48÷2(9+3) = ??? -- VOTE!! Poll Added

[rehmus]

I don't understand why the answer would somehow differ if the problem included unknown variables. 48/2(x+y) could just as easily be (24x+24y) as 48/(2x+2y). The use of x and y (vs. 9 and 3) don't have any bearing on the way that the question should be answered.

What I now realize is that in the absence of any authoritative rule that explicitly states 'multiplication by juxtaposition' is not given priority, most 2Heads will simply continue to argue that the answer is ambiguous. However, no true ambiguity exists. No rule speaks to the priority of multiplication by juxtaposition. Rules do, however, speak to the priority of left operators over rightward operators of equivalent priority (e.g. multiplication and division).

The answer should and would not differ if unknown variables were used, it was just put out there as an explanation of how some of us got to 2. Your first statement here makes me think you perfectly understand the ambiguity that exists (though 48/2(x+y) =(24x+24y) looks terribly ugly and wrong to me).

Algebra is a language. When we write 6 x 7, we are not physically stacking items in rows of 6 and columns of 7. We are using a language to represent that stacking (in much the same way that our mouths utter words like "dog" to represent canines-the word is not the thing itself). When a teacher wrote this problem on a test, and saw that they consistently got an answer from their students that they did not expect, they realized that there was an ambiguity. I can understand two teachers who write 48/2(x+y) on a test, with one of them expecting (24x+24y) and the other 48/(2x+2y). Can you? You're ready to call the second a bad teacher? I would, first and foremost, ask the teacher to be consistent.

This ambiguity is one that is represented by the zeitgeist that this internet meme has become, and one that has multiple people who have studied mathematics for multiple years siding with different answers. There are people with math degrees disagreeing. There are math teachers disagreeing.

How do you define ambiguity?

i think i love you.

 

[maxpower]

I don't understand why the answer would somehow differ if the problem included unknown variables. 48/2(x+y) could just as easily be (24x+24y) as 48/(2x+2y). The use of x and y (vs. 9 and 3) don't have any bearing on the way that the question should be answered.

What I now realize is that in the absence of any authoritative rule that explicitly states 'multiplication by juxtaposition' is not given priority, most 2Heads will simply continue to argue that the answer is ambiguous. However, no true ambiguity exists. No rule speaks to the priority of multiplication by juxtaposition. Rules do, however, speak to the priority of left operators over rightward operators of equivalent priority (e.g. multiplication and division).

The answer should and would not differ if unknown variables were used, it was just put out there as an explanation of how some of us got to 2. Your first statement here makes me think you perfectly understand the ambiguity that exists (though 48/2(x+y) =(24x+24y) looks terribly ugly and wrong to me).

Algebra is a language. When we write 6 x 7, we are not physically stacking items in rows of 6 and columns of 7. We are using a language to represent that stacking (in much the same way that our mouths utter words like "dog" to represent canines-the word is not the thing itself). When a teacher wrote this problem on a test, and saw that they consistently got an answer from their students that they did not expect, they realized that there was an ambiguity. I can understand two teachers who write 48/2(x+y) on a test, with one of them expecting (24x+24y) and the other 48/(2x+2y). Can you? You're ready to call the second a bad teacher? I would, first and foremost, ask the teacher to be consistent.

This ambiguity is one that is represented by the zeitgeist that this internet meme has become, and one that has multiple people who have studied mathematics for multiple years siding with different answers. There are people with math degrees disagreeing. There are math teachers disagreeing.

How do you define ambiguity?

I understand what you are saying, and yes, I recognize that to a great many people, the answer is ambiguous. However, I tend to think of ambiguity as describing a situation where there is a systematic lack of clarity (conflicting rules, for example). The way I view this situation is as one not of true ambiguity, but rather confusion.

I get why people zero in on '2' as an answer, but given the widely accepted rules in existence (and can we agree that PEMDAS is a universally accepted rule of mathematical construction?) as well as the lack of anything approaching agreement on a 'multiplication by juxtaposition' rule, I tend to think that '288' is far and away the best answer, and the one I'd give if my life depended on it. '2' is a fine proposal as the start of a discussion why multiplication by juxtaposition should be accepted as a rule, but to me, it falls short of being authoritative. A nice idea and one I can see the benefits of, but to me, still not a defensible 'right answer'.

As for the zeitgeist, I submit that it exists not because of ambiguity in the rule, but rather because the problem itself is so sloppily written. You are right that algebra is a language, one that should communicate clearly. This problem as written utterly fails to do so, and so to your question about the two teachers, I'd say that BOTH are bad =)

 

[UMich92]

*bangs head*

Hi, I'm a pension actuary. I have a BS in actuarial math from one of the top 25 institutions in the land (ok, now it's 29, I think, but it was 24 when I was there). I got a perfect score on my math SAT way back when.

The answer is 288.

You people that are saying 2 are putting the 12 in parenthesis as more important than the order of operations, when it's simply saying multiplication. Multiplication using parenthesis is just another way of writing it. Does writing it with an x take precedence over writing it with a dot? No.

48/2(9+3)
is the same as saying
48/2*(9+3)
48/2*12
According to order of operations, multiplication and division are equal, (as are addition and subtraction) and should be read from left to right. This also explains why, when I've seen this post, people have PEDMAS OR PEMDAS.
Thus, 48/2 = 24 * 12 = 288.

My math degree says 288. Multiplication and division are equals in order of operations as are addition and substraction. Division is multiplication of an inverse and subtraction is addition of a negative.

The formula 48÷2(9+3) can be rewritten as 48x(1/2)(9+3) by substituting x(1/2) for ÷2. And 48 x 0.5 x 12 = 288.

2 out of 2 University of Michigan actuaries know that the answer is 288.

And in response to the question about how one would evaluate 48÷2(X + Y), the process remains the same. Parentheses first, then exponents, then multiplication and division ordered left to right, then addition and subtraction.

= 48÷2(X + Y)
= 24(X + Y)
= 24X + 24Y
= 24 x 9 + 24 x 3
= 216 + 72
= 288

 

[cgilmo]

This thread makes my head hurt.

 

[bradical]

Time to move this sucker to the Hall of Shame.

I have my acceptance speech already on my note cards.