Here is the same formula 48/2(9+3) typed in two calculators from Texas Instruments, the TI-85 and the TI-86, and the answers are different:
The TI-85 gives the answer as 2.
The TI-86 gives the answer as 288.
This topic was debated by a number of people and apparently about half the people got 2 and half the people got 288.
Pretty much everyone agrees that the expression could benefit from parentheses to clarify it, which depending on how the parentheses were applied
would give you one answer or the other.
I did a little research into this to see if there really is a right answer and thought I'd post a few things I found that might be of interest.
If you found this expression in a journal published by the American Physical Society, the expression is not ambiguous and has a defined value. Here is
the convention used for manuscripts submitted for publication in their journals:
Physical Review Style and Notation Guide (pdf) p21
That convention is prominent in physics in other literature like prominent physics textbooks:
Order of operations
the manuscript submission instructions for the Physical Review journals state that multiplication is of higher precedence than division with a
slash, and this is also the convention observed in prominent physics textbooks such as the Course of Theoretical Physics by Landau and Lif#z and the
Feynman Lectures on Physics. Wolfram Alpha changed in early 2013 to treat implied multiplication the same as explicit multiplication (formerly,
implied multiplication without parentheses was assumed to bind stronger than explicit multiplication).
I am postulating that one reason they may
teach physics students an unambiguous interpretation is to get them thinking about how to write expressions clearly. In this context, when you see the
slash as a "divide by" symbol, what appears to the left of the slash is the numerator, and what appears to the right of the slash is the
denominator. When I typed the formula into Mathway, it interpreted and displayed the formula as such:
I find it hard to interpret the expression differently from how it displays in that mathway screenshot, because of the bias in the physics textbooks
since physics was one of my majors. To put it another way, if you wanted the term (9+3) to be part of the numerator, why not write it to the left of
the slash symbol, so the expression then becomes: (9+3)48/2 or 48(9+3)/2, which again using the convention in APS journals and prominent physics texts
gives you the unambiguous answer of 288.
There is another interpretation which is pretty common that states the answer is 2, based on the implied multiplication having priority over explicit
multiplication issue mentioned in the quote above regarding a change in 2013 in Wolfram Alpha. I couldn't find much documentation on this, but one
reason it's not even an issue at the American Physical Society is, they don't even use multiplication signs for multiplication, rather they are used
only for vector products so all multiplication is the "implied" type:
48/2(9+3) - implied
48/2x(9+3) - explicit
This can make a difference in some calculators:
The answer is 2 when implied multiplication takes priority, and 288 when the multiplication is explicit and not implied.
I did find out some things about how and why people interpret it differently, and I think there are some misconceptions, which I'll mention here, by
some acronyms:
PEMDAS is common. It stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. PEMDAS is often expanded to "Please
Excuse My Dear Aunt Sally" with the first letter of each word creating the acronym PEMDAS. Canada uses BEDMAS. It stands for Brackets, Exponents,
Division, Multiplication, Addition, Subtraction...Since multiplication and division are of equal precedence, M and D are often interchanged, leading
to such acronyms as BOMDAS.
In this context, the "O" in BOMDAS means "Order" and is referring to exponents or powers, not the order in which
terms appear, and there seems to be some confusion about that which I'll come to shortly.
There is some confusion about PEMDAS that M coming before D implies it's consistent with the APR journal standards potting a higher priority on
multiplication than Division, but for some reason, the proponents of this mnemonic have chosen not to resolve such ambiguity and say that M coming
before D actually doesn't imply M has a higher priority (maybe it should, but it doesn't).
Further, I found that some people have this "left to right" mentality about the order in which operations should be performed, but I don't think
this is a sign of clear thinking and may be a sign of a fault in the educational system, as explained here by someone who has given this subject a lot
of thought:
“Order of operations” and other oddities in school mathematics
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
However this idea of left to right is actually extraneous, but at least
now I understand why so many people have this extraneous idea, because that's what they were taught:
(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.
So in conclusion, we can't say that either 2 or 288 is the wrong answer. However if you're following prominent physics textbooks or reading journals
from the American Physical Society, if you said the answer is 2, you would be right, because of the conventions followed in those sources. One reason
you may want to adopt such convention yourself even if you're not a student of physics is that you will confuse yourself less often if you follow an
unambiguous convention (like expressions to the left of the slash are the numerator and expressions to the right of the slash are the denominator). Or
if all that school stuff is behind you, maybe this can help your children if they are still in school, since the author of the "school mathematics"
source above points out we are probably not doing the best job we can of educating our children.
If you want to run this by your kids to see if they know the answer, it would be interesting to see if they even notice it's ambiguous (unless they
started studying physics in which case it might not be as ambiguous, depending on what textbook they use).