Friday, May 6, 2011

6÷2(1+2)=1 & 30÷2(2+3)÷5=0.6

(A real life problem:
A millionaire passed away and left down 6 millions. He had 2 wives, and each wife had c children.
In his will, the wealth would be divided equally among all his children. So, how much would each of the children receive?
Naturally we will come out with this:
6millions ÷ 2c, since the total number of children = 2*c=2c
If c=(1 girl + 2 boys),
option (1) => 6millions ÷ 2(1+2) = 6millions * 1/2*1/3 = 1 million
option (2) => 6millions ÷ 2(1+2) = 6millions ÷ 2*3 = 9 millions

by option 1, each child would get 1 million,
by option 2, each child would get 9 millions.
Which one is correct?
(From one comment below, someone says it should be 6millions ÷ [2(1+2)] instead.
6millions ÷ [2(1+2)] is true, but since [2*(1+2)] = [2(1+2)] = 2(1+2), we may just drop the parenthesis. The detail is included in the proof below.

It is so clear that
6÷2(1+2) = 1
But... How could this confuse so many people? And there are so many people arguing that the correct answers should be 9 for the first problem and 15 for the second problem. Even the news and some professionals argue that the correct answers are 9 and 15 respectively.
However, I am quite sure most students from PRC will get the correct answers, which are 1 and 0.6 respectively.

In fact, there are many students gave the correct answers, but they were told that they were wrong. What the heck?!!

*[To my dear readers, if you have forgotten some properties of function, please read the 'Extra' at the end of the post before you proceed. Thanks =). ]

Why is 6÷2(1+2) = 1, not 9?
Let c=1+2. Compute 6÷2c.
Then 6÷2c = 6*1/2c = 6/2c = 3/c .
( How could they get 3c??? Sigh... they do it in this way: 6÷2c = 6÷2*c = 3c .
Well, assume to the contrary that this is true, then
6÷2c = 6÷2*c = 6*1/2*c = 6*c*1/2 = 6c/2 = 6c ÷ 2, which is a contradiction.
So, 6÷2c ≠ 6÷2*c for c≠1).

Why? Let a,b,c be any real numbers.
Then a÷b*c = a*1/b*c = ac/b ≠ a/bc = a*1/bc = a÷bc , for a≠b, and c≠1.
(Note: bc is a number, while b*c is the product of two numbers b and c, and b*c=bc.
If we write b=2, c=3, then 2*3 = 6, so bc=6 is a number, while 2*3 is the product of two numbers 2 and 3.
If we write a=6, b=2, c=3, then a÷b*c = 9 , but a÷bc = 1.
If we assume to the contrary that a÷bc=9, then from the previous argument, a÷bc=6÷6=9, which is a contradiction.)

(From one comment below, someone is confused when I assert b*c=bc but on the other hand I said 6÷2c ≠ 6÷2*c. I know this is trivial and I didn't want to be so nagging on this part. Anyway, since some people might not know, let me give a little bit explanation. As from the example above, 2*3 = 6, then 6÷(2*3) = 6÷6. Generally for b*c=bc, then a÷(b*c)=a÷bc. But, a÷b*c≠a÷bc if c≠1 and a≠b.
Still not convinced? Well, if alternatively, we write a÷b², we know that (b*b)=bb=b², how do you interpret a÷bb? )

Hence 6÷2(1+2) = 1,
but 6÷2*(1+2) = 9.

Most students never learn the proof above, including myself. Why?
Because it is so clear that we don't necessarily need to go through the proof to get 6÷2(1+2) = 1.

While for the second problem, 30÷2(2+3)÷5, it follows the similar argument that the answer is 0.6.

In fact, any qualified scientific calculators would give us the correct answers.

All the news below are bull shits!!!

Even the math puzzle in the end of the news below was incomplete/not the real one we learned, which again, misled the students.
The puzzle given by the teacher was:
3 ppl went to a hotel, the initial cost was $10 per person. But they were given 10% discount, which means the total cost = 3*9=27. However the waiter put $2 into his own pocket. Then $27+$2 =$29, where is the missing $1? (Is it a puzzle? Nope, it is so obvious that if you paid 27, while the waiter kept $2, the resulting payment the hotel received was $25. Who would go and calculate $27+$2 ?)

A more accurate puzzle should be:
3 ppl went to a hotel and rent a bedroom, the room initially cost $30. Each of them gave $10 to a waiter who helped them made the payment. But the casher told the waiter that the room cost $25 only for that day and returned $5 to the waiter in coins. But the waiter realized that he could not divide the change equally btw the 3 ppl. So he decided to put $2 into his own pocket and give $1 to each of the 3 ppl.
Now each of the 3 ppl had been given a dollar back, so each of them paid $9. Then 3*$9 = $27.
The waiter had $2 in his pocket. So, $2+ $27 = $29. The 3 ppl originally handed over $30, where is the missing dollar?

The following news is even more ridiculous. The higher quality calculator indeed gives the correct answer, while the other cheaper calculator gives the wrong one. But the broadcaster told a different story.

Some people argue that by PEMDAS rule, the answers should be 9 for the 1st problem.
P - parentheses
E - exponents
M - multiplication
D - division
A - addition
S - subtraction
PEMDAS is true, but using this rule without following other rules is not the right way to do mathematics.

When we learn function, we ordinarily write xy=z for x*y=z, where x and y are independent variables while z is the dependent variable.
We can see that for every real number x and y, x*y = xy. So, for each x and y, z=xy.
Writing xy=z is better than writing x*y=z because it is unambiguous and convenient.
Let say we are given the values of x and y, and we now want to compute a/z, where a is some constant.
By convention, for each real number x and y, we compute a/xy to get a/z.
We don't write a/x*y.
This is why when we write 6/2(1+2), we don't write 6/2*(1+2), they are different from each other.)


  1. I've seen a lot of arguments on this topic but I have to admit that yours is by far the most ignorant.

    First of all, 6/2(1+2) does not represent your real life problem. It should be 6/[2(1+2)]. Parentheses exist for a reason. Use them.

    Also, you seem to understand that bc = b*c which puzzles me as to why you would even think that 6÷2(1+2) is not the same as 6÷2*(1+2).

    Btw, are you aware that your proofs are a total mess?

  2. sk51244, 2(1+2)= [2(1+2)]. Please go and check ur textbook. Have u learnt function before? Go and learn what bc means. You will get an explanation that I think you haven't seen before, otherwise you wouldn't have made the comment above.

  3. sk51244, may be you didn't learn so much about mathematics, because you will see 6÷2(1+2)≠ 6÷2*(1+2)in almost all Mathematic textbooks. E.g. please look out 'Introduction To Real Analysis', 3rd Edition, by Robert G. Bartle and Donald R. Sherbert, page 53, 9th lines counting from the bottom, 1/2n = 1/(2n).
    I just simply open any book to get an example for you. To persuade yourself, please go library and borrow and Math book.
    Thank you for being interested in Math. I am sure you will learn more if you go explore more.

  4. By the way sk51244, I am not teasing you and my apology if you feel so. I really appreciate your comment here. And I thank you for the comment or else I wouldn't have touched the book above which I have never touched for a long time. =)

  5. haha I missed out one point you made. We should instead use parentheses when necessary.
    And also, 2(1+2) = [2*(1+2)].
    if we write [2(1+2)], you may just drop the parenthesis.

  6. If you want a person to understand your views, provide an explanation. Do not point them in the direction of random books. Not everyone has access to the books you use. Besides, it gives one the impression that you do not really know your stuff.

    Anyway, I do not see any relation between the author's preference in writing 1/(2n) as 1/2n and your claim that 6÷2(1+2)≠6÷2*(1+2).

    I notice that you said "6÷2(1+2)≠6÷2*(1+2) in almost all Mathematic textbooks". Could you give me an example of one such book? But then again, I might not have access to the book, so could you explain in your own words why 6÷2(1+2)≠ 6÷2*(1+2)?

    Btw, the author's decision to write 1/(2n) as 1/2n is not even a valid argument as to why 1/2n = 1/(2n). Authors have their own preferences. An author's preference does not make it a universally accepted fact.

    I could provide an example for you too. The Real Analysis book that I used during my undergraduate years was Analysis: With an Introduction to Proof (4th Edition) by Steven Lay. If you have access to the book, you might want to look at page 196, under 'Answers to Practice Problems', Q20.7. The author says "...let delta = 1/(6n)." Now, if 1/6n and 1/(6n) are universally accepted to be the same, why would he bother including the brackets?

    Also, I notice that you said "almost all Mathematic textbooks", which means there are some that say 6÷2(1+2)=6÷2*(1+2). How do you explain that? Which author is right? The majority? Such a statement would earn you lawsuits from all directions.

    As for the parentheses, I'm not an idiot. I know that 2(1+2) = [2(1+2)]. All I'm saying is that the extra set of parentheses is needed if you want to tell people that the 2(1+2) has to be evaluated first before performing the division.

    There isn't a universal rule saying that multiplication by juxtaposition has a higher precedence than regular multiplication. It's simply a matter of opinion and one person's opinion does not make it absolute.

    If you intend to argue, first try to understand what the person is actually saying, and then provide an explanation. Arguing about maths is only fun when the arguments are strong.

    Lastly, you might feel that I "didn't learn so much about mathematics" -and I won't be so arrogant to deny that accusation-, but we're dealing with elementary maths here. Basic maths is sufficient and audacious accusations are uncalled for.

    A little open-mindedness wouldn't harm anyone.

  7. Hi sk51244,
    Good to see your comment again and this time you give a more comprehensive explanation.

    Firstly, what I wanted to say is you will see why 6/2(1+2)≠ 6/2*(1+2)in almost all Mathematic textbooks.Those textbooks might not directly deal with 6/2(1+2) particularly, but u would be able to deduce a/b*c≠a/bc for some real numbers a,b,c. And I said 'in almost all Mathematic textbooks' because I myself never and am not able to read all the textbooks in the world. More accurately, I should have said all the textbooks I have ever used so that my judgement is generalized to all textbooks (simply because I shouldn't make judgement on textbooks which I never read before).

    1/2n = 1/(2n) for some constant n and I claim 6÷2(1+2)≠6÷2*(1+2). The relation is shown in my post.

    Well, my apology I just gave you an example randomly and I certainly realized not everyone has access to the textbook. But that is not the only book. You may look for other books. As I was reading your comment, I believe that you have used more than 1 book before. The example you gave me is good. And 1/(6n) is clear. As for the reason why he never use 1/6n, I think author has the right to choose either using 1/6n or 1/(6n) for the same purpose. Why not you get some other books to clarify? Btw, ur example doesn't tell me 1/6n ≠ 1/(6n). Did the author say he wrote 1/(6n) because 1/6n ≠ 1/(6n)? This is what I expected u to say when I read ur example.

    And yea, I have explained it in my own words why 6÷2(1+2)≠ 6÷2*(1+2) in the post.

    You are right every author has their own preferences which might not be valid. But if a method is considered as valid universally, the author's preference of using that same method should be valid. Btw, if you think it is the author's invalid preference to use 1/2n = 1/(2n), aren't u saying there is another invalid author's preference of using 1/2n ≠ 1/(2n)?

    Also, as I said "almost all Mathematic textbooks", it doesn't mean there are some that say 6÷2(1+2)=6÷2*(1+2), but because I don't read all the books and I can never finish reading all the books available to human beings, so I shouldn't say all Mathematic textbooks. But I can say I saw 6÷2(1+2)=6÷2*(1+2) on Facebook and news recently. I'm sorry but I have to say I never see 6÷2(1+2)=6÷2*(1+2) on any textbook.

    Next, the statement should be written in this way. It is true that extra set of parentheses could be added if we want to tell people that the 2(1+2) has to be evaluated first before performing the division. On the other hand, it is true that extra set of parentheses is needed if we want to tell people that the 2*(1+2) has to be evaluated first before performing the division.

    I don't think I say there is a rule saying that multiplication by juxtaposition has a higher precedence than regular multiplication. Instead, I said 2(1+2) is 6. [In the post, I wrote bc=6...]. PEMDAS is only for (), ^, *, ÷, +, -. Do you see * in 2(1+2)? But I saw many people added * into it themselves and manipulated it into 2*(1+2). (I have stated why it is different in my post, though 2*(1+2)= 2(1+2).)

    From your first comment, I really felt that you didn't learn so much about Mathematics. However, from your second comment, I have to change my mind as you give comprehensive arguments, which is very much better than others on Facebook and some forums. You are correct that we are dealing with elementary maths. Everything I mentioned in my post is basic math. I hope this time it is clear for you and I look forward more arguments from you. In fact, I am waiting for someone who can prove that I am wrong, if I am indeed wrong. Most explanations I have seen for both 6÷2c = 6÷2*c and 6÷2c ≠ 6÷2*c are wrong, invalid or incomplete.

    I agree with you that a little open-mindedness wouldn't harm anyone. So please comment here if you have found any correct proofs.

  8. Hi sk51244, please write down the proof here if you have found any correct proofs to show that 6÷2c = 6÷2*c, if you want to prove my proof is wrong. Thanks.

  9. error: that my judgement is not generalized to all textbooks ...

  10. I decided to give your post another detailed reading and came across this:

    "someone has no idea why I say b*c=bc, but then I assert 6÷2c ≠ 6÷2*c" - Please choose your words wisely.

    This is getting interesting but it feels like we're not speaking the same language.

    Please try to understand this: I did not provide the example to show that 1/6n ≠ 1/(6n).

    Le me reinforce my statement: An author's preference does not make it a universally accepted fact.

    I was just trying to make a point, which is: Your example does not prove 1/2n = 1/(2n). Both authors have their own preferences. If a universally accepted rule existed, such ambiguities wouldn't exist.

    That's all. Btw, I never mentioned anything about "invalid preferences".

    Anyway, I notice that the arguments are getting longer and I believe we are both people with busy schedules, so I think we should focus on the main problem here. Please provide a clear proof of the following statement:

    a÷b*c≠a÷bc if c≠1 and a≠b

    Thank you.

    A side note:
    It might have been a typo, I can't assume, but please never use 'math' and 'maths' in the same paragraph. Forgive the nitpicking.

  11. Hi again sk51244,

    I edited the sentence edy. I am sorry for that. is either 1/6n ≠ 1/(6n) or 1/6n = 1/(6n). If you did not provide the example to show that 1/6n ≠ 1/(6n), then why did you give me the example? Please tell me.

    You said an author's preference does not make it a universally accepted fact. I never deny this statement, but I do not agree with the way you assert the author's preference on using 1/2n=1/(2n). What I see from your argument is, you do not believe 1/2n=1/(2n), so you say the it is the author's own preference to write 1/2n=1/(2n). Why not you go and check if 1/2n=1/(2n) is a universal fact first?
    And I think you still remember that an example is not used to prove a true statement, but disprove a false statement. I didn't use the example to prove 1/2n = 1/(2n). Instead, 1/2n = 1/(2n) is the example itself which I gave to disprove the statement '6÷2(1+2) = 6÷2*(1+2)' in your 1st comment.
    Yes, you did not mention anything about "invalid preferences", but it is implicitly pointed out in your 2nd comment.

    A proof for a÷b*c≠a÷bc if c≠1 and a≠b, which I think is clear enough is already shown in my post. I won't say it is the best proof, because there is still a lot more for me to learn in order give a clearer proof. And I made the proof myself, which means it is an individual opinion and might be wrong. But if you want to deny it, please disprove it.

    As for your side note, I thank you for pointing out the typo. Having typos is common in my articles. I am certainly aware of this as it has existed for a long time.

  12. This comment has been removed by the author.

  13. My comments seem to have disappeared.

  14. Hi there,

    Here it is.

    I provided the example to show you that not everyone readily accepts 1/2n to be equal to 1/(2n).

    If I had not checked whether 1/2n=1/(2n) is universally accepted, do you think I would have the nerve to argue here? Definitely not. I have done my research and I can tell you that at the moment, it is definitely not a universal fact.

    I mentioned that 1/2n=1/(2n) is not universally accepted. Perhaps it would have been clearer to say 'neither 1/2n=1/(2n) nor 1/2n≠1/(2n) are universal'. It is still regarded as a matter of opinion.

    You're right, I do not like to take for granted that 1/2n is equal to 1/(2n). To me, it is not good practice to write 1/2n when you intend it to mean 1/(2n). That's my opinion, one shared by many others. However, I acknowledge the fact that many people, like you, accept 1/2n to mean 1/(2n). It is a matter of opinion, and definitely not invalid.

    When I first decided to comment on this post, the reason that led me to it was simply because you claimed that 6÷2(1+2)=1 and that those who answered 9 were wrong. I just wanted you to open up your mind a little and consider the possibility of both answers being acceptable. It is, once again, a matter of opinion. Those who accept 1/2n to mean 1/(2n) will answer 1, while those who do not, will answer 9. Both answers are not wrong. However, those who say '1 is wrong' or '9 is wrong' are indeed wrong. As long as there isn't a universal fact that forces everyone to accept 1/2n=1/(2n), the answer will remain ambiguous.

    As a matter of fact, I did seek a second opinion on a maths website in regards to your views and I was fortunate to have been provided with a brilliantly insightful explanation. Since the explanation was sent directly to my email, I'm still trying to figure out the link to share with you. It is a rather lengthy explanation and I do not wish to spam your comments any further. Once I get it figured out, I will drop by again. Or do you prefer me to forward it directly to your mail?

    Anyway, wrapping up this discussion, it has been wonderfully entertaining and I thank you for your patience and time in replying my comments.

  15. Hi sk51244,

    Although I don't change my mind but I have to say...

    I like this part:
    "Perhaps it would have been clearer to say 'neither 1/2n=1/(2n) nor 1/2n≠1/(2n) are universal'. It is still regarded as a matter of opinion."

    and this part:
    " I acknowledge the fact that many people, like you, accept 1/2n to mean 1/(2n). It is a matter of opinion, and definitely not invalid."

    Perhaps people from my place and calculators & books we use only allow 1/2n=1/(2n), so I never agree with 1/2n ≠ 1/(2n). But if what you say is true, I would like to apologize to you for asserting that 1/2n ≠ 1/(2n) is wrong in my post.

    I look forward receiving the link from you and I thank you in advance =).

    It has been great to discuss with you. If you don't mind please add me as your friend on Facebook! =)

    (uh...I have no idea why your last comment fell into the spam box, haha! Anyway this case isn't the first time happening in my blog.)

  16. Whoa, now it looks like I'm really spamming your comments. Could you do me a favour and remove the previous duplicates? Thanks.

    Anyway, sorry for the late reply. It has been a hectic week.

    It's frustrating but I couldn't work out the link, so I'm just going to paste the main explanation here.

    Q: 1/2n = 1/(2n)?
    A: The fact is that the order of operations rules that are commonly taught are nothing more than an attempt to codify how mathematicians interpret expressions in practice, and this kind of expression, not being used by mathematicians when they can avoid it, has never
    developed a standard interpretation. It is ambiguous, and we just write

    - n




    Most books that teach the subject just say that multiplication and division are done left to right. By that rule, a/bc means (a/b)*c. But such an expression LOOKS as if it should mean a/(bc), both to many
    students and to some mathematicians, so some authors include an extra rule to make it so. Of course, that means we have people who have been taught both interpretations, so you can't trust that anyone else will
    interpret it the same way you do. Therefore, you just don't write it, and you don't get into arguments about what it means. That's as silly as arguing about whether an "American history teacher" is an American
    who teaches history, or anyone who teaches the history of America. Neither is wrong, neither is right either.

    I'm not an avid Facebooker, but I'll add you when I'm on. =)

  17. Btw, just to clarify, the explanation ends at "...neither is right either."

  18. I see. Thanks a lot for the explanation =).

  19. Please do no not create FALSE Knowledge. Check this out.

  20. Boyi, that is 6(2+1)/2. It is different. :)

  21. Geek arguments are funny the answer is 1 why are people getting dumber. This is why I hated showing my work in school. Follow pemdas and it still rings true in this equation.People often divide before multiplying and need to remember you only go left to right in the terms of fractions otherwise follow pemdas. This equation has no fractions only real numbers. Fractions are in fact division problems within the equation used to usurp multiplication rules and force division to be taken into account prior to multiplication. Merely to make something simple seem difficult. If you use only real numbers and infinite possibilities reducing fractions to decimals you will find the same answers as if you used pemdas and the rule of Mult./Div. fractions. Just use decimals and we wouldn't have this problem with fractions at all. Fractions are just easier for idiots to understand because they don't have to move the decimal around during division & multiplication. The first proof that by making things easier we in fact make them dumber and to use a universal constant like math is nice for a change. We say something is unchangeable then we try to make it easier to understand therefore effectively changing it so as it does not stay constant. But we tell everyone it is and they go on believing it until some moron realizes that, when the constant is so far off from its starting point, it no longer exists optimistically at all and may never have to begin with and we have lived our lives on mans theory of an eternal constant. In short math tries to prove God by eventually disproving itself. Take for instance imaginary numbers, negative integers, imperfect squares, non-repeating decimals. All numerical designs that lead to infinite results when plugged in and one small step is missed in the calculation process. Thus resulting in a paradox of colloquial knowledge and arithmetic where 1+1=i3 because I imagine there be something else that was not calculated prior to be seen as a variable. Hence why word problems are wrong and you lose a dollar in translation but the work checks out. hint: the waiter stole it. You see even if you factor in all known variables the human element will always play a factor and what is accepted as correct will be what is correct so if enough people begin to believe the answer is 9 instead of following traditional laws man will eventual flip the d and M in pemdas and division will be done before multiplication and a constant will have been fundamentally changed and therefore lead to proof that there is no constant except the eternal search for a constant. and in that eternal search for a constant once your cup is empty and all questions have left you will you find the eternal answer you sought. Alas you will have no use for the eternal knowledge granted and should you fill your cup your knowledge will be lost. The universal answer to everything is 7. To understand true math you must first forget all that has been taught and merely say I have 1 and he has 1 for the 1 you have is the 1 you have and the 6 he has may be the 1 he has. More is not always more and to add measure is to assume that life is fair and it is in fact neither fair nor predictable and even psychics/physicists aren't right all the time but often enough to give credible consideration. Anything in society with more than 50% accuracy is given weight until someone can discredit it bellow that mark. Why do you struggle of trivial things such as numbers when numbers mean nothing in an eternal existence. And one constant is sure to be broken soon E=MC^2 as some objects in motion may end up diverted when not impacted by an outside source but by diversion of the original source prior to launch hence the theory of time travel being in fact logical and even more plausible than 1+1=2. Think Corkscrew and then answer.otherwise you aren't even in the same league of intelligence and your knowledge is all book learned and computer fed.

  22. So if I showed my work it would look like this.
    anything divided by itself is 1. Simple but i am sure most peoples papers after this problem looked like this.
    pr some real morons out there got it right the wrong way
    3/1(1+2)<Reduced Fraction
    simply put the one you get two out of the three possible ways to answer is the right answer... there will always be multiple answers to a problem you just have to plug it in and find which answer is right for you. problem is your right answer may be another's wrong one. point in fact type in another's in your post box. It gets redlined but it is accepted English for the possessive tense of "another person's"

    1. You make the same error that most other have...
      6/2(1+2) become 6/2(3) correct

      Your error which leads to the incorrect answer of 1 happens because you treat multiplication as greater importance than division. This is wrong. Multiply and Divide are of EQUAL importance and come before Addition/Subtraction.

      So, your calculation of 2*3 first is where the error occurs. The order is left to right, so 6/2 should be your first calculation THEN 3*3 :)

    2. 6÷2(1+2)

  23. And now back to the idiot box I came out of to fuck up all yall's day.... Peace homies... keep doing math and maybe one day you will be able to equate why you never got laid in high school, college, and now struggle with your soon to be estranged wife. All part of the plan guys don't worry zip up your pants pay the prostitute and get back to your postulations of why math teachers are truly evil. And money, even gold, is worthless unless used the proper way and that is as fire kindling, wiring, weapons, safety and hull protection of extra atmospheric crafts. All things pointlessly made to fall back to earth in a fiery display of glory and destruction. Math = Destruction Thru the Catalyst Monetary Value. Hence the purpose of math for math is only used to divide wealth and explain what went wrong. and why is divide not spelled devide it would make more sense since devision is a form of a deviation not a diversion.Unless Division is meant to divert attention.

  24. And RomanPeter, if these geeks didn't know what they know you would not exist because it is their knowledge that has kept the average bumpkin from having to live in the wild and being eaten by wild beasts. You must bow before them, as will I, and know that they know things more beautiful than your tiny brain is capable of understanding.

  25. ...and of course, RomanPeter got it wrong, because of the ambiguity we must bow to social custom and it is clear from my wanderings that the answer 9 is not only the most prevelant but in fewer cases was I able to prove or argue that the answer was incorrect.

  26. 1/2x is just that: (2x)^-1.
    It is not 'one half" x. That would be written x/2.
    1/2x has ONLY ONE meaning, and x/2 has only one meaning, just as 1/2π means something and π/2 also meaning something else. Thank you.

    1. 1÷2x is not the same as 1/2x. When you say half of something what do you write? If you say one third of something what do you write? 1/2. 1/3. So that is half x. 1/2x. same as .5x.

      In the end the answer is 1 and people that are argueing until the cows come home that it is 9 needs to go back to school.
      The money problem described it perfectly. 9 can never be the answer.
      And for anyone saying "left to right" you are correct but you still forgot about PEMDAS and Parenthesis is first. The () dont just go away just because you feel like it.
      so it is still 6÷2(3) and since () comes first, it is 6÷6. it is not 2x(3) that is completely different.

  27. I stopped reading after a while. His logical way of thinking about it 100% here. We have math to translate real world problems into numbers. QUOTE 6÷2c = 6÷2*c = 6*1/2*c = 6*c*1/2 = 6c/2 = 6c ÷ 2, UNQUOTE
    The problem with this is getting someone to believe that 6 ÷ 2c = 3/c. This is what I will do, and follow it up with an online algebra lesson I found.
    6 ÷ 2(2+1) = ??

    People are incorrectly separating the 2 from the parentheses. It is a factor of the 2 numbers inside it. People are trying to solve this as a calculator would, but that is the wrong way. Solve it using algebra techniques with a pen & paper (See URL at the bottom)

    Now, think of the Identity Law: a = 1a = 1(a)

    Ask yourself: what is a ÷ a = ?? Ans: 1
    a ÷ a = 1

    Now, Identity Law says any variable that has no coefficient actually has a 1, whether it is written or not. So,

    a ÷ 1a = 1 ALWAYS.

    It is NOT a ÷ 1 * a = a² That is ridiculous.

    This proves that 1a = (1a) and (1 * a). 1a is not the same meaning as 1 * a.
    a ÷ 2a = 1/2
    6 ÷ 2a = 3/a

    Let a = 2+1.

    6 ÷ 2a = 3 /a
    6 ÷ 2(2+1) = 3/(2+1)

    6 ÷ 2(2+1) ? Ans: 1 Proven with the laws of math.

    6 ÷ 2(2+1) = 1

    6 ÷ (2+1)2 = 1

    (6 ÷ 2)*(2+1) = 9
    6(2+1) ÷ 2 = 9

    Here is an algebra Lesson:

  28. How about this as well:
    what is n+n+n+n+n+n+n ?
    Do we want to write things like this in every equation?
    No, of course not. We write "7n".
    7n = n+n+n+n+n+n+n
    what is 14 ÷ 7n ?
    7n has one and only one meaning, which is n+n+n+n+n+n+n
    14 ÷ 7n = 14 ÷ (n+n+n+n+n+n+n) = 2/n
    the RESULT of 7n is 7*n, but the MEANING is a
    remember the rule about a group of variables and coefficients? All factors of a product are the coefficients of the remaining factors, eg:
    7abc = 7bac = a7bc = abc7 etc etc.
    Everyone is thinking we MUST use redundant parentheses when we do not. When entering expressions into a calculator or computer program, then yes, you have to use them to force that program to give the answer you are looking for.
    2(1+1) ÷ 2(2+1) = 1
    This is the same as
    6 ÷ 6 = 1 and,
    2(2+1) ÷ 6 = 1 and
    6 ÷ 2(2+1) = 1

  29. What Caper26 said. End of story. No ambiguity. The answer has been, is and will always be: ONE.

  30. Refer to your example number 1,How would you solve this?:
    The speed of a car is half the speed of a bus.The speed of the bus is 6 km per hour.The car travels for 1 hour towards north and for 2 hours towards east.What is the total distance covered by the car?

    Isn't this correct:
    distance=speed x time

    1. Yes that is correct, But the problem is you are putting an operator "*" in-between the 2 and the (1+2). Since in the problem there is no operator the 2(1+2) is one term. 2*(1+2) is two terms. The original problem is one because it is 6 divided by 2(1 plus 2). the lack of an operator makes it a single term. Single term divided by single term.

      The assosiative property could also shed light on this. If you had 6/(2+4) what would the answer be? now factor out the a 2. 6/2(1+2). those are the dame equation because it is still a single term.

      Hope that helped!

    2. How does the lack of a "*" make it a whole term? There's no real difference between the implied multiplication with the explicit one. Some calculators give the implied multiplication priority, but this isn't universal and some simply don't. Without going further, google's calculator fills in the blanks in another ways.

  31. Who posted this problem was trolling, in fact lots of math people change ÷ to / instantly, with 6/2(2+1) maybe people could think that author didnt have an option to write as fraction and that infact is 6/2 (fraction) which is 3 and them multiply the other result which is 3 and the final result is 9, on the other hand, who posted this troll question used ÷ to know/have fun on how many people will fall under the premise of changing ÷ to a fraction instantly, saying all this is very clear that you need to solve everything after ÷ first which is 2(2+1) which is 6 and then you have 6 ÷ 6 = 1 . My 2 cents.

    1. This comment has been removed by the author.

    2. I think exactly the opposite. A / could suggest that everything after it goes "below" the 6. A ÷ is a lot less ambiguous. It's only a division and there's no reason to believe you should solve it after everything.

      Still, it's true that the author was obviously trolling. This doesn't have an universally accepted way to solve it, and a lot of the ambiguity here could be gone simply by being more specific and adding yet another parenthesis around 6/2 or around 2(2+1), according to what the author wanted to convey.

      Also, I need to say that the real life problem suggested by this guy here is simply false because it assumes that the actual operation is 6/(2(2+1)) and not (6/2)(2+1).

  32. I only feel I am correct (that the answer should be interpreted as 9) because I also feel I would not have been able to go as far as to comelette calc III & differential equations without interpreting a text equation correctly.

    I interpret 6÷2(2+1) as (*6)(÷2)(*3),
    and I do this because of the ÷, which is an operator associated with 2 (not (2+1) necessarily).

    If it were written as 6/2(1+2), I would get 1, because I would interpret that 6 is the numerator & everything after the slash is the denominator.

    Some simply state to move from left to right, it helps in this case, but isn't a law. Just like "pemdas" isn't a law; it's just a tool to "try to" help you get the right answer.

    Multiplication and division are the same, so neither have precedence over one another... you can multiply 2 (1+2) first, but you must do it correctly, and to do it correctly means you cannot forget the ÷
    ÷2(1+2) = 0.5(1+2)
    == 6(1.5)=9