1. As per title's question, is the answer 9 or 1?

2.

3. Depends where you are putting your brackets. If there is a bracket around the 6 divided by 2 then the answer is 9.

But I am not sure how you do your notation.

You have to be clear in your own mind how you write the calculations before you can know what the answer is.

4. I would say it is 1.

First get rid of the brackets (Order of operations - Wikipedia, the free encyclopedia) and it becomes 6 / 6

5. 9

6. Originally Posted by Mayflow
9
You should either explain your answer, or else stay out of the hard science section of the forum. Now, why do you think it's 9?

7. 9. Multiplication and division have equal priority and are performed left to right. Thus, 6÷2(1+2) = 6÷2×3 = 3×3 = 9

8. Fair enough, apparently wrongly I always tend to treat things like 2(1+3) as a single term to evaluate first. TBH though if anyone gave me something like the question in the OP at work I'd tell them to make it more clear what they meant. As in writing so too in maths, if there is room for ambiguity you need to express yourself better

9. Originally Posted by PhDemon
Fair enough, apparently wrongly I always tend to treat things like 2(1+3) as a single term to evaluate first.
I'll admit that was my first inclination.

10. What is interesting is that if the problem had been "6÷2×(1+2) = ?", it would have been obvious, but the removal of "×" provided a closeness between "2" and "(1+2)" that incorrectly suggested a change in the order.

11. Originally Posted by Harold14370
Originally Posted by Mayflow
9
You should either explain your answer, or else stay out of the hard science section of the forum. Now, why do you think it's 9?
What exacTly do you think 3 * 3 equals? You divide 6 by 2 and you get 3. Then you multipy it by (2+1) WhaT DO YOU GET?

12. If you enter it as written into Mathcad the answer is 1.

To get "9" you have to deliberately and consciously add extra parentheses.

13. 9 is correct.

14. Originally Posted by Dywyddyr
If you enter it as written into Mathcad the answer is 1.

To get "9" you have to deliberately and consciously add extra parentheses.

What happens if you include the explicit multiplication sign?

15. It is simple math guys. Now KJ, what is this implicate order thing?

16. Originally Posted by KJW
What happens if you include the explicit multiplication sign?
1 again.
It merely places a "." to explicitly indicate the multiplication.

17. The answer is still 9.

18. According to BODMAS:

Brackets off
Divide
Multiply
Subtract.

However since some people freely interchange Divide and Multiply; in the OP example you can get an answer of 1 or 9. Therefore the question is ambiguous and needs more brackets.

OB

19. The answer is still 9. It is not complicated.

20. However since some people freely interchange Divide and Multiply
From KJW's post it seems they are given equal priority and carried out in order left to right.

21. Originally Posted by Mayflow
Originally Posted by Harold14370
Originally Posted by Mayflow
9
You should either explain your answer, or else stay out of the hard science section of the forum. Now, why do you think it's 9?
What exacTly do you think 3 * 3 equals? You divide 6 by 2 and you get 3. Then you multipy it by (2+1) WhaT DO YOU GET?
Well, as you have seen, some people have given reasons why they think it should be 1. If those people just said "It's 1, period, end of discussion" and other people say "it's 9 period, end of discussion" then we wouldn't get very far towards arriving at an answer, would we?

You like to defy moderator requests, don't you?

22. Harold read the posts., I already explained - You divide 6 by 2 and you get 3. Then you do the simple math in the paragraphs to get 3. Then you multilpy 3 * 3 and you get 9.

23. What happens if you include the explicit multiplication sign?
1 again.
It merely places a "." to explicitly indicate the multiplication.
So 6÷2*(1 + 2) = 1?
I thought you did multiplication and division left to right, so first you must simplify the (1 + 2) to 3, so we get 6÷2*3 = 3*3 = 9.
PEMDAS is more like PE(MD)(AS). You don't do multiplication and then division, you do them both together left to right, and the same with addition and subtraction. Otherwise 100 - 1 + 1 would be 100 - 2 or 98.

24. Originally Posted by anticorncob28
What happens if you include the explicit multiplication sign?
1 again.
It merely places a "." to explicitly indicate the multiplication.
So 6÷2*(1 + 2) = 1?
I thought you did multiplication and division left to right, so first you must simplify the (1 + 2) to 3, so we get 6÷2*3 = 3*3 = 9.
PEMDAS is more like PE(MD)(AS). You don't do multiplication and then division, you do them both together left to right, and the same with addition and subtraction. Otherwise 100 - 1 + 1 would be 100 - 2 or 98.
Good example. You do the division of 6/2 first with a resultant of 3. Then you do what is in the (parenthisis) which also results in 3, then you multiply 3*3. Good god even a second grader could get that.

25. Yeah, you can go looking for notational ambiguities if you really want to find them, but most mathematicians would have no difficulty interpreting the OP as

As to whether introducing, say, or any other convention for arithmetic multiplication would alter the result I remain puzzled why anyone should think so........

26. But Mayflow (and everyone else at a second grade maths level) knows you are wrong

27. What? you just said 1 = 1 = 1=1 but it had nothing to do to do wth 6/2(1+2) which is just 3*3=9.

28. Originally Posted by PhDemon
But Mayflow (and everyone else at a second grade maths level) knows you are wrong
He wasn't wrong- but it had nothing to do with 6/2*(1+2)

29. My two cents: multiplication and division are equivalent (e.g. 2 x 2 = 2 ÷ 0.5) and addition and subtraction are equivalent (e.g. 1 - 2 = 1 + (-2)), so they are just done left to right.

So I get: 6÷2(1+2) = 6÷2x3 = 3x3 = 9

(Sorry Mayflow post #23, the parenthesis comes first. (However the result still 9, in this case.))

But I agree with KJW post #9 that showing the multiplication as 2(...) not 2 x (...) does seem to imply an ordering. So I wouldn't be very surprised if a "real mathematician" told me that by convention (which BODMAS or its variants is after all) it does comes first, giving 1.

30. Yeah, the kitty is correct, but really, people on a science forum could not get this?

31. I have long ago abandoned relying on conventions to determine the order of operations. Use parentheses whenever there is a possibility of ambiguity.

32. Good greif - how much ambiguity is there to 3 times 3?

33. The ambiguity isn't 3 time 3. It's whether the terms, as written, resolve to 3 time 3.

34. Originally Posted by mathman
I have long ago abandoned relying on conventions to determine the order of operations. Use parentheses whenever there is a possibility of ambiguity.
There speaks a pragmatically minded man.

*tries to click the 'Like' button*

35. So how is 6/2 not 3, and 1+2 not 3?

36. If you evaluate the terms first in the parenthesis and then from right to left, you get 6 / 2 * 3 which resolves to 6 / 6.

The discussion is about what order you resolve the operations in.

37. Originally Posted by Mayflow
So how is 6/2 not 3, and 1+2 not 3?
This has to do with the order that the operations are performed in, Mayflow. The link that PhDemon posted early on describes which functions get done first, depending on how the equation is written. It is not a case of simply starting at the left and continuing to the right.

Here is the link once again.

Order of operations - Wikipedia, the free encyclopedia

38. Originally Posted by Mayflow
So how is 6/2 not 3, and 1+2 not 3?
You REALLY need to visit an optometrist.
It's because there's the possibility that it's NOT (6/2)x(1+2) but, as shown very clearly in PhDemon's post #7 and my post #11, 6/(2x(1+2)).
Do try to keep up.

39. So I wouldn't be very surprised if a "real mathematician" told me that by convention (which BODMAS or its variants is after all) it does comes first, giving 1.
AFAIK Guitarist is the only real mathematician who has posted so far so I'll take his word for it (especially as it agrees with my post#3 )

40. [QUOTE=AlexG;579859]If you evaluate the terms first in the parenthesis and then from right to left, you get 6 / 2 * 3 which resolves to 6 / 6.

The discussion is about what order you resolve the operations in. you go from left to right. Order of Operations - PEMDAS

41. Originally Posted by PhDemon
So I wouldn't be very surprised if a "real mathematician" told me that by convention (which BODMAS or its variants is after all) it does comes first, giving 1.
AFAIK Guitarist is the only real mathematician who has posted so far so I'll take his word for it (especially as it agrees with my post#3 )

42. Pathetic troll is pathetic...

43. And the answer is still 9. I am sorry you could not make second grade, but you need to be able to read from left to right, and understand simple math first.

44. Even when you try to civil to May, it doesn't work.

45. True, it seems she can't go more than 5 posts without trolling or being an asshole...

46. And the answer is still 9.

47. Her poetry stinks as well.

48. And the answer is still 9.

49. 6/2(1+2) = 6/2(3)

However the lack of operator between the 2 and the 3 does seem to imply that it is supposed to be done before any other mathematical operation, (at least from what I remember of algeba) so in this case it seems to me that it should read 6/6 which =1.

50. I see what my second grade teacher must have gone through now. People have a difficult time knowing how to read from left to right.

51. Originally Posted by Mayflow
And the answer is still 9. I am sorry you could not make second grade, but you need to be able to read from left to right, and understand simple math first.
Yeah.
You seem to be conveniently forgetting that the rules of maths dictate that the operation on the right should be done first.

52. Where did that right to left rule come from? Mars?

53. As a mathematician I would not count on either answer. If this were in a paper, I'd look at the authors other work. If it was something from a student, I would ask them to fix it. If it showed up on a test, I'd make my decision clear by writing in some extra parentheses.

As a computer scientist (this almost never comes up with hand-written math), the answer is 9 because computers follow a rigid set of rules and would parse that as (6/2)*(1+2). You'd write 6/2/(1+2) or 6/(2*(1+2)) to get 1.

In between the two, I do often bump in to this simply due to the limits of a keyboard for dealing with math notation. (I prefer the long bar for division as it's mostly unambiguous, but that's usually not an option on a computer.) What I do most of the time is just include possibly extraneous parentheses to make the meaning clearer.

And yes, the way it's written it looks like the 2(1+2) was intended to be a single term, but without the author telling you, I wouldn't count on it.

54. [QUOTE=Mayflow;579863]
Originally Posted by AlexG
If you evaluate the terms first in the parenthesis and then from right to left, you get 6 / 2 * 3 which resolves to 6 / 6.

The discussion is about what order you resolve the operations in. you go from left to right. Order of Operations - PEMDAS
How Do I Remember It All ... ? PEMDAS !
 P Parentheses first E Exponents (ie Powers and Square Roots, etc.) MD Multiplication and Division (left-to-right) AS Addition and Subtraction (left-to-right)
So even the instructions you claim to be following tell you to do the parentheses (brackets) first before you start distinguishing multiplication and division from addition and subtraction or going from left to right.

Following the instructions you claim to rely on

First step. Parentheses out.
The term containing the parentheses is 2(1+2).
2x1 + 2x2 = 2+4
2+4= 6.

Put the expression back together and it now reads 6÷6.

Which is 1.

55. It all comes down to the implied multiplication, and whether that's given any particular significance in order of evaluation.

Did some googling, and found a Dr Math post on it: Math Forum - Ask Dr. Math

That says, in reference to a question from someone:

Some texts make a rule, as in your second solution, that
multiplication without a symbol ("implied multiplication") should be
done before any other operations in an expression, including "explicit
multiplication" using a symbol. Following this rule, you would
multiply a by x, then multiply b and y, then divide one by the other.
Some (probably most) texts don't mention such a rule - but some of
those may use it without saying so, which is far worse.

I don't know of a general rule among mathematicians that implied
multiplication should be done before explicit multiplication. As far
as I'm concerned, all multiplications fit in the same place in the
order of operations. It's not an unreasonable rule, though, since it
does seem that implied multiplication ties the operands together more
tightly, at least visually; but the idea of Order of Operations (or
precedence, as it is called in the computer world) is supposed to be
to ensure that everyone will interpret an otherwise ambiguous
expression the same way - so if some texts change the rules, or if
people do what feels natural, the purpose has been lost.
... so for now I'll stick with my 9. But if someone said "1" I wouldn't laugh and point, nor slap them with a fish.

56. Originally Posted by Mayflow
Where did that right to left rule come from? Mars?
You're still having trouble reading aren't you?
Nowhere did say there was right to left rule.
But, as you SHOULD have learned at school, the FIRST thing you should do (as has been stated numerous times in this thread by more than poster) is "remove the brackets".
They happen to be AT THE RIGHT.
Ergo, you start at the right. Which is exactly what I said.

... The term containing the parentheses is 2(1+2). ...
I don't agree with Mayflows' rudeness, but I don't agree with this, either. In PEMDAS (or its variants) with "Parenthesis first", that refers to what's inside the parenthesis, and does not in itself justify treating the initial question as: 6÷(2x(1+2)) versus (6÷2)x(1+2).

i.e. The part of the equation containing parenthesis is "(1+2)", not "2(1+2)".

If implied multiplication is going to be treated as having special precedence, then that needs to be made a standard, not something assumed.

58. Originally Posted by Dywyddyr
Originally Posted by Mayflow
Where did that right to left rule come from? Mars?
You're still having trouble reading aren't you?
Nowhere did say there was right to left rule.
But, as you SHOULD have learned at school, the FIRST thing you should do (as has been stated numerous times in this thread by more than poster) is "remove the brackets".
They happen to be AT THE RIGHT.
Ergo, you start at the right. Which is exactly what I said.

I don't think it was succinct (or at least, not correct).

By that logic, if you saw 2 x 2 + (1 + 2) and treated anything inside - and outside the brackets - as having precedence, then you'd do that as (2 x (2 + (1 + 2))) and get 10; and that would surprise me a lot. (I'd expect ((2 x 2) + (1 + 2)) getting 7.)

59. All removing the brackets does is give A 3 that times another three is still 9.

60. Originally Posted by pzkpfw
I don't think it was succinct (or at least, not correct).
Tsk, you missed the point.
Regardless of whether or not you do the implied operation (and, as I showed, Mathcad at least DOES take that route), the simple fact that the parentheses are done first (by the basic rules of maths) puts the lie to Mayflow's repeated "read from left to right".

Originally Posted by Mayflow
Originally Posted by AlexG
If you evaluate the terms first in the parenthesis and then from right to left, you get 6 / 2 * 3 which resolves to 6 / 6.

The discussion is about what order you resolve the operations in. you go from left to right. Order of Operations - PEMDAS
How Do I Remember It All ... ? PEMDAS !
 P Parentheses first E Exponents (ie Powers and Square Roots, etc.) MD Multiplication and Division (left-to-right) AS Addition and Subtraction (left-to-right)
So even the instructions you claim to be following tell you to do the parentheses (brackets) first before you start distinguishing multiplication and division from addition and subtraction or going from left to right.

Following the instructions you claim to rely on

First step. Parentheses out.
The term containing the parentheses is 2(1+2).
2x1 + 2x2 = 2+4
2+4= 6.

Put the expression back together and it now reads 6÷6.

Which is 1.
No the term in the brackets is (1+2) the 2 before the brackets is a divisor of 6, and 6 becomes 3 before multiplyng the other 3 to make 9. Trust me, I am not a doctor! But... I am right.

Besides, how the heck could you have ever get 6/6 out of this?

62. Originally Posted by Dywyddyr
Originally Posted by pzkpfw
I don't think it was succinct (or at least, not correct).
Tsk, you missed the point.
Regardless of whether or not you do the implied operation (and, as I showed, Mathcad at least DOES take that route), the simple fact that the parentheses are done first (by the basic rules of maths) puts the lie to Mayflow's repeated "read from left to right".
Sorry, but that's a rather contrived example. And Mayflow is correct that you do the math left to right, on (yes, the bit he or she usually misses out) the bits of equal precedence. Yes, the brackets here get done first, and they are on the right, but after that the real question is whether the divide or multiply is done next, and left to right, I would say the divide (unless, as noted, the implied multiply is given special treatment) is first.

The guts of your example (post #55), was a reference to the post by adelady, which I believe was wrong. Your example was not, ahem, succinct. (Your post #50 needed more context; as stated, it seems like a general rule, not a reference to a specific example.)

63. Originally Posted by Mayflow
...

Besides, how the heck could you have ever get 6/6 out of this?
In turn, I don't think it's hard to accept that some people do see the implied multiplication as having special precedence.

i.e. some people see 6 ÷ 2(1 + 2) as 6 ÷ (2 x (1+2)) giving 6 ÷ (2 x 3) giving 6 ÷ 6 giving 1.

You may not agree with these people, but you don't need to think of them as idiots. Take a breath.

64. Originally Posted by MagiMaster
As a mathematician I would not count on either answer. If this were in a paper, I'd look at the authors other work. If it was something from a student, I would ask them to fix it. If it showed up on a test, I'd make my decision clear by writing in some extra parentheses.

As a computer scientist (this almost never comes up with hand-written math), the answer is 9 because computers follow a rigid set of rules and would parse that as (6/2)*(1+2). You'd write 6/2/(1+2) or 6/(2*(1+2)) to get 1.

In between the two, I do often bump in to this simply due to the limits of a keyboard for dealing with math notation. (I prefer the long bar for division as it's mostly unambiguous, but that's usually not an option on a computer.) What I do most of the time is just include possibly extraneous parentheses to make the meaning clearer.

And yes, the way it's written it looks like the 2(1+2) was intended to be a single term, but without the author telling you, I wouldn't count on it.

Well the math seemed good until whatever you derived 2*3 from, where did you get that? The 2 was clearly a divisor of 6 as evidenced by a divisor sign between them.

65. Hey all,

Try solving this :

6÷2(x+2) = 1

and

6÷2(x+2) = 9

And we'll see whether 1 or 9 is correct.

66. Let's try this again. As introductory algebra.

6÷2(1+2)

Rewrite this with all the numbers replaced with a,
b, c.

It becomes
a÷b(c+b).

Which should really be written properly, but I'll just do it without fussing - but with biiiig spaces for thinkies.

a
_______

b (c + b)

which becomes
a
_______

bc + b2

We can now substitute whatever values we like for the unknowns.
6
_______

2 x 1+ 22

which equals
6
_______

2+ 4

which equals
1

In fact. For those who don't really get it, it's worth doing with a few different sets of numbers for the
a, b, c or plugging in different numbers into the original 6÷2(1+2) and seeing how different approaches turn out. My own view is that if you get different results from the algebra and the arithmetic, the algebra approach is always the better option.

67. Whoever said math was unambiguous?

Let's try this again. As introductory algebra.

...
You are still making the assumption that the division comes after the multiplication. You are trying to solve the real question (about precedence) by making an assumption about precedence. That's circular.

69. Originally Posted by molecool
Hey all,

Try solving this :

6÷2(x+2) = 1

and

6÷2(x+2) = 9

And we'll see whether 1 or 9 is correct.

I am not sure how to solve 3X+6 = 1 I suppose X would have to be A negative number equal to 1/3 of -5. If you could multiply 3X you could get -5 to add to 6 to get 1. I can work on this tomorrow. Right now my pickup has a dead battery!

Now let me think ... 3x+6= 9 that's too easy, x would be 1..

70. Originally Posted by pzkpfw
Let's try this again. As introductory algebra.

...
You are still making the assumption that the division comes after the multiplication. You are trying to solve the real question (about precedence) by making an assumption about precedence. That's circular.
The argument is outside of the domain of pure mathematics and within the domain of conventions. There being no fundamentally wrong or right about such things, I don't see a problem, as long as one is careful to specify which convention is being used. Or, by appropriate use of parentheses, force the expression into a form that is evaluated the same way, regardless of which convention is used.

I was taught two different conventions (I get 9 for the OP's question if I follow what I was taught in grade school, and 1 if I follow what I was taught at university). That has made me watchful about adding parentheses to my own equations, just to avoid the very sorts of arguments we're seeing here.

71. Originally Posted by pzkpfw
Sorry, but that's a rather contrived example.
Not at all, simply because... we're discussing THIS EXACT example.

(yes, the bit he or she usually misses out)
I.e. she was being less than accurate/ precise 1.
I.e the rule is NOT left to right, it is, as has been stated, PEMDAS (or whatever the new-fangled thing is. BODMAS is what I learnt).

The guts of your example (post #55), was a reference to the post by adelady, which I believe was wrong.
Again, no.
Adelady's post showed that (regardless of whether or not it's considered "correct" it was a lucid and succinct explanation of how the answer "1" was obtained).

1 Which is "normal" for her.

72. Yep it's about a really poor question and conventions. Sadly seen similar obscure examples on test, but most often embedded the work of students trying to work out a solution and confusing themselves which of course leads them to the wrong answer.

73. Originally Posted by Dywyddyr
...
I.e the rule is NOT left to right, it is, as has been stated, PEMDAS (or whatever the new-fangled thing is. BODMAS is what I learnt).
...
The thing is you're being imprecise when complaining about someone being imprecise. It's correct to say (paraphrased) "the rule is not 'left to right'". But it's also incorrect to imply it isn't relevant at all. See "left to right" in the quote in the post by adelady (#53) who used the link provided by Mayflow in post #39.

"BODMAS", as you learned, puts Division and Multiplication ahead of Addition and Subtraction. That's normal. But because "D" is before "M" do you think that puts division ahead of multiplication? Does that contradict "M" before "D" in "PEMDAS"?

... of course not. In normal usage, multiplication and division are of equal precedence - so are calculated left to right, as in the quote that adelady used.

So yes, Mayflow (who I agree behaves like an arse) is imprecise when making posts that only mention "left to right", but you are hardly better in saying things like "it is, as has been stated, PEMDAS" - as though that makes left-to-right irrelevant.

74. Originally Posted by pzkpfw
But it's also incorrect to imply it isn't relevant at all.
Not what I stated, nor what I implied.
I'll leave it at that.

75. Just to be a little silly...

6÷2(1+2) = 6÷2(3) = 6÷23 = approx. 0.26

This thread need a little silliness before the arguments get too heated.

76. Um, the answer here is undefined as the normal orders of precedence are not fully specified (as One beer mentioned may posts ago). So you are free to do the multiplication and division in either order and get 9 or 1. In fact, feel free to throw that expression into your favorite REPL shell and see what you get.

• Python gives 9
• Scheme gives 1
• Matlab gives 9
• Lua gives 9

I think there was a whole minute physics episode on this silly fact a while ago.

77. Well, I can see the answer 1 now but then it should not be presented so ambiguously. How about 6/(2*(1+2))=1?

The math God Excel asked if I would like to correct the equation to 6/2*(1+2) - which is what I interpreted it to mean in the first place

I consider it like Implicate order. See Neils Bohr.

78. So after yesterdays trolling you now try and pretend to be reasonable to avoid a possible suspension. You are so transparent it is laughable...

79. Originally Posted by PhDemon
So after yesterdays trolling you now try and pretend to be reasonable to avoid a possible suspension. You are so transparent it is laughable...

80. After yesterdays trolling you have the gall to tell people to stay on topic

Learn. To. Read. I gave my answer (and some more discussion on it) earlier in the thread. Tell me do you actually read what others post before you troll?

81. Originally Posted by PhDemon
After yesterdays trolling you have the gall to tell people to stay on topic

Learn. To. Read. I gave my answer (and some more discussion on it) earlier in the thread. Tell me do you actually read what others post before you troll?

82.

83. Yeah so much for a math discussion, huh?

84. Originally Posted by Mayflow
Yeah so much for a math discussion, huh?
Indeed.

Post #3, page #1.

Originally posted by PhDemon.
July 14th, 2014, 08:51 AM
I would say it is 1.

First get rid of the brackets (Order of operations - Wikipedia, the free encyclopedia) and it becomes 6 / 6

PhDemon posted a link right from the start that explained why this equation was ambiguous. I generally sit on the fence regarding these discussions. Congratulations on convincing me that I should venture into one, not once but three times.

There is nothing more for me to contribute on this topic.

85. LIKE for the above. It is pointless trying to communicate with her, I give up, hopefully she'll carry on in the same vein and get perma-banned soon...

86. I leave the math here To Ph and Nothern Horse Whisperer. The topic has lost my interest now.

87. I patiently await the day the forum as a whole loses your interest and you piss off back to your world of woo.

88. Mod note I think this thread has about run its course

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