# 48÷2(9+3) = ??

• ### 288

• Total voters
510

#### sureshs

##### Bionic Poster
Again, I NEVER called you a "strikingly dumb" as you claim (to enforse your argument)
She is strikingly good looking, she can do whatever she wants

#### Hidious

##### Professional
I'm also doing advanced math on a daily basis, soon to be teacher, and, even thought this could be wrong, i would also have said 2.

Why? Because i too see a dinstinction between 48÷2(9+3) and 48÷2*(9+3). When a coefficient is linked to another value without the * sign, i will naturally assume they cannot be separated. For example, if you replace (9+3) with C, you get 48÷2C which is different from 48÷2*C.

I'm not saying anybody is wrong here, just that it's completly natural to assume 48÷2(9+3)=2 and not even close to being idiotic, uneducated or dumb.

Anyboding changing the notation to prove their point (48*1/2*(9+3) or entering 48/2*(9+3) in a program doesn't prove anything because THE NOTATION IS THE PROBLEM. If you change it, the problem disappears.

#### BobFL

##### Hall of Fame
She is strikingly good looking, she can do whatever she wants
Mr. Hefner, is that you?

#### sureshs

##### Bionic Poster
^^^ Yes it is what people are used to.

As a teacher, you should teach 288 because that is the modern interpretation with the more widespread use of computers.

#### BobFL

##### Hall of Fame
Hidious, machines REQUIRE certain rules to be followed when you input values and define equations. That is what separates humans from the rest of the known universe. We do not operate like that and we do not come with a manual as opposed to your cute little Sharp

#### dr325i

##### G.O.A.T.
She is strikingly good looking, she can do whatever she wants
I would say...the looks are subjective...and leave on that

#### dr325i

##### G.O.A.T.
I'm also doing advanced math on a daily basis, soon to be teacher, and, even thought this could be wrong, i would also have said 2.

Why? Because i too see a dinstinction between 48÷2(9+3) and 48÷2*(9+3). When a coefficient is linked to another value without the * sign, i will naturally assume they cannot be separated. For example, if you replace (9+3) with C, you get 48÷2C which is different from 48÷2*C.

I'm not saying anybody is wrong here, just that it's completly natural to assume 48÷2(9+3)=2 and not even close to being idiotic, uneducated or dumb.

Anyboding changing the notation to prove their point (48*1/2*(9+3) or entering 48/2*(9+3) in a program doesn't prove anything because THE NOTATION IS THE PROBLEM. If you change it, the problem disappears.
Mr. Teacher,

I would never call someone uneducated for "assuming" what you mentioned above, but when they keep arguing...
Again, the notation is NOT a problem, it may be assumeed to be a problem.
a*b = ab, the end.

As for your "calculator proof", ifyou used it to prove something, it is scary that you will be teaching some kids soon

#### Hidious

##### Professional
Hidious, machines REQUIRE certain rules to be followed when you input values and define equations. That is what separates humans from the rest of the known universe. We do not operate like that and we do not come with a manual as opposed to your cute little Sharp
I don't see your point but you are right; humans are subject to what we call interpretation as opposed to machine and we can clearly see the 2 possible interpretation opposed here.
Mr. Teacher,

I would never call someone uneducated for "assuming" what you mentioned above, but when they keep arguing...
Again, the notation is NOT a problem, it may be assumeed to be a problem.
a*b = ab, the end.

As for your "calculator proof", ifyou used it to prove something, it is scary that you will be teaching some kids soon
I'm sorry but for me and a lot of people, a/bc is not the same as a/b*c.
And how is the way my calculator was programmed a proof of anything? Where did i say that?

I'll also add Prague to my "avoid list", unbelievable how unpleasant some people can be.

#### kevoT

##### Semi-Pro
It's 2. You add the two in the brackets and write it underneath. You then divide 48 by 2 which is equal to 24 (write that under neath 48/2). Then you have the equation to work out the answer, which is 24/12, and that is is equal to 2.

#### cucio

##### Legend
What you are saying is that MD is wrong because a/b*c will become a/bc instead of ac/b

That is why the variant PEMDAS of PEDMAS is wrong, and a previous poster already made that mistake.
DM on its own can also be wrong when you don't couple it with a strict left-to-right rule, that's what the acronym PEDMAS/PEMDAS painfully obviates after you have forgotten the small letter. Subtraction and division operators are not commutative, so order is vital.

E.g.: 4 / 2 / 2

(4 / 2) / 2 != 4 / (2 / 2)

That's why what you call the "old method" is suboptimal (if such method exists, I have never seen it used anywhere, even by old teachers or old books): the moment you introduce a second division in the expression you have to set a precedence order (left to right). It is easier to teach and remember (and automate) positional precedence from the get go, rather than adding it as an afterthought when things start getting ambiguous.

Arithmetic operations are binary, in the end you need to precisely specify which two values you must operate. Implicit rules of precedence are only syntactic sugar to save ink (simplify codification and decodification) omitting implicit parentheses that define the operators precisely. As any other syntax, it is not God given, these are rules that must be agreed among a community for the sake of efficient and precise communication.

#### kevoT

##### Semi-Pro
Does anyone else see that 69 people voted for 288 ;P That makes me worried, though...

#### sureshs

##### Bionic Poster
It is about how much someone is used to computer notation. There is no "2 line" fraction as compilers parse symbols in a linear sequence across lines, so a / needs to be followed by parantheses for the same effect. There is no such thing as ab, only a*b, because compilers parse ab as a variable whose name is ab. Some languages like C do not have exponential operators, so a^b or a**b is not available, only the math library function pow(a,b).

#### sureshs

##### Bionic Poster
DM on its own can also be wrong when you don't couple it with a strict left-to-right rule, that's what the acronym PEDMAS/PEMDAS painfully obviates after you have forgotten the small letter. Subtraction and division operators are not commutative, so order is vital.

E.g.: 4 / 2 / 2

(4 / 2) / 2 != 4 / (2 / 2)

That's why what you call the "old method" is suboptimal (if such method exists, I have never seen it used anywhere, even by old teachers or old books): the moment you introduce a second division in the expression you have to set a precedence order (left to right). It is easier to teach and remember (and automate) positional precedence from the get go, rather than adding it as an afterthought when things start getting ambiguous.

Arithmetic operations are binary, in the end you need to precisely specify which two values you must operate. Implicit rules of precedence are only syntactic sugar to save ink (simplify codification and decodification) omitting implicit parentheses that define the operators precisely. As any other syntax, it is not God given, these are rules that must be agreed among a community for the sake of efficient and precise communication.
I did mention that in a previous post. Also, that exponentiation goes right to left, so a further clarification is needed.

#### sureshs

##### Bionic Poster
Does anyone else see that 69 people voted for 288 ;P That makes me worried, though...
Why, because they are right?

#### sureshs

##### Bionic Poster
No because they're wrong...
You have a lot to catch up on this thread - it is 20 pages already

#### dak95_00

##### Hall of Fame
I'm also doing advanced math on a daily basis, soon to be teacher, and, even thought this could be wrong, i would also have said 2.

Why? Because i too see a dinstinction between 48÷2(9+3) and 48÷2*(9+3). When a coefficient is linked to another value without the * sign, i will naturally assume they cannot be separated. For example, if you replace (9+3) with C, you get 48÷2C which is different from 48÷2*C.

I'm not saying anybody is wrong here, just that it's completly natural to assume 48÷2(9+3)=2 and not even close to being idiotic, uneducated or dumb.

Anyboding changing the notation to prove their point (48*1/2*(9+3) or entering 48/2*(9+3) in a program doesn't prove anything because THE NOTATION IS THE PROBLEM. If you change it, the problem disappears.

That calculator is the problem. Who uses a Sharp brand calculator? Where do you buy one of those? There is no distinction between 48/2(9+3) and 48/2*(9+3).

There is a distinction between 48/[2(9+3)] and 48/2(9+3).

Other than that, it appears that most don't feel that division is as important as multiplication or that they're equal.

As a teacher (one who knows the answer is 288), I like to use the demonstration of smashing a bag of chips. My question always follows: Are there more chips? Multiplication or division? Both are simple questions w/ simple answers. There are more chips and it is division. Is there more to eat? No! It is possible to divide and get a larger number because when you divide, you multiply by the reciprocal. Dividing by 2 is the same as multiplying by 1/2. Therefore, 48*1/2*(9+3) does follow ALL of the laws of mathematics which state that you must complete multiplication or division from left to right by doing whichever comes first; in this case division.

#### dak95_00

##### Hall of Fame
Wow! Look another computer problem. Why does TT sometimes turn an 8 into a smiley icon? It should've read 288!

#### kevoT

##### Semi-Pro
You have a lot to catch up on this thread - it is 20 pages already
Sorry! just realised that if there is no symbol between a number and a bracket it should be *. So in that case it would be 24 x 12 which is equal to 288.

#### olliess

##### Semi-Pro
As a teacher (one who knows the answer is 288 ),
Being a teacher, have you never encountered the convention in some mathematics literature of treating the bc in a / bc as the denominator of a fraction?

48*1/2*(9+3) does follow ALL of the laws of mathematics which state that you must complete multiplication or division from left to right by doing whichever comes first; in this case division.
Notation conventions don't have a huge amount to do with the "laws" of mathematics, do they?

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#### tennisnoob3

##### Professional
Ah, I forgot that Harvard is the only "a halfway decent school"...
Your logic, just like "with *" and "without" is just...disturbing...

Give up, becoming too embarrassing...

Again, this is a 3rd (in the US) and 2nd (in Europe) grade level math... Most of us should not even discuss it...
are you trying to say europe is more advanced than usa in elementary school? thats very false. USA is #1 in elementary education, they lag in high school years.

#### Hidious

##### Professional
That calculator is the problem. Who uses a Sharp brand calculator? Where do you buy one of those? There is no distinction between 48/2(9+3) and 48/2*(9+3).

There is a distinction between 48/[2(9+3)] and 48/2(9+3).

Other than that, it appears that most don't feel that division is as important as multiplication or that they're equal.

As a teacher (one who knows the answer is 288), I like to use the demonstration of smashing a bag of chips. My question always follows: Are there more chips? Multiplication or division? Both are simple questions w/ simple answers. There are more chips and it is division. Is there more to eat? No! It is possible to divide and get a larger number because when you divide, you multiply by the reciprocal. Dividing by 2 is the same as multiplying by 1/2. Therefore, 48*1/2*(9+3) does follow ALL of the laws of mathematics which state that you must complete multiplication or division from left to right by doing whichever comes first; in this case division.
Now the calculator is the problem and you're trying to make me look bad for owning a sharp calculator? It's not a problem, it was programmed that way because many people actually favor the convention a / bc. What kind of calculators do you use? Is Texas Instrument good enough?

Both answers can be justified. As a teacher, you should accept that instead of beeing so close minded.

#### tenis1

##### Banned
thats what i would do?

calculators read left to right which confuses things
Lol. Your math skills are worse than your poll making skills. Unbelievable.

##### Legend

Both answers can be justified. As a teacher, you should accept that instead of beeing so close minded.
I've also seen the old TI-82s give the answer that the TI-85. I got curious and decided to check on my TI-30X. Sure enough, it gave 288. I think Texas Instruments has fixed the problem, and all their new calculators would give 288 as an answer.

I'm used to seeing this variation of the problem each semester. What is 12/2*6? The first time I answered 1, but the answer in the back of the student's text was 36. So since then, I always say the answer is 36. I could say that my answer is 1, and the textbook is wrong b/c blah blah blah, but I know the students wouldn't accept it. This kind of question is only asked in very basic math classes, but the correct answer 288. Students won't like it if you say the answer is 2 when the correct (accpeted) answer is is 288.

For that reason the answer is 288. It has nothing to to with being closemanded.

#### olliess

##### Semi-Pro
I'm used to seeing this variation of the problem each semester. What is 12/2*6? The first time I answered 1, but the answer in the back of the student's text was 36. So since then, I always say the answer is 36...For that reason the answer is 288. It has nothing to to with being closemanded.
As I'm sure you can see, your example as stated does not raise the same questions of notation that this thread seems to be about.

##### Legend
As I'm sure you can see, your example as stated does not raise the same questions of notation that this thread seems to be about.
Sorry to let you down, but they seem to be about the same thing, i.e. many assume that PEMDAS means that multiplication comes BEFORE division, so they replace the 2*6 with a 12. Others, going left to right divide 12 by 2, get a 6 and multiply that 6 by the other 6.

And, again, a TI-83/4 will display 36, while the old TI-82 will display 1. This kind of question rarely comes up in math higher than algebra though, because textbooks use a horizontal fraction bar instead of a forward slash, this makes it clear what's in the numerator and what's in the denominator.

#### olliess

##### Semi-Pro
Sorry to let you down, but they seem to be about the same thing, i.e. many assume that PEMDAS means that multiplication comes BEFORE division, so they replace the 2*6 with a 12. Others, going left to right divide 12 by 2, get a 6 and multiply that 6 by the other 6.
Before you say you're sorry, be clear that the ambiguity I perceive is whether the implicit multiplication has higher precedence than "ordinary" multiplication, not whether ordinary multiplication has higher precedence than division (which I do not take to be true).

I also do not see how PEMDAS settles this question definitively.

And, again, a TI-83/4 will display 36, while the old TI-82 will display 1. This kind of question rarely comes up in math higher than algebra though, because textbooks use a horizontal fraction bar instead of a forward slash, this makes it clear what's in the numerator and what's in the denominator.
The forward slash is very common well beyond the algebra level, and I readily find examples in textbooks (two examples: Boyce and DiPrima, Diff. Eq..., and Bender and Orszag, Advanced Mathematical Methods...) as well as mathematical papers which DO clearly expect the reader to give the "juxtaposed" multiplication the higher precedence.

By the way, this thread is really quite fascinating, not because I particularly care about the problem, but just seeing the variety of responses and the level of "evidence" on which people seem to be forming (extremely strong) opinions one way or the other...

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##### G.O.A.T.
Interesting question, and I have not read every post, but I tend to agree with Hidious here. On the surface, the answer is 288, but depending on the convention for a particular computing system, omission of a multiplication sign can be construed as grouping (i.e., an extra set of parentheses around the multipliers), and then the answer is 2. Personally I would not condone the latter practice, so 288 is easier to defend.

##### Legend
Before you say you're sorry, be clear that the ambiguity I perceive is whether the implicit multiplication has higher precedence than "ordinary" multiplication, not whether ordinary multiplication has higher precedence than division (which I do not take to be true).

I also do not see how PEMDAS settles this question definitively.
Despite several readings, I don't know what you mean in the top paragraph.

As to the second, PEMDAS does NOT settle the question. Some math teachers I know hate PEMDAS and use GEMAL because of situations like this. GEMAL has only one M, it cover both multiplying and division, now there isn't a misleading feeling that multiplying should always be done before dividing. I'm not a PEMDAS enthusiast either.

BTW, GEMAL stands for (G)rouping (E)xponents (M)ultiplying (A)dding (L)eft to right. Grouping is better than parentheses because it includes braces, brackets and parentheses. Not only does the M cover multiplying and dividing in GEMAL, but the A covers adding and subtracting. GEMAL is less redundant. Finally, I don't say the answer is 288 because I like that answer better. It's not my choice. I know that if you're taking a beginning math class, your teacher will count 288 as right and mark you down for saying 2. It's not my choice, I'm just the messenger.

#### olliess

##### Semi-Pro
Before you say you're sorry, be clear that the ambiguity I perceive is whether the implicit multiplication has higher precedence than "ordinary" multiplication, not whether ordinary multiplication has higher precedence than division (which I do not take to be true).

I also do not see how PEMDAS settles this question definitively.
Despite several readings, I don't know what you mean in the top paragraph.
Nothing more than what people have been discussing for pages and pages: is the "implied" multiplication between the 2 and the (9+3) understood to have higher precedence than "ordinary" multiplication, e.g., 48/2*(9+3), which is more like the example you posted.

GEMAL stands for (G)rouping (E)xponents (M)ultiplying (A)dding (L)eft to right. Grouping is better than parentheses because it includes braces, brackets and parentheses. Not only does the M cover multiplying and dividing in GEMAL, but the A covers adding and subtracting.
The question isn't whether M comes before or after (or at the same time as) D in PEMDAS -- put another way, it's whether "multiplication by juxtaposition" falls under (G)rouping or (M)ultiplying in GEMAL.

Finally, I don't say the answer is 288 because I like that answer better. It's not my choice. I know that if you're taking a beginning math class, your teacher will count 288 as right and mark you down for saying 2. It's not my choice, I'm just the messenger.
In light of the differing conventions which seem to be out there, the teacher should probably be marked down for posing the question that way.**

**) Even in the case that the teacher said, "Hey class, I know it is sometimes taken the other way, but we are going to stick to this convention, which also happens to be the one in your textbook..." I would still wonder why the question was asked in the first place.

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##### Legend
In light of the differing conventions which seem to be out there, I'd give the teacher poor marks for posing a question like that.
The local textbook asks it. And I agree with you, it's a poor question. The textbook teaches PEMDAS and then, almost immediately, tries to trick them with this contrived question.

But I tutor math on the side, and I've seen this one for a long time. My tutees want to know the answer that the teacher will like, so that's how I coach them. To them, correct = what the teacher believes.

#### F-T-S

##### Rookie
Keep reading on the next page at that site, http://www.purplemath.com/modules/orderops2.htm, where a similar example is treated. Notice that the convention chosen there is the one that gives 2 in the current problem -- that is, the opposite convention of what some of you think are the "rules of mathematics."
Interesting, you're right. Here's the part in question:

This next example displays an issue that almost never arises but, when it does, there seems to be no end to the arguing.

Simplify 16 ÷ 2[8 – 3(4 – 2)] + 1.
16 ÷ 2[8 – 3(4 – 2)] + 1
= 16 ÷ 2[8 – 3(2)] + 1
= 16 ÷ 2[8 – 6] + 1
= 16 ÷ 2[2] + 1 (**)
= 16 ÷ 4 + 1
= 4 + 1
= 5

The confusing part in the above calculation is how "16 divided by 2[2] + 1" (in the line marked with the double-star) becomes "16 divided by 4 + 1", instead of "8 times by 2 + 1". That's because, even though multiplication and division are at the same level (so the left-to-right rule should apply), parentheses outrank division, so the first 2 goes with the [2], rather than with the "16 divided by". That is, multiplication that is indicated by placement against parentheses (or brackets, etc) is "stronger" than "regular" multiplication. Typesetting the entire problem in a graphing calculator verifies this hierarchy:

Note that different software will process this differently; even different models of Texas Instruments graphing calculators will process this differently. In cases of ambiguity, be very careful of your parentheses, and make your meaning clear. The general consensus among math people is that "multiplication by juxtaposition" (that is, multiplying by just putting things next to each other, rather than using the "×" sign) indicates that the juxtaposed values must be multiplied together before processing other operations. But not all software is programmed this way, and sometimes teachers view things differently. If in doubt, ask!

(And please do not send me an e-mail either asking for or else proffering a definitive verdict on this issue. As far as I know, there is no such final verdict. And telling me to do this your way will not solve the issue!)
I have a feeling this may be the older thought on it.

#### jaggy

##### Talk Tennis Guru
Happy to have been an English major and the answer is 288

##### Legend
As I'm sure you can see, your example as stated does not raise the same questions of notation that this thread seems to be about.
This is about 12/2*6 being different from 12/2(6).

I can see that it makes a difference to some calculators. Some claim that it matters in their industry. All I can say is I've never seen a textbook claim that multiplying comes before dividing when there's a parentheses but not with a times sign.

P.S. My TI-84 doesn't distinguish.

#### krz

##### Professional
purplemath.com

And please do not send me an e-mail either asking for or else proffering a definitive verdict on this issue. As far as I know, there is no such final verdict
I wouldn't exactly take what she says as fact. As she herself states that she's never seen proof either way. Things getting even more blurry now.

She does have a M.A. in Math from Washington Univ at least.

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#### BreakPoint

##### Bionic Poster
What argument? By your definition, everything after / has to be in the denominator. That * does jack.
Wrong. The * clearly separates the equation into two operatives. The * is easily typed with a computer keyboard. So if you don't use it, it means you didn't want it there and the accepted convention of an equation like 48÷2(9+3) is that the (9+3) is in the denominator because it is to the right of the ÷ sign or the / sign and next to the 2 with no mathematical operative sign between them.

Like I said earlier, if the (9+3) was in the numerator, the equation would have been written: 48(9+3)÷2. But it wasn't, was it?

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#### BreakPoint

##### Bionic Poster
did you get the joke? i was referring to fedace's standford, something you should know about with 30k posts. halfway decent school doesnt even make sense(halfway to where?). you probably went a lower ivy anyway or donated a building, big whoop.
What the heck is a "lower Ivy"????

Oh yeah, you must mean one of those schools you had zero chance to get into? OK, I see.

Oh, and what does either "Standford" or Stanford have to do with Ivy schools? Oh, yeah, nothing! :???:

#### BreakPoint

##### Bionic Poster
Ah, I forgot that Harvard is the only "a halfway decent school"...
Your logic, just like "with *" and "without" is just...disturbing...

Give up, becoming too embarrassing...

Again, this is a 3rd (in the US) and 2nd (in Europe) grade level math... Most of us should not even discuss it...
Then I guess you must still be in the 1st grade then. :lol: LOL

##### Hall of Fame
This internet forum is killing you guys.

#### BreakPoint

##### Bionic Poster
Again, the notation is NOT a problem, it may be assumeed to be a problem.
a*b = ab, the end.
And a/bc = a/bc and NOT ac/b. The end.

#### dr325i

##### G.O.A.T.
And a/bc = a/bc and NOT ac/b. The end.
WHAT????
You have no clue about the BASIC math...
Or, you are confused about the "*" sign that is exactly the same function if shown or not.
Similarly as a+b-c = a-c+b = b-c+a
a/bc = a/b*c = ac/b = a*c/b

#### dr325i

##### G.O.A.T.
I don't see your point but you are right; humans are subject to what we call interpretation as opposed to machine and we can clearly see the 2 possible interpretation opposed here.

I'm sorry but for me and a lot of people, a/bc is not the same as a/b*c.
And how is the way my calculator was programmed a proof of anything? Where did i say that?

I'll also add Prague to my "avoid list", unbelievable how unpleasant some people can be.
WOW, what an irony...
Avoiding the city/country because of pointing out your flaws. This just re-inforces my argument above -- please do not teach our kids and put them deeper in the ****-hole... Telling them to avoid the Country because someone challenged the views will create many little George W's around the USA and we DO NOT need that crap!
I am an American Expat by the way, maybe you should move to Argentina?

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#### BreakPoint

##### Bionic Poster
I'm used to seeing this variation of the problem each semester. What is 12/2*6? The first time I answered 1, but the answer in the back of the student's text was 36.
But that equation has a * sign, which means the 6 can be in the numerator.

What if the equation was 12/2(6)? Which is more like the equation under discussion here.

#### BreakPoint

##### Bionic Poster
WHAT????
You have no clue about the BASIC math...
Or, you are confused about the "*" sign that is exactly the same function if shown or not.
Similarly as a+b-c = a-c+b = b-c+a
a/bc = a/b*c = ac/b = a*c/b

Wrong. If bc is in the denominator, most engineering textbooks write it as a/bc. The / is the horizontal divide line and bc is below this line, making it the denominator.

How can you be so sure that the c in a/bc is in the numerator? You can't!

#### dr325i

##### G.O.A.T.
Wrong. If bc is in the denominator, most engineering textbooks write it as a/bc. The / is the horizontal divide line and bc is below this line, making it the denominator.

How can you be so sure that the c in a/bc is in the numerator? You can't!
be sure???
Do you know the difference between the rule and assumption.
This is the rule.
What you're talking about is: a/(bc) = a/(b*c) = a/b(1/c) = a/b*(1/c)

#### dr325i

##### G.O.A.T.
But that equation has a * sign, which means the 6 can be in the numerator.

What if the equation was 12/2(6)? Which is more like the equation under discussion here.
"*" or lack of it is confusing you. However, in both cases, the outcome is the SAME!

#### BreakPoint

##### Bionic Poster
be sure???
Do you know the difference between the rule and assumption.
This is the rule.
What you're talking about is: a/(bc) = a/(b*c) = a/b(1/c) = a/b*(1/c)
And exactly how many engineering textbooks have you studied? I thought so.

What you're saying is that 48(9+3)÷2 = 48÷2(9+3), which is not true. One has (9+3) above the horizontal divide line and the other has (9+3) below the horizontal divide line.

##### Professional

That's what you get for use an unreleased software, 2011a is just out and u are using 2011b?? My 2009b tells me it's 288 which is correct.