Order of Operations: Fractions, Evaluating, and Simplifying

(An archive problem of the week)

Last time we looked at the subtle distinction between the order of operations, which defines the meaning of an expression, and properties that allow us to do something other than what an expression literally says. Here I want to look at one longer discussion that brings out these issues nicely.

Left-to-right meaning, ad-hoc evaluation

Here is the first half of this long question, from Terri in 2010:

Fractions: On the Order of Operations and Simplifying

The 2nd rule in the order of operations says to multiply and divide left to right. I've been thinking that the only reason for this "left to right" part is so I don't divide by the wrong amount. 

For example, in the problem 3 / 6 * 4, if I didn't follow the order of operations, but instead did the 6 * 4 first, I'd get a wrong answer.

Now, my text says I can avoid having to work left to right if I convert division to multiplication by the reciprocal. This makes sense.

My question is: when I write a division problem using the fraction line, do I ever have to worry about following the left to right rule, or does writing it as a fraction void the need for this rule just as writing division as multiplication of the reciprocal did? It seems that in my math text, when it comes to fractions such as ...

   24(3)x
   ------
    8(3)y
   
... they cancel and do the division and multiplication within a fraction in any order. For example, I would cancel the 3's and divide the 24 by 8, which isn't doing division and multiplication from left to right, nor does that treat the fraction line as a grouping symbol. Even multiplication of fractions doesn't seem to go by the left to right rule, because we're multiplying numerators first before we're dividing the numerator by the denominator of each particular fraction. I can write the problem above as multiplication by the reciprocal and see that I can divide and multiply in any order.

So I'm wondering if I can make this a general rule: in fractions, the left to right order is not an issue.

The question was long enough that I want to pause here and look at my answer to this part.

First, on avoiding left-to-right:

Yes, I've said the same thing; in a sense this is the reason for the left-to-right rule, since a right-to-left or multiplication-first rule would give different results.

I discussed this in Order of Operations: Common Misunderstandings.

Next, on fractions:

You're partly confusing order of operations (which applies to EVALUATING an expression -- that is, to what it MEANS) with techniques for simplifying or carrying out operations in practice. Properties of operations are what allow us to simplify, or to find simpler ways to evaluate an expression than doing exactly what it says. For example, the commutative property says that if the only operation in a portion of an expression is multiplication, you can ignore order.

This is the main topic I discussed last time, in Order of Operations: Subtle Distinctions. Simplifying (including canceling in fractions) is a step taken after understanding how an expression would be evaluated literally, as written, and involves changing the expression to one that is equivalent. That is, once we know what an expression means, we can find alternative ways to evaluate it that will take less work. In particular, this includes canceling in a fraction, and “multiplying across” to multiply two fractions.

Shortcuts to evaluation

Terri’s question continued:

Of course, it seems that just when I think I can generalize about something, there's a case where it doesn't hold true, and I'm wondering why, if this is the case, I've never seen it written anywhere.

I've been looking on the Internet and in algebra books to see if anyone addresses this particular part of the order of operations in detail, and it seems that most just generalize about the order of operations. I'm wondering if there is an unwritten rule that when you write division using the fraction line, you no longer need to do the division and multiplication from left to right.

Another math website stated the order of operations and then said there are a lot of shortcuts that a person can use because of the associative and commutative rules, but the site didn't elaborate. Is writing division using the fraction line one of these shortcuts that allows you to avoid the left to right rule when multiplying and dividing?

Thank you for taking the time to read this problem. Sorry to be so long-winded. I appreciate your time and help very much.

It’s true, as I said last time, that this is something not often discussed explicitly. It is not discussed under order of operations, because it is not really part of that! Rather, it’s part of the overall context. My answer continues:

In a fraction, the bar acts as a grouping symbol, ensuring that you evaluate the entire top and the entire bottom before doing the division. Thus, the division is out of the "left-to-right" picture entirely. In fact, since here the division involves top and bottom rather than left and right, I'm not sure what it would even mean to do it left to right.

About the other site’s comment on shortcuts:

Yes, that's what you're talking about -- shortcuts that essentially rewrite an expression (without actually doing so) as an equivalent expression that you can evaluate easily. Again, that is outside of the order of operations.

As an example, multiplying fractions is explained here in terms of the properties on which it is based:

  Deriving Properties of Fractions
  http://mathforum.org/library/drmath/view/63841.html

The idea here is that whereas a multiplication like \(\displaystyle\frac{4}{15}\cdot\frac{35}{64}\) as written means to multiply the first fraction by the second (left to right), in actually carrying it out, we can get the same result by canceling common factors anywhere in a numerator and a denominator, without regard to location, and then multiply all numerators and all denominators separately: \(\displaystyle\frac{4}{15}\cdot\frac{35}{64} = \frac{1}{3}\cdot\frac{7}{16} = \frac{7}{48} \). But what we are really doing is applying properties to rewrite the original product as the single fraction \(\displaystyle\frac{4\cdot 35}{15\cdot64}\), and then applying further properties to rewrite that by dividing the entire numerator and the entire denominator by the common factor 20.

Grouping and canceling

Terri quickly wrote back:

Thank you for your time in answering my question. I appreciate it.

If you have time, I have just two more questions to make sure I can get this straight in my head...

You mentioned that, for a fraction, the division is out of the "left-to-right" picture entirely. So, I'm guessing that I can safely say that the left-to-right rule applies only to division that is written on one line.

Last question: another website says that if I have the problem ...

   4(12)
   ----
     3

... then I need to multiply the 4 and 12 first before dividing by the 3, according to the order of operations, using the fraction line as a grouping symbol. But when I cancel, of course, I'm not doing it in this order. So is canceling one of those "properties of operations" you mentioned that allows us to evaluate this without having to stick to the order of operations?

I answered the first question:

Right. When division is written as a fraction, the order is forced by the grouping-symbol aspect of the fraction bar; it's as if division were always written like

   (a * b) / (c * d)

Mathematicians rarely write division in the horizontal form, probably because indicating it vertically makes it so much clearer what order is intended.

Fraction bars, like parentheses, override default rules about order, and make a visible division (no pun intended) between the numerator and denominator. There is no left and right except within the numerator and the denominator separately.

As to the second question, on canceling as a property:

Again, canceling is not the same thing as evaluating; the order of operations only applies to what an expression MEANS, not to how you must actually carry it out.

To EVALUATE this expression, in the sense of doing exactly what it says, I get 48/3 which becomes 16. I followed all the rules.

To SIMPLIFY the expression, I can follow the rule of simplification. This says that if I divide ANY factor of the numerator (wherever it falls -- it doesn't matter because of commutativity) and ANY factor of the denominator by the same number, the resulting fraction is equivalent. The reason I can use the properties is because the canceling is equivalent to this sequence of transformations:

  4(12)   4 * 4 * 3       4     4     3     4     4
  ----- = ---------  =   --- * --- * --- = --- * --- * 1 = 16
    3     1 * 1 * 3       1     1     3     1     1

All sorts of properties of multiplication come into play here, but the idea of canceling wraps it all into a simple process in which, again, the order doesn't matter. But that only works when it is ONLY multiplication in either part.

This is what I demonstrated above, but expressed a little differently.

Fractions vs divisions

Terri responded again the next day:

Thank you very much for your help.

I guess my questions must have sounded very confusing; I was confused, looking at the expression ...

   10
   --- * 2
    5
      
... as being 2 steps in the order of operations -- a division of 10 by 5 and a multiplication -- like the expression 10 divided by 5 times 2 written all on one line (with no fractions). But now I see that in my first example above, the fraction is considered to be just one number for the purposes of the order of operations so there is just 1 step -- a multiplication of the fraction times 2. Even though the fraction line means division, it doesn't count as division in the order of operations.

Hope I got this right. A HUGE thank you for taking the time to make sense out of my confusion!!! Have a great week!!

(Terri’s expression was accidentally modified when the question was archived, making the question and my answer a little confusing; I have fixed it here to match the original.)

I answered,

For many purposes it is easiest to say that a fraction is just treated as a number in the order of operations (in fact, I usually do that); but you don't have to, and that isn't what I've been saying, because I don't think it's what you've been asking about.

Your example certainly CAN be treated as a division followed by a multiplication, and it doesn't violate anything; you are still working left to right. What's different from the horizontal expression 10 / 5 * 2 is just that everything isn't left or right of everything else, so left-to-right isn't the only rule applied.

In \(\frac{10}{5}\cdot 2\), the division must be done first simply in order to get a value that will then be multiplied, because the fraction/division is the first operand; there is no way to read it so that the multiplication would be done first. In \(10\div 5\cdot 2\), an order of operations must be invoked.

The fraction bar primarily serves to group the numerator and the denominator, as I've said; I suppose, though I haven't said this, that it also groups the entire division relative to anything to its left or right, since it forces you to do the division first. A clearer example would be ...

      10
  2 * ---
       5

... which amounts to 2 * (10 / 5), where we technically have to divide first (so in a sense we are deviating from the left to right order). However, this is one of those cases where it turns out not to matter, because the commutative property and others conspire to make that expression EQUIVALENT to ...

  2 * 10
  ------
     5

... and therefore if you multiply first and then divide, you get the same answer. But this is NOT really left-to-right, because the 5 is not "to the right of" the division in the original form. It's just a simplified version -- a NEW expression that has the same value, not the way you directly evaluate it. And that's been my main point: HOW you actually evaluate something need not be identical to WHAT the expression means, taken at face value.

Rewriting is a feature of most of what we do with fractions, if you think about it.

Observe here that in \(2\cdot\frac{10}{5}\), the fraction must be treated as a single quantity (as if it were in parentheses), simply because of the typography: the entire fraction is written as the second factor. Writing on one line, \(2\cdot 10\div 5\) would not have that same constraint, so in order to have the same literal meaning, it would have to be written as \(2\cdot (10\div 5)\). But, again, this still gives the same value as \(2\cdot 10\div 5\) evaluated left-to-right, because both mean \(2\cdot 10\cdot \frac{1}{5}\), to which the associative property applies. (That’s what I meant by my mistaken reference to the commutative property.)

But these examples, instructive as they are, don’t have the same features as in the original examples.

Your questions until now were about something different -- where the numerator or denominator was not just a single number -- so it couldn't really be considered a mere fraction. For example, you asked about

  4(12)
  -----
    3

There, you can't just say the fraction is treated as a single number; you have to use the grouping properties of the fraction bar to determine the meaning of the expression.

Indeed, until the mention of \(\frac{10}{5}\cdot 2\), we saw no fractions except in the technical sense of an algebraic fraction, which is really a division expressed in a certain way.

To summarize, the fraction bar groups at two levels, first forcing the numerator and denominator to be evaluated separately, and then forcing the entire division to be done before anything to the left or right. Thus, this expression ...

      2 + 3
  1 + ----- * 6
      4 + 5

... means the same as this:

  1 + ((2 + 3) / (4 + 5)) * 6

In simple cases, where the numerator and denominator are single numbers, this implies that the one will be divided by the other before anything else, so for all practical purposes you can think of the fraction as a single number (the result of that division).

Terri concluded:

Thank you for your patience in answering my questions which I'm guessing were a headache to answer. I apologize for my inconsistency and confusion in writing them. I have not seen "spelled out" in my algebra books the relationship between order of operations and evaluating versus shortcuts like simplifying.

I've read and reread your answers, and I think I'm hopefully understanding it.

Thanks again. Have a good week!

There is a lot hidden in the way we write expressions, isn’t there?