It’s there.
8+16 ÷ 1.5 means you do division first.
So:
8 + 10 2/3
= 18 2/3 (or 18.66666_)
There seems to be a common misconception that a ÷ sign has an implicit parentheses- in other words, that everything before it is in the numerator and everything after in the denominator.
This is erroneous but the misconception is reinforced by the use of both a ÷ and a / in the math problem. And based on the intended solution, the teacher made the same mistake.
Poorly written question all around. I asked my wife what her opinion as a teacher was; she said she’d want the kids to let he know so she can make that question not count.
How does 8+16/1.5 change things?
Order of operations says you do the division first so it would look like:
8+(16÷1.5)
(I was called out for doing that earlier)
NOTE: I agree the equation in the OP was very poorly written and clearly wrong given the answers available on multiple choice.
Oh! OK, now I get it. Hmmm, it didn’t even occur to me to do it that way. I just read it as everything to the left of ÷ divided by everything to the right. Yes, now I see how that can be confusing.
Where does the notion come from that ÷ works that way? I don’t think it’s taught that way in any schools.
No clue, this is the first time I’ve encountered it
FTR I never held this conception either. I just worked it out backwards from the OP’s assertion that the intended answer was to be a whole number.
You know Google does math: 8 + (4)^2 ÷ (7/2 - 2) - Google Search
In other words, teaching the order of operations must follow a certain necessary order of teaching the order of operations.
I would almost certainly interpret this as
2
--
3a
even knowing fully well that this isn’t strictly following the rules of order. The problem is that using either / or ÷ to write a division out on one horizontal line (like you would do in FORTRAN) just doesn’t have the same “intuitive” semantics as using the horizontal line with a numerator above and denominator below. Instead, putting two factors together with no visible operator between them, as in 3a seems to bind them more strongly than, say, writing 3×a – This is what Babale is saying in Post #48.
Algebraic notation evolved over several centuries, and it evolved the way it did to make things as clear as possible – e.g., the notation like 3a for multiplication, and the horizontal bar with numerator above and denominator below (and where the horizontal bar itself also serves as a symbol of grouping), all seems to make unnecessary the extensive use of nested parentheses. Contrast the strictly linear layout of programming languages, where layers of parentheses are more necessary, and even where not, still necessary to make expressions readable.
ETA: Reading a few more posts, I see there are questions about the use of ^ for exponentation. This again has been a convention since the early days of computer programming, where you couldn’t write a line of code with superscripts and subscripts.
Similarly, variable names in programming could be multiple letters long, so you can’t just write ab to mean a times b, you had to write a*b . (Because early computer didn’t have the × symbol either. And you had to write ** instead of ^ because they also didn’t have the ^ symbol.)
And in this case especially, if “a” was meant to be in the numerator, it would have been written as 2a/3. That it was written as 2/3a implies that something different was intended.
Mathematical notation isn’t supposed to be some kind of puzzle; it’s the expression of some underlying concept. There is a ton of ambiguity once you get to more advanced concepts, and have to use contextual clues to figure out what was meant. The underlying ideas are not ambiguous even if the notation is, or even if the notation is “wrong” by some strict standard.
I notice that Google not only does the math, but it also displays the expression with all the implied parentheses written explicitly, and even added white space around some of the operators (but not around the ^ symbol), and replaced the ÷ with /
My initial knee-jerk instinct would be to interpret it exactly the same way as Senegoid did. And then I would (hopefully) catch myself, and remind myself of the rules, and interpret it the other way.
Something that I don’t think we’ve discussed is WHY Senegoid and I would do that. I would like to suggest this as the reason: When we see “2/3a”, my instinct would be to read it aloud as “two over three a”, NOT as “two divided by three times a”.
If someone WOULD read it as “two divided by three times a”, then you have preserved the ambiguity that pushes you to the rules. In other words, when you read it as “two divided by three times a”, you must make a conscious choice between “two divided by (three times a)” and “(two divided by three) times a”.
But - and this is critical - “two over three a” doesn’t have that blatant ambiguity. And the reason why it doesn’t have such ambiguity is because “three a” is perceived as a distinct entity. What I am trying to say is that PEMDAS and it’s relatives fail to address the situation of the implied operation. When there is no visible operator at all, such as with “3a” or with “(a+b)(c+d)”, we are taught that the implied operation is multiplication. This is fairly obvious with adjoining parentheses, as in “(a+b)(c+d)”, but it is far more subtle, and too easy to forget, with “3a”.
What rules? 
Your billion-dollar satellite is a goner for sure if you assume anything.
3a is clear and unambiguous, no subtleties. The problem is with 2/3a, which is why the standard says thou shalt not write that.
I think schools should teach an entire unit on recognizing good versus poor/ambiguous notation, shortly after introducing PEMDAS or whatever they call it. They should cover alternatives like RPN at the same time. They can demonstrate several of those “gotcha” math puzzles that are all over the internet. The student should come out of it with a clear understanding about how the way we notate math is an arbitrary invention and be willing to question a teacher when handed a problem that’s ambiguous. The notation is all about communicating the underlying mathematical concept, nothing more, and nothing less. Communication can break down.
I’m not sure that every middle school student is capable of understanding this. I could see it as part of an honors hs math course or as college math but if you tried putting that in the middle school courses half your class will flunk out.
Suppose that I have an expression that, when fully typeset, looks like
2
— a
3
And I want to write that all on one line. Surely, the proper way to do that would be 2/3a, no?
And yes, I know that I could instead have written the original expression as
2a
— ,
3
which transcribes to one line as 2a/3 . But there may well be perfectly good reasons why I chose not to do that. Maybe, for instance, I have a lot of different expressions that consist of a numerical coefficient times a, and I want to call attention to their similarity (for instance, the moment of inertia of a round, symmetric object is some coefficient times mr^2, and what the coefficient is depends on the shape).
One might argue, of course, that there’s no one single universal standard for how to interpret that expression… but that’s not really true any more. Everyone nowadays uses computers (of some form) to do calculations, and nearly all modern computer system will interpret 2/3a as meaning “two thirds of a”.
And if you don’t include it in the classes, half your class will still flunk out. Oh, they might get the letter grade on their report card that lets them take the next course, but they’ll come out of the class without the essential knowledge and skills that were the purpose of taking the class.
Based on my daily experience with a heck of a lot of people, many of whom are professionals in their fields who really should know better… the percentage of people that leave school (especially if they didn’t go on to college) while retaining hardly any of the so-called “essential” knowledge you are talking about, and go on to get high paying positions at prestigious companies while lacking basic skills necessary for their job performance, is astoundingly high.
I would be shocked if even 50% of the general population could get you either 18 2/3rds or 16 from the first question; I’d be shocked if even 20% could have an intelligent conversation about why those answers are those answers and what the different chains of logic leading there were.
I don’t think that’s a problem you solve by changing school curriculums. That’s not the problem. The problem is how low American culture places “intelligence” and “education” on the great cultural lists of positive traits. Wealth, fame, attractiveness, and ability in sports all come in light-years ahead of a good education in the American consciousness. That affects how much value children (and just as importantly, their parents) place on school.
Like I said earlier – my wife has had kids interrupt her during a lecture saying “can you repeat the last 5 minutes? My mom had me LEAVE CLASS TO HELP HER BRING IN THE GROCERIES” (emphasis mine). I don’t care if your mom is disabled; if she needs your help unloading groceries, she should wait to go shopping until you aren’t in class.
Further, my wife can monitor the kids’ screens, and just in the last two weeks she’s caught this kid:
-
turn on his Webcam, then leave his computer and lie down in bed, tuck himself in, and go to sleep – all on camera with the other kids cracking up.
-
routinely watching YouTube videos when he should be working
-
pull up assignments, not touch the mouse or keyboard for all class, then turn in an empty assignment.
Despite all this, she sits through her lunch 3 times a week to tutor this kid because:
Meanwhile, that same mom is emailing the administration telling them my wife’s class, as well as her boy’s math and art(!!!) classes, are going too fast, and the kids can’t possibly keep up. And sure, Admin has been dealing with them for years and knows not to take them seriously. But how exactly is my wife supposed to get through to this kid when his own mom is telling him he can’t possibly understand it, the teacher “goes too fast” (she’s going at maybe 2/3rds speed due to Corona if not 1/2)…
And the majority of kids in her class, while not quite that bad, are much closer to the level I described than the level that would benefit from the more comprehensive deep dive you described.