48÷2(9+3) = ???

[imsotyerred]

If you're saying the first division symbol encountered in the equation groups all operations after it, then 12/4+2=2. I'm wondering where in your third grade math book it says that.

no

__12__
....4..... +2 = 3+2 = 5

__12__
.(4+2)...= 12/6 = 2

two completely different equations

[tonywonder]

48/2(9+3)=2

(48/2)(9+3)=288

[Big Willy]

48/2*(9+3) =288
48/(2*(9+3)) =2
Order of operations for multiplication and division is left to right.

[reelizmpro]

48/2(9+3)
48/2(12)
2 is actually DISTRIBUTED into parentheses as it's the only way to get rid of parentheses at this point. Which is not the same as 2x12. Technically, it's multiplication through distribution.
48/24
2

another way to do it...

48/2(9+3)
48/(18+6) 2 is Distributed into parentheses but they remain UNTIL the two terms have been added! This takes priority over the DIVIDE symbol.
48/24
2

yet another way to do it...
48/2(9+3)
24/(9+3) expression is simplified by the factor 2. Parentheses still remain!
24/12
2

You must resist the temptation to divide into 48 without first getting rid of the parentheses.

[frankenbeemer]

A better explanation below. Note, it does throw a bone to those who would get the answer 2, but that is not a rigorous use of PEDMAS. In the absence of a strict rule that says that implied multiplication has a higher priority than regular multiplication (which he alludes to, but no one has yet supplied despite repeated requests) I would conclude that 2 is a sloppy answer.

Order of Operations

Date: 05/19/99 at 13:54:24
From: Stephanie Wu and Meghan Heil
Subject: Algebraic expressions and order of operation

The problem was presented like this:

a = 1.56
b = 1.2
x = 7.2
y = 0.2

ax/by = ?

Here are two ways that I solved it:

1) I first rewrote the problem as [1.56(7.2)/ 1.2](0.2). Second, a was
multiplied by x. The product was 11.232. Then, since no parentheses
were present, I followed the order of operations and divided 11.232 by
b, which was 1.2. The quotient was 9.36. Then I multiplied 9.36 by y,
which was 0.2. The final answer was 1.872.

2) The other way, the first thing I did was multiply a by x. The
product, which was 11.232, was set aside for the time being. Then b
was multiplied by y, which gave the product of 0.24. The problem was
now solved by dividing 11.232 (or ax) by 0.24 (or by) to reach a final
answer of 46.8.

Can you please tell us which answer is correct and why?

Date: 05/19/99 at 17:03:49
From: Doctor Peterson
Subject: Re: Algebraic expressions and order of operation

Hi, Stephanie and Meghan.

You are not alone in wondering about this. We have had several other
questions about expressions similar to yours, from confused teachers
and students who have found that different books or teachers have
different answers, and even calculators disagree.

As written, your expression

ax/by

should be evaluated left to right: a times x, divided by b, times y.
The multiplication is not done before the division, but both are done
in the order they appear. Your first solution is right.

Some texts make a rule, as in your second solution, that
multiplication without a symbol ("implied multiplication") should be
done before any other operations in an expression, including "explicit
multiplication" using a symbol. Following this rule, you would
multiply a by x, then multiply b and y, then divide one by the other.
Some (probably most) texts don't mention such a rule - but some of
those may use it without saying so, which is far worse.

I don't know of a general rule among mathematicians that implied
multiplication should be done before explicit multiplication. As far
as I'm concerned, all multiplications fit in the same place in the
order of operations. It's not an unreasonable rule, though, since it
does seem that implied multiplication ties the operands together more
tightly, at least visually; but the idea of Order of Operations (or
precedence, as it is called in the computer world) is supposed to be
to ensure that everyone will interpret an otherwise ambiguous
expression the same way - so if some texts change the rules, or if
people do what feels natural, the purpose has been lost.

The problem here is that the expression looks as if it were meant to
be

ax
----
by

In the Dr. Math FAQ about writing math in e-mail, one of our
recommendations is to use parentheses wherever possible to avoid
ambiguity, even where the rules should make it clear, because it can
be easy to forget them in some situations:

http://mathforum.org/dr.math/faq/faq.typing.math.html

(click on the Fractions link).

So in e-mail we would write it like this:

ax/(by) or (ax/b)*y

depending on what is intended.

In my research for another Dr. Math "patient," I found that some
calculators have experimented with this rule. Calculators have
somewhat different needs than mathematicians, since they have to take
input linearly, one character after another, so they are forced to
make a decision about it. On the TI Web site I learned that they
deliberately put this "feature" into the TI 82, and then took it out
of the TI 83, probably because they decided it was not a standard rule
and would confuse people. Take a look at their explanation:

http://www.ti.com/calc/docs/faq/83faq039.htm

They also talk about a similar issue for exponentiation of the form
a^b^c, and give the same conclusion we give: always use parentheses
where a statement is ambiguous without special rules:

http://www.ti.com/calc/docs/faq/83faq058.htm

So to answer your question, I think both answers can be considered
right - which means, of course, that the question itself is wrong. I
prefer the standard way (your first answer) when talking to students,
unless their own text gives the "implicit multiplication first" rule;
but in practice if I came across that expression, I would probably
first check where it came from to see if I could tell what was
intended. The main lesson to learn is not which rule to follow, but
how to avoid ambiguity in what you write yourself. Don't give other
people this kind of trouble.

- Doctor Peterson, The Math Forum

[frankenbeemer]

I didn't check links before posting. Looks like at least one is useless, sorry. Nevertheless, a good explanation of the ambiguity here. To arrive at that ambiguity you must either violate PEDMAS or add additional rules to PEDMAS.

[frankenbeemer]

yet another way to do it...
48/2(9+3)
24/(9+3) expression is simplified by the factor 2. Parentheses still remain!
24/12
2

This clearly violates PEDMAS as you must get rid of the parentheses first.

[frankenbeemer]

no

__12__
....4..... +2 = 3+2 = 5

__12__
.(4+2)...= 12/6 = 2

two completely different equations

http://mathforum.org/dr.math/faq/faq.typing.math.html Look at how to write fractions. This is you confusing ÷ with /.

[frankenbeemer]

the idea of Order of Operations (or
precedence, as it is called in the computer world) is supposed to be
to ensure that everyone will interpret an otherwise ambiguous
expression the same way - so if some texts change the rules, or if
people do what feels natural, the purpose has been lost.

If you can't produce the rule that says 2(9+3) is treated differently than 2x(9+3) or 2*(9+3), you are violating PEDMAS. Show me the exception.

[imsotyerred]

http://mathforum.org/dr.math/faq/faq.typing.math.html Look at how to write fractions. This is you confusing ÷ with /.

how can I be confusing them? they are the SAME FUCKING THING\

also, as stated previously by another member, you are not done with the parenthetical part of the equation...the 2 is attached to the (9+3).

[frankenbeemer]

how can I be confusing them? they are the SAME FUCKING THING

Hint: It is not a vinculum.

[KenC]

When you add 9 + 3 you get 12, then you are done with the parenthetical element. At that point, you return to the left because multiplication and division are equal operators. I have yet to see the rule which states that 2(some operation) is different from 2*(some operation) or 2 x (some operation).

You don't add 9+3, you distribute the 2 through and it becomes (18+6)

[frankenbeemer]

Another hint: 48÷2*(9+3) is not the same as:

48
2*(9+3)

Sorry for the poor format. It is supposed to reflect the equation written as a fraction. The fraction bar is a vinculum. All operations are to be performed under the vinculum before proceeding. Why do you treat the / symbol as a vinculum in some cases and not in others?

If you follow PEDMAS in the equation 48÷2*(9+3) do you not get 288? What rule says 48÷2(9+3) is different?

[frankenbeemer]

You don't add 9+3, you distribute the 2 through and it becomes (18+6)

Where is this rule written down? I don't see it in PEDMAS. How do you know that you must distribute this? Is it just a personal preference? What compels you to violate left to right order here if division and multiplication are equal? Where is the rule that says implied multiplication must take precedence over PEDMAS?

[frankenbeemer]

also, as stated previously by another member, you are not done with the parenthetical part of the equation...the 2 is attached to the (9+3).

What law says the 2 is attached? Is it not implied multiplication?

The info I posted above says that some textbooks treat implied multiplication as higher order. Does your third grade book do so? Please cite. Apparently, not all texts do (according to Dr, Math, whoever he is). It's ok with me if you want to do so, but I see no reason to violate PEDMAS to do it. It seems to me if you are going to ignore the PEDMAS order, you should be able to provide a reason that supercedes PEDMAS order. You have not done so.