive Property”, as well as Implicit Multiplication, and Juxtaposition. Research what the obelus is and means (÷). Define the “Terms” of the equation. Read up on Algebra and what Parentheses are used for. Research “Factorising”. Now that I have stated all the actual Mathematical principles, let’s prove what the answer is.
====================
For starters: What does 1/2x mean? It clearly means 1/(2x), but we just write 1/2x. It does NOT mean (1/2)x or “one half x” since the notation would clearly be x/2 for that.
So 1/2x = 1/(2x), just as 6/2x is 6/(2x). When x = 2+1, the answer is 1.
====================
Method 2: Let’s take the number 6. Do you agree that 6 = 4+2?
Now what is a common factor of 4 & 2? 2 works. So factorise the 2 from 4+2 and you get 2(2+1).
6 = 4+2 = 2(2+1). When we do math, we all know you have to use the WHOLE term, and not a portion of it, with an operator. Therefore, 2(2+1) is an entire term with a value of 6, and you cannot use a portion of it with another operator, such as doing the 6/2 =3, and them multiplying by (2+1). That is incorrect.
Therefore, 6÷2(2+1) = 6÷6 = 1.
====================
Method 3: Distribution. Similar to the above but a different principle;
a(b+c) = (ab+bc). 2(2+1) = (4+2). Notice how all these math principles are proving the SAME thing?? 6 ÷ (4+2) = 1.
====================
Method 4: Using parentheses as grouping. Think about this equation as a real world problem. I have 6 apples. i want to divide among 2 groups of children, each group with 2 boys and 1 girl. How many apples does each child get? Now look at the question again, while thinking of that situation: 6÷2(2+1). They each get 1 apple.
=====================
Method 5: Juxtapostion. The 2 that is “in front” of the brackets is there by this principle and some say this holds a higher precedence than explicit multiplication (and division). Answer = 1.
=====================
Method 6: The “obelus” ÷
While researching the exact meaning of this, I found these results. The ‘dot’ on the top of the symbol represents the dividend and the bottom dot represents what comes after the ÷. The 6 precedes it, and the 2(2+1) comes after it. Again ,yielding 1.
=======================
Notice how I didn’t once use ‘pedmas’ or ‘my calculator said z was the answer’ ??
Now I will show you why those people who get 9 as an answer, will get different answers for the same values. Remember factorising?
6 = 4+2 = 2(2+1)
also:
6 = 3+3 = 3(1+1).
Therefore,
3(1+1) = 2(2+1) = 6. All the same value: 6.
So, if 6 ÷ 6 = 1, then 6 ÷ 2(2+1) = 1.
however, the “9” people, will strictly use “left to right” like this:
6 ÷ 2 = 3
3*(2+1) = 3*(3) = 9.
Now if we replace 2(2+1) in the equation with 3(1+1), the EXACT same value, we get:
6 ÷ 3(1+1), and making the mistake of “left to right” we INCORRECTLY get
6÷3 = 2
2*(1+1) = 4.
How is it possible to take
6÷2(2+1) = 9
6÷3(1+1) = 4
where 2(2+1) and 3(1+1) both = 6, and get different answers? Because the laws of mathematics were not adhered to and both answers of 9 & 4 are wrong.
The answer to the original question is 1.
I kinda don’t understand this cos im just a 9 year old kid, but as i have learnt BODMAS (Brackets Order Division Multiplication Addition Subtraction) which is the same a PEMDAS ( Parenthesis Exponents Multiplication Division Addition Subtraction) I can understand quite a bit, but I haven’t learnt other maths terms yet obviously. but am I correct to say the answer is 9? that’s what I got when I tried to work it out. if im right then I have got to say that’s pretty good! really i did it quite simply.
(the / is divided by)
6/2 (1+2) =
well according to BODMAS i am supposed to do Brackets first so i did (1+2) first and i got 3. then also according to BODMAS im supposed to do division next, and so i did the 6/2 and i got 3 and then as i had learnt from my mum, if there is nothing in between the 2 equations, in this case, 6/2 and (1+2) as in no times sign or divide sign and so on, then that means i multiply it.
so that was 3 times 3, which equals 9!
short and simple!
Jayjay,
Unfortunately, you bump into an issue then… Because bodmas says do division first.. But pemdas says do multiplications first…
As explained earlier, the answer has to be 1, because by solving out the parens, you solve the second half to 6 and then perform the division of 6/6
Will Wagner,
I think you are missing something here it does not matter if it is BEDMAS or PEMDAS or PEDMAS or PODMAS or whatever the order is ALWAYS Parenthesis/Brackets THEN Exponents/Order THEN Division OR Multiplication THEN Addition OR Subtraction. It is this way because Multiplication and Division are essentially the same operation and the same goes for adding and subtracting.
I have always come up with the answer of 9 for this problem and that is because of the way I was taught to do simple math problems and this IS a simple math problem. I feel like people are just trying to use higher math on lower math problems.
Many people are translating the popular Order of Operations problem going around on the internet, which is actually stated as 6 ÷ 2(1+2). You cannot simply rewrite this problem as 6/2(1+2) as these are two completely different problems. Anytime you place an expression above or below a fraction bar, this is the same as enclosing that portion of the expression with parentheses. Consider that this slight difference, 6÷(2(1+2)) would correctly be translated to 6/2(1+2) which is what the original problem was translated to. In the original problem, leaving the fraction bar out of the expression and following established mathematical rules for Orders of Operation, the problem would be correctly solved as such:
the (1+2) is completed first as parentheses take precedence over all else. This gives a new expression, 6÷2(3) or 6÷2×3. Both multiplication and division have identical precedence within the rules of Orders of Operations, the operations are performed left to right, therefore the division is performed first and the expression is now 3×3. Last step is completed to give us 9. It is true that 6/2(1+2)=1, but you cannot simply exchange a division with a fraction bar when we are dealing with expressions and operations within the expression. So, for all of those people arguing this questions, your differences are because of the neglect to the rules of expressions translated from horizontal to fraction form.
@Ray: Did you read what I said about the obelus and how it was meant to be used? And yes you can replace the obelus with a ‘slash’ as long as it not used as a solidus, which as you said, means a fractional line. Therefore 6÷(2(1+2) equates exactly to 6 / 2(2+1) as long as we know we are using the ‘/’ for divisional purposes only. With this in mind, 1÷2n = 1/2n = 1/(2n).
The references for the obelus and slash equality is: http://en.wikipedia.org/wiki/Slash_(punctuation)
And the reference for the 1/2n = 1 “all over” 2n is:
http://faculty.ksu.edu.sa/fawaz/481files/Bartle-Introduction-to-Real-Analysis.pdf on page 53.
Thereby, 6/2n = 6/(2n). Let n = 2+1 and the answer is 1.
@ Melissa – thanks for the vote of support, I appreciate it! It is time to mosey on to another post isn’t it? I think the powers that be need to deal with this issue…clearly, it’s an issue!
@PSU Wayne – LOL Don’t do that to yourself!! But this was an interesting diversion from the craziness of life!
@caper26 – children schmildren!! I’ll use PEMDAS forever! haha…But if I had a hat on, I would tip it for you because that was a very detailed explanation you posted – excellent!
Absolutely, I read what you said and I agree. But you failed to read my initial statement. If the original question that spawned the first post was 6÷(2(1+2)) then the answer, would indeed, be 1. This would translate to 6÷(2×3) or 6÷6 or 6/6 = 1. The question that was actually posted that spawned all of this was not as stated here, but rather the following: 6÷2(1+2). The lack here of the second set of parentheses changes the translation. In the first case above, the inner parentheses is computed first and then the second set second and then the division is executed. In the second case, 6÷2(1+2), the parentheses are computed first. We are all in agreement to this. Let us all call the resultant n. It is understood that n=3, but for purposes of this explanation, we will hold off evaluating for n once we reach the end. The problem now looks like this: 6÷2xn. We are now at the point where we have only a quotient and a factor present, or, if you will, division and multiplication. In standard Order of Operation, or PEMDAS, or whatever you wish to use, multiplication and division has identical precedence and, in the case of identical precedence, the two operands will be executed left to right. In this case, 6÷2 is first and results in 3. The multiplication is next and we have 3n. Evaluating for n=3, as we agreed upon in the beginning, we have 9 as the answer. Beware of the parentheses or lack thereof when translating from obelus to slash. In summary, I agree with your explanation that 6÷(2(1+2)) translates to 6/(2(1+2)), but 6÷2(1+2) does not.
Ahhhhhhhhhhhhhhh!!!!!!
Those who don’t get 9 fail to understand that there is NO DISTRIBUTION HERE! Someone already said, if you want to distribute into the parentheses you have t take the 6/2 and not just the 2. If it was addition instead of division you could distribute, although it would be more work than necessary, since THERE ARE NO VARIABLES! Parenthese first means do wat is insite of the parentheses first. It does not mean that multplication when implicated with parentheses takes precedent over division. They are always done left to right. 2(2+1) is the exact exact exact same thing as 2*(2+1). If you distribute you are doing it wrong and failing to properly utilize the preceding division sign.
Next week I’ll be 60 so it has been quite a few years since school. But after reading some of these post I’m reminded of a thought I had over 45 years ago, that is due to the fact that every math instructor teaches different rules — there is simply no way that math any higher than MDAS can be an exact science — so in this case I will go with my granddaughters answer of ” 1″ it makes as much sense as any of the above. One can always know that 2+2=4 it never changes even though — every day you will see people that will try to make it “5” it was rather fun to see the passion that some of these entries contained.
If 6÷2(2+1)=9, then X÷2(2+1)=9 would have to be X =6. But that does not work.
@caper:
Your first post seems to be circular. You have shown that the answer is undoubtedly 1 if the problem is read as 6/(2(1+2)), but whether or not the problem is read that way is the debate in the first place. This isn’t something that can be shown to be 1 or 9 through manipulating equations because you have to choose a convention to use when doing so.
You are right that the obelus and slash (inline slash- not fraction bar) are1equivalent. However if they both implied some kind of grouping (as a fraction bar) that would leave us without any symbols for standard, nongrouped division. An obelus and inline slash infer no more grouping than does a standard multiplication sign.
@Ray: Sorry it has been a while, but my posts were not getting through, from what I believe to a URL pasted into the “Website” area of the form. That aside, yes, I did read your post. I followed you right up until you posted: “The problem now looks like this: 6÷2xn” There should be no ‘x’ there. If I asked you the question over coffee: “What is 18 divided by 3a, and a = 2”, how would you answer? Regardless of that here is my next point, which was already pointed out in my first post, but since then, I found a lot of references and examples and wanted to further it:
Simplifying Equations. I hope everyone is reading this. We are allowed to simplify an equation before we even begin to solve it. We use the order of operations to solve; and we use combining like terms, and the distributive property, to simplify. Remember the difference? It is so obvious, that everyone is overlooking that simple aspect. Keep in mind that the distributive property is an “equality” statement, and therefore, can be used to make substitutions without changing the value of the original notation:
a(b+c) = ab + ac
Once we have an equation simplified, we can then begin calculating/solving/combining constants using the order of operations.
This is why I rebuild the original equation from scratch, without introducing any operators outside of the numerical value itself, as a proof. Factoring is the opposite of the distributive property:
ab + ac = a(b+c), where a is a common factor
Therefore, I can start with 6 ÷ 6 = 1
Now, I can expand 6 to 4+2,
6 ÷ (4+2) = 1
and factor out a 2
6 ÷ 2(2+1) = 1
I didn’t do any operations in this little factoring exercise.
Now we can show that there are implied parentheses as well:
6 = (6)
4+2 = (4+2)
2(2+1) = (2(2+1)).
In summary, the distributive property, by definition, is a mathematical equality, and is never limited to any part of the Order of Operations. The distributive property is also the reverse of factoring.
I believe the author of this equation was a smart fellow and did this unnecessary factoring to throw everyone off. I saw another equation similar to this on a Chinese youTube news report.
I won’t tell you what the original question is, just for fun, but I can recreate a similar expression for you, and show you how to create this mess 🙂
3/5
= 3 ÷ 5
= 30 ÷ 10 ÷ 5
= 3(5+5) ÷ 2(2+3) ÷ 2(1 + 3/2) <— simple factoring
Notice that I didn't do any computations, only manipulate the original values with simple arithmetic and factoring.
Now if a person were to strictly follow 'pedmas' or "left to right", and attempt to solve this, they would get:
3(10)/2 * 5/2 * 5/2 = 750/8 ?? and not 3/5.
The more you use factoring, the larger the wrong answer becomes…
The reason people "fall" for this, in my opinion, is they follow the 'pedmas' and strictly work "left to right" without being familiar with any other math principles and/or properties.
Best Regards,
~K.M.
K.M.
I have to agree with everything you have posted, but wish to just interject with this: This is a very ambiguous expression and no mathematician would ever write the expression in this manner unless he/she wished to spawn controversy. That being said, let me throw this example out to you just to show you that an ambiguous problem can be argued to several different points. Order of Operations, like most rules in math, are arbitrary determinations that were decided and agreed to by mathematicians thousands of years ago to prevent what we have seen in these posts above from occurring. Now, one of the established Orders of Operation Rules is that multiplication and division have equal precendence. The rule also states that, in the event that two operations within an expression have equal precedence, then precedence will occur left to right. I did not write this rule, didn’t make it up, but rather it was established thousands of years ago. Now, I agree with your Distributive Property arguement, but you need to incorporate this rule. The correct translation or distribution should look like this: 6÷2(2+1)=6/2(2+1) and then distributed, it would appear like this, and still adhering to the established rule: (6/2)(2)+(6/2)(1) as, because of precedence, it is six halves that needs to be distributed and not two. Completing this proof, (6/2)(2) simplifies to 6, (6/2)(1) simplifies to 3, so we can now write 6+3=9. As further proof, the scientific calculater must be, and is, built with the rules of Orders of Operation as one of many established mathematical rules. Enter the original expression of 6÷2(1+2) into the scientific calculator just as written here and you should now understand the reason the calculator correctly evaluates the expression as 9.
K.M.
Just a quick point for clarification, had you expanded your Distributive Property argument and the expression would have been something like 6+2(2+1), then you would have been absolutely correct in the 2(2+1) grouping as you did, but you actually disregarded the “6÷” portion of the expression in applying the Distributive Rule. Even though we see division, division is still a form of multiplication and is still a valid factor.
And finally, using all previous arguments, the problem broken down using the Distributive Property:
9=6+3
=(6/2)(2) + (6/2)
=(6/2)(2+1)
=6(1/2)(2+1)
=6÷2(2+1)
Otherwise, the only agreement we will come to is disagreement.
Ray: I totally agree that this question was definitely made to “troll”, ie, cause a bunch of online emotional responses. You bring a great point to the table and this has been a common rebuttal in other forums that I have read. However, that is the “only” rebuttal that holds any water. The reason why I firmly believe it not to be the case though, is that by performing the operator “÷” first, you thus begin the order of operations and the combining of constants, and the value of “6” is no more.
Consider 6 = 4+2 = 2(2+1). I do not have to put the 2 in front. I can write (2+1)2 just the same. This holds true to all my previous statements, and will still solve to 1, even if I recreate the equation from scratch, however, your way will yield 4. What I am saying is that by doing distribution first, you maintain the integrity of numbers themselves. That Chinese example, for instance, I can recreate numerous “equations” from 3/5, which will maintain that value if you simplify first, but inserting a multiplication sign will give a different answer for each “equation”.
I actually found some references today that said “distribute before pedmas”. I am not just siding with them because they said that, since I had other evidence prior to reading it, but it was nice to actually see it written there.
Also consider 6 = 3+3 = 3(1+1) or (1+1)3
this means 3(1+1) = 2(2+1) and can be interchanged, and MUST give the same result because of the equality; as does (1+1)3 and (2+1)2.
I would like to reiterate the implied parentheses ‘proof’ in my previous post.
So again, distributing is the reverse of factoring, and, as in 100% of the examples I have read over the last few days, I have never seen (a÷b) ever factored from a set of terms. When you start combining constants by operator (division) you are no longer simplifying.
Maybe some more persuasion:
2(2+1) = [(2+1) + (2+1)] I swear I remember that from some algebra class in university but I can’t find a property or axiom for it.
I found a definition of grouping/distributing: “2(3 + 5) ; we mean “add 3 to 5 first, then multiply the result by 2.”
Last thing, calculators are split 1 or 9 depending on which one you use, and they all come with a disclaimer saying that “you must verify the results before relying on them” 🙂 Cheers,
K.M.
In your last post, you eliminate necessary parentheses around the (6/2), in my opinion.
(6÷2)(2+1) to 6÷2(2+1)
@Jayjay – You should always, ALWAYS go to your teacher first! Ask what this problem equals from your teacher. We are basically discussing whether or not the answer should be 1 or 9 based on what we know about math – we are questioning the mathematicians that make up the rules!! But you go by whatever your teacher shows you because that’s what is accepted at the moment. You have to pass those exams, and we will probably just confuse you!
@Ray Bilkie – you broke a few math rules to come to that conclusion. When you apply the distributive rule ie, a(b + c) = ab + ac, you only multiply the item attached to the parentheses. So with the division sign, or even with the slash, it will only be multiplied by the 2. A division sign clearly defines the problem – that is NOT a fraction 6/2. They really confused the problem by using the slash when it was originally with the division symbol.
This is so fun…. Is everyone enjoying this? I was a believer of the solution equal to 1. But after doing some research, I have changed my opinion. This can be a fractional expression and as I will demonstrate, there is no real difference between the slash or obelus operator. (6/2)*((2+1)/1) = 6*(2+1)/2*1 = 18/2 = 9/1 = 9. Why did I add all of the parenthesis? Well, for the same reason we wanted to expand the brackets with the distributive property rule. To better define the order of operations.
A term is defined as a number, a variable, or a product / quotient of numbers. Examples in order, 5, m, or 2x^2. When we learned distributive property rules it was to help simplify expressions with variables and like terms. Parenthesis have nothing to do with terms. Example, we cannot distribute 6/(2+1). We are not taught to handle division with DP, because it does not work the same way. 6/3 = 2 or is it (6/2+6/1) = 9. Case end point! So with that in mind, and while multiplication and division retain equal priority from left to right, we must distribute 2nd, not 1st. 6/2 = 3. 3(2+1) = 6+3 = 9.
I have been back and forth on this solution since the ball dropped on New Years. The one overwhelming lesson that helped me prove the solution is 9 was the difference between Associative, Commutitive, and Distributive Properties. They just don’t work with division at all. Which is why we must eliminate it first in order to properly distribute, or rather simply follow the order of operations which intentionally left DP out of the process for a well known reason to me now.
This exercise has given me a better appreciation for all of the rules and not just the ones we assume are implied. I set out to prove these experts wrong, but in the end, we must conclude; this is just simple arithmetic. 6/2*3=9. KISS. Yeah, I called myself stupid, but not anymore. I am a born again mathematician.
And one more thing guys: If you read some online textbooks, you will see fractions in full fraction form (ie, using a horizontal line instead of a slash) OR the fraction has parentheses around it, ie (6/2)x if it meant to be used as a fraction. Always.
@Intrigued: That is also incorrect:
Obelus: The obelus is primarily used as a symbol for division
Slash: Used between numbers slash means division, and in this sense the symbol may be read aloud as “over”.
Solidus: The solidus /ˈsɒlɪdəs/ or a shilling mark is a punctuation mark used to indicate fractions.
Now, the obelus and slash can be used interchangeably as long as the slash is interpreted as division NOT mistaken for a solidus.
In that regard 6÷2n = 6/2n, which MEANS 6/(2n). Let n = 2+1.
I only opened 2 books, and both calculus books use the notation 1/2n = 1/(2n). I have used this notation throughout my engineering studies. It is not something you can ignore.
There is also other explanations which break down what each part of the obelus means, and there is a text where the first obelus was used, and in fact 6÷2+1 implied brackets around the 2+1 but were not written. Here is what the obelus means:
http://www.freeimagehosting.net/rqr8m
Open any math text, and a faction will either use a horizontal line so there is no ambiguity, OR they use parentheses around the fractional coefficient like (6/2)x. They are not implied where you can just leave them out.
6/2x is not (6/2)x.
(6/2)x without parentheses is 6x/2
check some books 🙂
2=1+1, right?
so….
6/(1+1)(2+1)
applying FOIL to distribute the parenthesis:
6/(2+1+2+1)
Solve in parenthesis:
6/(6) =1
?
Okay, this is my last response. There have been several posts where the form of the expression had been changed to make the expression fit within the constraints of certain properties, i.e. Distributive Property. First, there is no math rule, principle, or anything else that states that only the numeral immediately ahead of the parentheses is involved in the Distribution. The form, a(b+c) is meant to show how the distribution occurs. There is no mention of how this is handled if you have an additional factor preceding the numeral immediately preceding the parentheses. Since others have decided to sway away from the original expression and use variables to prove their cases, allow me to do the same. First, we all in agreement that the numerals within the parentheses is not at issue, so we can simplify them to 3.
Now, using variables and established equalities:
Our original expression is 6÷2(1+2). Agreeing that 1+2 is, in fact, 3, let us instead write this as 6÷2(3). Rules involving parentheses as implied multiplication allows us to rewrite the 2(3) as 2*3. We can also use the same equality to show that 2*3=2(3). So now, with established mathematical equalities, here we go, variables and all:
From a college pre-algebra book: a÷b=a(b^-1), defining division as a form of multiplication. Now, we can use variables and established definitions to rewrite the entire expression as factors.
Let a=6, b=2, and c=3, then.
a÷b(c) =
a(b^-1)(c)
This correlates three factors in the original to the three factors in this representative expression.
This can now be simplified to ac/b.
We substitute the integers back in for the variables and we get (6)(3)/2=9.
Stop trying to make a(b+c) fit this problem. If you want it to fit, find a property that gives a rule for a÷b(c+d). In a(b+c), the “a” represents ALL factors immediately preceding the parentheses. If you are going to use the property, use it correctly.
There is rule of Mathematics that the complicated items get solved first …
In 6/2(1+2) obviously the most complicated item is the brackets … everyone knows that 1+2 is 3. You are left with 6/2*3. Note: here is where you need to be careful.
you cannot assume the sum 2*3=6 and hence 6/6=1. You need to follow the logical sequence of numbers 6/2=3 3*3=9
48/2(9+3) is similarly 48/2*12 which is not 2 but 288.
Remember LOGIC says left to right not solve the term next to the brackets with the brackets first.
Well, if I were to use variables, I would put them everywhere:
z ÷ a(a+b) = z ÷ (a^2 + ab)
Also I have several references now, that say ‘eliminating’ parentheses is done by using the distributive law. Your parentheses are not eliminated in your problem, they are still there, thus you break the order of operations by doing the division first? Parentheses must be eliminated first
6÷2(3) = 6÷2(3+0) = 6÷(6+0) = 1
Here are some references for some interesting points for removing parentheses. I have full links if you so desire:
“We use the distributive property to help us find a way around the order of operations while still being sure that we keep the value of the expression.” -wikibooks
“(the first step to) follow to simplify an algebraic expression: remove parentheses by multiplying factors” -math.com
“Get Rid of parentheses with Distribution” -helpalgebra.com
“When simplifying expressions with parentheses, you will be applying the Distributive Property.” -Purplemath
” If there is some factor multiplying the parentheses, then the only way to get rid of the parentheses is to multiply using the distributive law.” -jamesbrennon.org
“The PEMDAS rule says we have to do all operations inside parentheses before we can multiply. However, the DISTRIBUTIVE PROPERTY allows us to VIOLATE the PEMDAS rule to a certain extent!!! It allows us to multiply first WITHOUT doing the operations of addition or subtraction inside the parentheses first.” -algebra.com
Just to re-iterate, ever book I look in, if they use a slash for a fractional coefficient, they use parentheses 100% of the time, so if you wanted 6÷2 to mean (6/2), then parentheses must be used, or a horizontal fraction line. Otherwise 6/2n is 6 “all over’ 2n. I listed the ref already for that.
Here is another example I showed to people:
Identity Law: a = 1(a) = 1a
a÷a = 1, so therefore a÷1a = 1 and a÷1(a) = 1, but I see people argue tooth & nail that a÷1a = a^2, because a÷1*a = a^2. what?!?!?! 🙂
One more time, solve the following 🙂 :
1) 30÷10÷5 = ?
2) 30÷2(2+3)÷5= ?
3) 30÷2(2+3)÷2(1 + 3/2) = ?
Best Regards,
K.M.
This will probably be my last post on this issue because at this point, I’m just repeating myself. So is everyone else actually!
I just assumed everyone knew what PEMDAS was all about. Order of operations, I learned it as PEMDAS tells you the order in which things are
PEMDAS is the acronym that tells you the order in which things are done. If you need a quick recap:
P = Parentheses
E= Exponents
M=Multiplication
D=Division MD are on the same level of importance
A=Addition
S=Subtraction AS are on the same level of importance
So with that said, your first goal is to remove the parentheses
6 / 2(2 + 1) = 6 / 2(3) notice, there are still parentheses here. At this point people ASSUME parentheses mean multiplication, so they remove them and read the problem like this…
6 / 2*3 and since MD are on the same level, go left to right. But this violates the rule of parentheses, not to mention, the distribution rule. The parentheses are still there. Parentheses TRUMP division SO…
6 / 2(3) must lose parentheses . only way to do that is to multiply the 2 which by the way, if you apply PEMDAS strictly, multiplication does come before division but that’s neither here nor there. The FACT is, there’s still parentheses there! And you can’t move on until they are GONE!
6 / 6 = 1
And…here is the link I posted earlier again, from a very reputable website:
go to the last problem on the page. They are well aware of this argument as you will see… It also explains why different calculators/software deliver different answers.
http://www.purplemath.com/modules/orderops2.htm
(I am arguing this because with that left to right thing, and the assumption you can replace parentheses with multiplication, students are going to be confused when they get to distribution and that’s no longer the case. I’ve seen them confused, it’s not pretty. Math should be consistent, not change a rule when you feel like it!)
PS – Ironically, I said I’m repeating myself and then I repeated myself lol…(“the order in which things are”). I did not plan that…that’s an edit fail!
nope wayne
still follow order till () is gone so in your example
you still use PEDMAS to decide when to use distributive property with regard to getting rid ot the () so you still do exponents before M/D
I hate this problem…
As defined.
6/2(1+2)
3(3)=9
However if you’re anal about parenthesis then,
6/2(1+2)
6/2(3)
6/6 =1
This is the way I see this problem. The answer is 9. If the author of the problem was anal about parenthesis then they wouldn’t have used them in the equation to begin with. The fact that the author is defining what we need to do first supercedes any other operation and we must assume that parenthesis are welcome. It’s quite simple. If the author had actually intended x=1 instead of x=9 then the author should have written the equation 6/(2(1+2)) or even more simply put 6/2*3.
9 is the correct answer. If the author intended it to be 1 then they are a troll because the they didn’t define the problem properly and on purpose.
Let me summarize the big points I already made:
First,
if you want to say 0.5x, then you HAVE to write (1/2)x with parentheses or, x “all over 2” with a horiztonal fraction bar, or write x/2. I have never seen (1/2)x before I researched this equation, but since searching online, I HAVE seen fractional coefficients written this way, only because computers are limited to the horizontal typing space. Therefore:
x/2 = (1/2)x = 0.5x
1/2n = 1/(2n) This sort of notation is used especially with pi, ln, or e. We have never had to say 1/(2pi). It was simply 1/2pi, or 1/2e^2. I am not one to refer to any calculator, however, even wolfram will recognise 6/2n = 3/n. (More below on this point).
I have always used ab/cd to mean (ab)/(cd) and I topped almost all of my calculus classes since high school through university. (moot point here, I know)
Just to re-iterate though, to use 6/2 as a fraction, parentheses are REQUIRED. Every book will tell you this.
Now consider the Identity Law:
a = 1a = 1(a)
We know there is ALWAYS an ‘invisible’ 1 as a ceofficient of a variable if no other number is there. Therefore:
a/a = 1, and if a is also 1a, then a/1a = 1. Blindly using ‘pemdas’, some folks would do this:
a/1a = a/1*a = a*a = a^2. I hope this drives home the silliness of this calculation.
Now, on to my second point:
consider: factoring, simplifying equations, and the distributive property.
Lets start with the number 6.
6 = (4+2). There is a common factor here: 2. So let’s factor it out of both terms.
(4+2) = 2(2+1). The outside 2 remains a part of of the 2 inner terms at all times. It cannot be used in an operation by itself without the rest of (4+2). The reverse of factoring is distribution, so, 2(2+1) = 6. This has to be true always. The argument I have seen to this is that (6/2) can be distributed. This is true ONLY if 6/2 is in parentheses, otherwise, the 6 and 2 are separated by a division slash or obelus, and the 2 is a factor of 2+1. So, let’s prove the initial equation:
6/6 = 1
6/(4+2) = 1
6/2(2+1) = 1
the same can be done for other factors:
6/6 = 1
6/(3+3) = 1
6/3(1+1) = 1
Distribution is actually a part of “Simplifying Equations” and is not bound to the order of operations as “multiplication”, since it is in fact “removing parentheses by distributing”. This can be googled and several references found.
Simplifying 2(2+1) + 3(2+1) = 5(2+1). We “combined like terms” here, by adding, and did not perform the “parentheses” part of order of operations, nor did we multiply, which is also higher priority than adding, because we only simplified.
If we try to prove that = 9
9 = 9
6 + 3 = 9
(6/2)2 + (6/2)1 = 9
(6/2)(2+1) = 9,
(6÷2)(2+1)
you end up with parentheses around the 6/2, as you should, changing the intial equation at hand.
Lastly, I hear the argument that “This is strictly numbers and you don’t use algebra rules since there are no variables”. That is the most asinine arguement I have heard yet. All axioms, laws, and properties use variables, meaning that they hold true for “any number”, hence the proofs with variables.
But “google says it is 9”! Google actually changes the equation to (6/2)*(2+1), and, wolfram contradicts itself with 2n/2n = 1, and 6/2n = 3/n, but then says 6/2(2+1) is 9. wolfram’s “terms” state that any answer should be verified with common sense and accuracy should also be verified.
Best regards as always,
K.M.
This is a great pictorial about how terms outside brackets belong with them, no matter what, in keeping with the intended value:
Area of a rectangle:
http://www.freeimagehosting.net/yys1h
Here is a problem: Assuming the unit is square feet.
Now, take 100 square feet, and divide it by the area of that rectangle which is 4(3+2) square feet. How many times will it go into 100 square feet ?
100 sq ft ÷ 4(3+2) sq ft = 5
4(3+2) is a SINGLE value and must be operated on as a whole…
100 ÷ 4(3+2) = 5
(100 ÷ 4) * (3+2) = 125
🙂
Great explanation. I am back on the # 1 team. It is how I originally processed the equation, it is how I was taught some 13 to 16 years ago and I was an A + math student and it just makes the most sense. Again Caper, great explanation. I think the people who think the answer is nine are simply cat lovers. However, I think that this has been proven to be one more than nine ways, so perhaps it’s time to let this cat die.
People are talking about this like its a debate. The answer is in fact 9. In order to get 1 the problem would have to written as 6/(2(1+2)). I have a degree in mathematics and use working on my masters in comp science, so i do have some knowledge in this area.
Let me address a few flaws with the proofs other people gave for some other answer:
From Sue:” 6 / 2(3) must lose parentheses . only way to do that is to multiply the 2 which by the way, if you apply PEMDAS strictly, multiplication does come before division but that’s neither here nor there. The FACT is, there’s still parentheses there! And you can’t move on until they are GONE!”
The parentheses can just be dropped because everything inside of them was already simplified as far as it could. They are only used to tell you what do do first them they are dropped. 6/2(3)==6/2*(3)= 6/2*3=9
Ida says”6/(1+1)(2+1)
applying FOIL to distribute the parenthesis:
6/(2+1+2+1)”
Problem is you cant foil as the 2 is tied to the 6 and technically in the denominator where as the (2+1) is in the numerator of the next expression.
Also, several people mentioned distributing the 2, well you can but you have to take the 6 with it so after distribution it would look like (6/2)*2+(6/2)*1=3*2+3*1=6+3=9
Also, to follow up with caper’s rectangle solution is right about most of it except the question he posed need to be written out as 100/(4(3+2)) you forgot a set of parathesis.
@TED Ted, I have already tried every which way to explain this. I, too, have a degree in mathematics and have tried many approaches to explain this. Seems every one on here wants to apply certain properties like Distributive but fail to take into account that it is not 2 times the parentheses, but rather 6/2 times the parentheses. Try as you will to convince them, but you will also become as frustrated as I have and stop posting on it. Just agree to disagree with them. At least you, I, and the original mathematician can agree. 🙂
And to clarify the point why the parenthesis were missed it is because the original equation asked how many times will the area fit inside 100 sq feet, so it should be written 100/A meaning 100/(A)=100(4(3+2)). The entire area of the rectangle needs to be divided by 100 which is why the whole area equation needs to be in parenthesis.
6 ÷ 2n = 1; where n = 2+1
Parentheses are not actually required, and there is no debate on this.
(6 ÷ 2)n = 9
6 ÷ 2n = 1
The logic is shown with the Identity Law:
a = 1a
a ÷ 1a = 1
a ÷ (1a) = 1
Parentheses are not required since a is always 1a with or without the 1 actually written.
a ÷ 1a is not a ÷ 1 * a = a²
Simple Algebra lesson to show grouping:
http://cstl.syr.edu/fipse/Algebra/Unit2/parenth.htm
Here is an example:
{4 [2xz⁴ (2c³v ÷ 2cv²)³ ÷ 2]} ÷ c⁶x • v³ = 4z⁴
Just in case exponents didnt work, here it is again:
{4 [2xz^4 (2c^3v ÷ 2cv^2)^3 ÷ 2]} ÷ c^6x • v^3 = 4z^4
Parentheses are just not required, as you say, Ted.
4x ÷ x² = ?
x² = xx or x * x, or x(x), and we dont do the maths as this: 4x ÷ x * x = 4x
There are implied ( ) around x²
As for the rectangle drawing, there are not required there either, for the reasons above. Check out the link on grouping, then try this test, but only questions 2 & 5:
http://cstl.syr.edu/fipse/Algebra/part4/revSelf.html
Regards
KM