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
http://mathforum.org/dr.math/
http://mathforum.org/library/drmath/view/54341.html
this is how you learn and become a better person. I am sure that if those who got the correct answer had been proven wrong, they would have accepted it. No matter which way you learned, there are still parentheses in the equation and you must deal with those first and get rid of them BEFORE YOU MOVE ON. Okay horses? 
