A problem that hit the Internet in early 2011 is, "What is the value of 48/2(9+3) ?"

Depending on whether one interprets the expression
as (48/2)(9+3) or as 48/(2(9+3)) one gets 288 or 2.
There is no standard convention as to which of these two ways
the expression should be interpreted, so, in fact, 48/2(9+3)
is ambiguous.
To render it unambiguous, one should write
it either as (48/2)(9+3) or 48/(2(9+3)).
This applies, in general, to any expression of the
form *a/bc* : one needs to insert
parentheses to show whether one means
(*a/b*)*c* or *a/*(*bc*).

In contrast, under a standard convention,
expressions such as *ab*+*c* are
unambiguous: that expression means only (*ab*)+*c*; and
similarly, *a*+*bc* means only
*a*+(*bc*).
The convention is that when parentheses are not used to
show the contrary, multiplication precedes addition (and
subtraction); i.e., in *ab*+*c*, one *first* multiplies
out *ab*, *then* adds *c* to the result, while
in *a*+*bc*, one first multiplies
out *bc*, then adds the result to *a*.
For expressions such as
*a*−*b*+*c*, or
*a*+*b*−*c*, or
*a*−*b*−*c*, there is also a
fixed convention, but rather than saying that one of addition
and subtraction is always done before the other, it says that when one
has a sequence of these two operations, one works from left to
right: One starts with *a*, then adds or
subtracts *b*, and finally adds or subtracts *c*.

Why is there no fixed convention for interpreting expressions
such as *a/bc* ?
I think that one reason is that historically, fractions were
written with a horizontal line between the numerator and denominator.
When one writes the above expression that way, one either puts
*bc* under the horizontal line, making that whole product the
denominator, or one just makes *b* the denominator and
puts *c* after the fraction.
Either way, the meaning is clear from the way the
expression is written.
The use of the slant in writing fractions is convenient
in not creating extra-high lines of text;
but for that convenience, we pay the price of losing
the distinction that came from how the terms were arranged
horizontally and vertically.

Probably another reason why there is not a fixed convention for order of multiplication and division, as there is for addition and subtraction, is that while people frequently do calculations that involve adding and subtracting lengthy strings of numbers, the numbers of multiplications and divisions that come into everyday calculations tends to be smaller; so there is less need for a convention, and none has evolved.

Finally, the convention in algebra of denoting multiplication by
juxtaposition (putting symbols side by side), without any multiplication
symbol between them, has the effect that one sees something
like *ab* as a single unit, so that it is natural to
interpret *ab*+*c* or
*a*+*bc* as a sum in which one of the summands is
the product *ab* or *bc*.
Without that typographic convention, the order-of-operations
convention might never have evolved.
When one has numbers rather than letters, one can't use juxtaposition,
since it would give the appearance of
a single decimal number, so one must insert a symbol
such as ×, and there is less natural
reason for interpreting 2 × 3 + 4 as
(2 × 3) + 4 rather than 2 × (3 + 4), but
I suppose that we do so by extension of the convention that
arose in the algebraic context.
Likewise, because addition and subtraction constitute one
"family" of operations, and multiplication and division another,
and perhaps also because the slant "/" doesn't seem to separate two
expressions as much as a + or − does, we are ready
to read *a/b*+*c* etc. as involving division
before addition.
But when it comes to *a/bc*, where the operations belong
to the same family, the left-to-right order suggests
doing the division first, while the "unseparated letters"
notation suggests doing the multiplication first; so
neither choice is obvious.

It is interesting that in the 48/2(9+3) problem, the last element was written 9+3 rather than 12. If the latter had been used, it would have been necessary to insert a multiplication sign, 48/2×12, and I would guess that a large majority of people would have then made the interpretation (48/2)×12. Perhaps we will never know where this puzzle originated; perhaps it was cunningly designed so that one interpretation would seem as likely as the other; or perhaps it came up as a real expression that someone happened to write down, not thinking of it as ambiguous, but that other people did have trouble with.

From correspondence with people on the the 48/2(9+3) problem,
I have learned that in many schools today, students are taught
a mnemonic "PEMDAS" for order of operations: Parentheses, Exponents,
Multiplication, Division, Addition, Subtraction.
If this is taken to mean, say, that addition should
be done before subtraction, it will lead to the wrong answer
for *a*−*b*+*c*.
Presumably, teachers explain that it means "Parentheses
— then Exponents — then Multiplication and Division
— then Addition and Subtraction",
with the proviso that in the "Addition and Subtraction" step,
and likewise in the "Multiplication and Division" step,
one calculates from left to right.
This fits the standard convention for addition and subtraction,
and would provide an unambiguous interpretation
for *a/bc*, namely, (*a/b*)*c*.
But so far as I know, it is a creation of some educator, who has
taken conventions in real use, and extended
them to cover cases where there is no accepted convention.
So it misleads students; and moreover, if students are
taught PEMDAS by rote without the proviso mentioned above, they
will not even get the standard interpretation
of *a*−*b*+*c*.

*Should* there be a standard convention for the relative order
of multiplication and division in expressions where division
is expressed using a slant?
My feeling is that rather than burdening our memories with
a mass of conventions, and setting things up for misinterpretations
by people who have not learned them all, we should learn how to
be unambiguous, i.e., we should use
parentheses except where firmly established conventions exist.
If expressions involving long sequences of multiplications and
divisions should in the future become common, then there may
be a movement to introduce a standard convention on this point.
(A first stage would involve individual authors writing that
"in this work", expressions of a certain form will have a
certain meaning.)
But students should not be told that there is a convention
when there isn't.

Incidentally, it is worth noting that in certain cases,
no convention is needed.
The meaning of *a*+*b*+*c* is unambiguous
even without the "left-to-right" convention, by the associativity
of addition, and similarly *abc* by
associativity of multiplication.
By further properties of the operations, the values
of *a*+*b*−*c* and
*ab/c* come out the same whichever order one uses.
In contrast, *a*−*b*+*c* and
*a*−*b*−*c* require
the "left-to-right" rule,
while in the absence of a corresponding rule for
multiplication and division, *a/bc* (as discussed above),
and likewise *a/b/c*, are ambiguous.