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Saying "they are always done left to right" is citing a rule, but strictly adhering to this rule is no more likely to give you a right answer with an ambiguous expression than typing the same expression into a calculator. Here is some more background on why this is so, which was only partially excerpted in the OP:
Masterjaden
You're just straight up wrong. This is NOT a physics equation, it is an algebraic equation and in algebra, order of operations does NOT give precedence to multiplication over division and they are always done from left to right unless there is a delineation with parentheses or brackets. The answer is unambiguously 288.
Hopefully you can understand "why it is unprofitable to pursue these rules vigorously in the arithmetic context the way it is done in most school classrooms" referring to the order of operations. If the expression is ambiguous, we really need to resolve the ambiguity, which as I said some physics journals do by saying multiplication before division. I didn't say it was a physics problem, but if you remove something like the "multiplication before division" rule in physics journals, and have no other way to resolve the ambiguity, the "left to right" really isn't a satisfactory way to resolve it, which you can hopefully understand from the explanation above, but if not that's ok too, you're entitled to your opinion.
An example is the convention known as the Rules for the Order of Operations, introduced into the school curriculum in the fifth or sixth grade:
(1) Evaluate all expressions with exponents.
(2) Multiply and divide in order from left to right.
(3) Add and subtract in order from left to right.
In short, these rules dictate that, to carry out the computations of an arithmetic expression, evaluate the exponents first, then multiplications and divisions, then additions and subtractions, and always from left to right. To sixth graders, these rules must appear random and therefore meaningless, and the meaninglessness of it all gives rise to the infamous Please Excuse My Dear Aunt Sally mnemonic device.
In this short article, I will first briefly discuss the mathematical background of these Rules in order to make sense of them, and then explain why it is unprofitable to pursue these rules vigorously in the arithmetic context the way it is done in most school classrooms. Finally, I will present an argument against using assessment items in standardized tests that assess nothing more than students’ ability to commit certain definitions and conventions (like Rules for the Order
of Operations) to memory.
The mathematical origin of Order of Operations
To understand how the Rules for the Order of Operations came about, one has to consider polynomial expressions in algebra. Look at the following polynomial function of degree 8 with coefficients 17, 2, 1, 0, 0, 6, 0, 3, 4:
17x^8 + 2x^7 + x^6 + 6x^3 + 3x + 4, (1)
where x is any number. The notational simplicity of this expression must be obvious to one and all, and this simplicity is the result of a common agreement that what this expression really says is:
(17(x^8)) + (2(x^7)) + (x^6 ) + (6(x^3 )) + (3x) + 4. (2)
To be precise, the symbolic notation in (1) is free of the annoying parentheses in (2) because the convention of performing the operations in the order indicated by the parentheses in (2) is universally accepted, namely,
(A) exponents first, then multiplications, then additions.
To drive home the point, the sum 17x^8 + 2x^7 does not mean
[[(17x)^8 + 2]x]^7 ,
which would be the case if the operations were uniformly performed from left to right, but rather
(17(x^8 )) + (2(x^7))
as indicated in (2).
The rule of “multiplications before additions” may sound simple, but these three words contain more than meets the eye. Because we are in the realm of algebra, “division by a (nonzero) number c” is the same as “multiplication by
1/c”. Moreover, “minus c” is the same as “plus (−c). Therefore if one rewrites what is in statement (A) above in the language of arithmetic, then one would
have to expand it to:
(B) exponents first, then multiplications and divisions, then additions and
subtractions.
Except for the stipulation about performing the operations “from left to right”, (B) is seen to be exactly the same as the Rules for the Order of Operations. It is important to note that this stipulation about “from left to right” is entirely extraneous, because the associative laws of addition and multiplication ensure that it makes no difference whatsoever in what order the additions or multiplications are carried out.
Yes a lot of people were taught this about implicit multiplication, and if you look at the picture of the Casio calculator in the OP, it looks like that rule is programmed into the calculator. So, that's one way to resolve the ambiguity, but it's not universally taught or accepted. Where it is taught and accepted, it's certainly valid and results in the answer "2" as you say.
abecedarian
Given the aforementioned equation: 48/2(9+3)....
I was taught that implicit multiplication, i.e. 2x, takes precedence over explicit multiplication, i.e. 2 * x.
In other words, the omission of the explicit function implies the implicit function occurs first.
The argument in favor of this would be to use something like this:
y = 48 / 2x where x = 9 + 3; solve for y. (literal mutation of the equation in the OP)
You cannot solve for y without the value of x, therefore 2x is evaluated first.
The result being 48 / (2 * 12) = 48 / 24 = 2.
AthlonSavage
reply to post by Arbitrageur
The general accepted rules of mathematics Multiplication, Division, addition and subtraction is that it occurs in that order. Therefore /2 operand takes place before 9+3.
mojo2012
what is 48/2(x+y)? i typically assume that 48 is being divided by 2(x+y). I guess it should be written like this 48/[2(x+y)].
Olivine
I guess I'm just a "everything below/behind the slash is in the denominator" kind of gal.
bigfatfurrytexan
reply to post by James1982
a lot of folks, i would say the majority, don't really have occasion to continue their practice with algebraic math once they leave school and enter the workforce.
And since school isn't about understanding, but rather memorization instead, it is easy to see how things can be forgotten or become more opaque in peoples memory.
James1982
AthlonSavage
reply to post by Arbitrageur
The general accepted rules of mathematics Multiplication, Division, addition and subtraction is that it occurs in that order. Therefore /2 operand takes place before 9+3.
Absolutely incorrect.
9+3 is in parenthesis, which means you absolutely must do the addition first.
Or you can distribute out the 2 over the parenthesis and get
48 divided by (18+6) which still gives you 24 on the denominator, and two as the final answer.
There is no debate here. 2 is the correct answer. The only debate is by people who forgot how to properly do math, or by people who don't realize calculators aren't perfect and you must be liberal with parenthesis to get the correct answer.
James1982
mojo2012
what is 48/2(x+y)? i typically assume that 48 is being divided by 2(x+y). I guess it should be written like this 48/[2(x+y)].
Great way of looking at it.
That would be 48/2x+2y
x=9
y=3
Then you get 48/18+6 or 48/24 or TWO.
I can't believe some people still think it's anything other than two. Sad state of education for sure.