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Fractions: On the Order of Operations and Simplifying

Date: 02/27/2010 at 19:48:22
From: Terri
Subject: how fractions fit in to 2nd rule in the order of operations

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.

Of course, it seem 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.



Date: 02/27/2010 at 21:02:19
From: Doctor Peterson
Subject: Re: how fractions fit in to 2nd rule in the order of operations

Hi, Terri.

As Terri wrote to Dr. Math
On 02/27/2010 at 19:48:22 (Eastern Time),
>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.

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.

>My question is when it comes to fractions... 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.

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.

>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.

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.

>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?

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 

If you have any further questions, feel free to write back.

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



Date: 02/27/2010 at 22:00:45
From: Terri
Subject: how fractions fit in to 2nd rule in the order of operations

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?

Thank you again. Have a good weekend.



Date: 02/27/2010 at 22:27:33
From: Doctor Peterson
Subject: Re: how fractions fit in to 2nd rule in the order of operations

Hi, Terri.

As Terri wrote to Dr. Math
On 02/27/2010 at 22:00:45 (Eastern Time),
>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.

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.

>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?

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.

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


Date: 03/01/2010 at 02:47:02
From: Terri
Subject: Thank you (how fractions fit in to 2nd rule in the order of operations)

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!!




Date: 03/01/2010 at 10:11:53
From: Doctor Peterson
Subject: Re: Thank you (how fractions fit in to 2nd rule in the order of operations)

Hi, Terri.

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.

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.

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.

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).

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



Date: 03/02/2010 at 01:29:59
From: Terri
Subject: Thank you (how fractions fit in to 2nd rule in the order of operations)

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!
Associated Topics:
Middle School Fractions

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