Fractions with expressions

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mmarcosbarbosa

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We have some discussions about the expression in the picture.

13021

What is the correct answer?
Option d) or option e)?

Please, justify your answer.
 
The only real "discussion" or "debate" to be had here is a philosophical one. At its core, this problem (as well as the "only 6% of people are smart enough to figure this out" nonsense posts that circulate on Facebook) is one of Order of Operations. Specifically, we can ask (1) is multiplication where we explicitly write the times operator a different thing from multiplication where we don't, and (2) does one way have a higher "priority" than the other way? One camp says that multiplication, when written without a symbol, causes the terms to become "grouped" together as one unit that must be "untangled" first, such that it is given higher priority when evaluating according to PEMDAS. The other camp says that multiplication is multiplication is multiplication, whether or not there's a symbol.

I'm firmly in the latter camp, and I believe the first camp to be wrong and completely misguided. The nature of order of operations requires everybody to be operating on the same page, and the people who designed calculators made the choice that multiplication without a symbol is the same as multiplication with a symbol. Consult with WolframAlpha to see what it says. You can also try inputting it in your own calculator and see that that answer agrees with WolframAlpha.
 
Unfortunately, although there should be a consensus on this, there is not, and never has been. Though most calculators today treat all multiplications alike, there have been some that did not. Though most textbooks in my experience don't distinguish a multiplication written with a symbol from one implied by juxtaposition, some have done so. Some have reported that among mathematicians one rule is standard, and others have reported the opposite. The reality, in fact, is that we almost always use the fraction bar rather than the division sign in the first place, and with good reason.

So in my opinion, though I might state a personal and logical preference, the only useful answer is to refuse to write this sort of expression, which is effectively ambiguous even if it should not be. Therefore the question itself, rather than any particular answer, is wrong. You should simply ignore such posts.
 
Hello Marcos. I'm adding a comment about multiplications and divisions together. After we finish evaluating inside the square brackets, we have:

24 ÷ 8[4]

We see both a division and a multiplication. The Order of Operations tells us to do multiplications and divisions in the order they appear, reading from left to right.

Therefore, we do the division first, and the final result is 12.

If the author intends for people to do the multiplication first (instead), then they would need additional grouping symbols to indicate that, and the final result would be 3/4:

(24 ÷ 8)[1 - (-2-4) ÷ 2]

?
 
I agree with all that Prof Peterson posted. I have always found the to be as useless questions as question which ask for the next term in some sequence. Thankfully, most testing companies have given up using either type.
 
I am not a professional mathematician, but symbols have meaning only if there is an extra-personal consensus about what the meaning should be. The correct answer depends upon how most people in the relevant community interpret the symbols. If people believe that "gluglog" means a fruit that, when ripe, has a yellow, peelable skin, you are very unlikely to get the fruit you desire if you ask for a "banana."

Questions about the meaning of symbols are really questions about implicit social agreements in a specific community.
 
JeffM said:
I am not a professional mathematician, but symbols have meaning only if there is an extra-personal consensus about what the meaning should be. The correct answer depends upon how most people in the relevant community interpret the symbols. If people believe that "gluglog" means a fruit that, when ripe, has a yellow, peelable skin, you are very unlikely to get the fruit you desire if you ask for a "banana."
Questions about the meaning of symbols are really questions about implicit social agreements in a specific community.
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The point is there is no consensus as to the order of operations period.
I can list at least four computer languages that use different conventions. There is no agreement among calculator brands.
There is even no agreement here on this site. I have been called a bias person by a know-nothing member for using \(\displaystyle \log(x)\) instead of \(\displaystyle \ln(x)\). That is been standard notation in the calculus reform movement for fifty years. That is a know-nothing comment.
 
In the case of "log", there are different "consensuses" in different communities and for different purposes. I would say that it is "bias" to insist on the convention of your own community (even if it's the best community!) in communications with a person of another. (Just as it would be rude to tell a British student his spelling and vocabulary are wrong because they don't match my American expectations - but it's okay to use my own spellings in writing back, as long as it doesn't cause confusion.)

At the high school level it has long been standard to use "log" for base 10 and "ln" for base e (at least in part because base 10 is easiest for beginners to understand, and formerly was needed in tables); calculators have surely helped to maintain this tradition. My understanding is that in calculus, base e is the default, while in programming, base 2 is the default. But in any classroom, the textbook's notation is the default (unless the teacher chooses to overrule it, and standardized tests don't make it necessary). I generally try to use the language of a student who is asking the question.

Back to the topic, this is also somewhat true of the order of operations. Although I prefer textbooks that never use a combination of multiplication and division, if a student asks a question about it based on a classroom problem, I'll want to find out what convention is used there. (I've heard of inconsistent textbooks that state one rule but follow another in exercises. That's another issue.) When a problem has no such context (like all those internet nonsense questions), it is the lack of context or community that is the cause of the arguments.
 
pka said:
The point is there is no consensus as to the order of operations period.
I can list at least four computer languages that use different conventions. There is no agreement among calculator brands.
There is even no agreement here on this site. I have been called a bias person by a know-nothing member for using \(\displaystyle \log(x)\) instead of \(\displaystyle \ln(x)\). That is been standard notation in the calculus reform movement for fifty years. That is a know-nothing comment.
Click to expand...
Actually, I think you are missing my point, which perhaps I did not make clear enough. The original question of "justify your answer" is inane. The question does not identify a specific community or standard and so could be answered ONLY if there were universal agreement. As the posts preceding mine made clear, no such universal agreement exists. (See for instance post #3.) Moreover, because the "meaning" of symbols is arbitrary, the only ultimate justification is simply social agreement (although with consciously created languages that consensus may be influenced by logical arguments related to convenience, conciseness, and clarity).

I do not see how you can have read my post as implying a universal consensus when it explicitly talks about "specific community."
 
We can sometimes discern intent. From this, one might construe the prevailing convention. One certainly should disclose the convention so construed. This can be done by the method of the next comment.

We can always show everything we did and thus, express the active intent and active convention. Or, perhaps, suggest where the issue might be discussed.

Certainly, if a particular convention is in one's established textbook, or discussed clearly in class, one should follow that convention - at least until the exam is concluded.

I have to wonder if the first response had been, "Please indicate the convention that should be used to establish a consistent result," would the useful conversation have ensued? In my view, it's a pretty useful group we have, here. :)
 
pka said:
There is even no agreement here on this site. I have been called a bias (sic) person by a know-nothing member for using \(\displaystyle \log(x)\) instead of \(\displaystyle \ln(x)\). That is been standard notation in the calculus reform movement for fifty years. That is a know-nothing comment.
Click to expand...

Of course you are a biased person! And now you have been down-graded to an irrational troll making an absurdly false statement
with "know-nothing member." That (notation) has not been "standard (exclusive) notation in the calculus reform movement
for fifty years." No, you have made the "know-nothing comment."

And while we are here, another irrational bias of yours is your insistence of using radian measures and dismissiveness of degrees notation.

You see, pka, you have shown more than once in your big arrogance and insistence that you do not know what you are talking
about. Accept you are wrong about a number of things and take a back seat. The more you are close-minded to the truth and
dig in, the more you are being left in the dust as "know-nothing."
 
As the author of this thread has marked it as solved, and the discussion appears to have devolved into insults, I'm going to close the thread.
 
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