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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


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 


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:   

(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:   

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:   

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   
Associated Topics:
High School Basic Algebra
High School Calculators, Computers
Middle School Algebra

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