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### Order of Operations Dispute

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Date: 09/26/2000 at 16:37:55
From: Terre and Robert Reidy
Subject: Order of Operations Dispute

The problem reads: N divided by (division sign)ml where n=12, m=6, and
l=3. I believe the correct answer should be .6666, as 12 divided by 18
equals this.  My husband agrees with me.

My son came home very upset from school, with a note from his teacher
that the answer was wrong. She indicated that I should have divided
the 6 (m) into 12 (n) before I divided the 3 (l) into the equation.

My son is very upset with me; his teacher told him I was doing "old
fashioned math."  Do I need to go back to school?
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Date: 09/26/2000 at 21:03:38
From: Doctor Peterson
Subject: Re: Order of Operations Dispute

Hi, Terre and Robert.

Let's write the problem as

n / ml, with n=12, m=6, l=3

I can give you some good news and some bad news. First, the bad news:
according to the usual order of operations rules now taught, your

http://mathforum.org/dr.math/faq/faq.order.operations.html

Basically, you do all multiplications and divisions before all
additions and subtractions; and when you have a string of
multiplications and divisions to do, you go from left to right in the
order they come. In your case, that means you divide n by m giving 2,
then multiply by l to get 6.

BUT...

You are not alone in your opinion. This part of the rule - doing
multiplication and division together - is probably the last rule to
have stabilized; I know that in the 1920's, at least, there was no
agreement. It seems that an agreement developed, but it is unraveling
now, as I hear from many students whose texts answer questions like
this the way you did. It appears that they are adding an unstated
rule, which seems entirely reasonable in this context, that an implied
multiplication (indicated by simply placing two variables or
expressions together, as in "ml") should be done first. It certainly
looks as if it should mean that. The problem is that, although I've
heard of this rule being followed frequently, I've hardly ever heard
of it being taught, so these texts are not following their own stated
rules.

Since this type of expression is so ambiguous, with people disagreeing
on the rules, and the rules being easy to overlook, my own opinion is
that neither your answer nor the teacher's is right: the question is
wrong. No responsible mathematician would write such an expression; we
would just say

n
---
m l

so there would be no question about its meaning. After all, the
purpose of rules is to allow us to communicate clearly, not to help us
trick students and start fights among families.

So you may in fact be "old-fashioned"; or you may be on the cutting
edge. In any case, I'm afraid you'll just have to learn how they are
doing it in class, and follow along. There shouldn't be many more
issues like this to worry about.

You can read more about these issues in the Dr. Math archives:

Order of Operations
http://mathforum.org/dr.math/problems/wuandheil.05.19.99.html

More on Order of Operations
http://mathforum.org/dr.math/problems/breitenbach.2.13.00.html

I hope this helps at least a little.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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