Order of Operations and Fractions

Date: 02/22/2002 at 17:02:41
From: Marlene Pealer
Subject: Not clear in order of operations

I KNOW the order of operations. I am quite clear that 105 / ab = 7 
when a = 3 and b = 5.  But my son has a sixth-grade math teacher 
equally convinced the answer is 175 because you do multiplication and 
division left to right, whichever comes first! She is teaching the 
kids that 105 divides by 3 = 35; 35 x 5 = 175. I know the answer has 
to be 7, but I can't find anything to tell me why. On the test, it 
was written 105 "line with a dot above and a dot below) ab.  Please 
give me the logic. I have crawled all over the archives (learned 
some neat stuff) .... is it because a division sign ... the line with 
two dots OR "/" throws parentheses around the two sides??

Date: 02/22/2002 at 18:08:53
From: Doctor Ian
Subject: Re: Not clear in order of operations

Hi Marlene,

The reason you can't find anything to confirm your answer is that your 
answer is incorrect, and the teacher's is correct.  

What's going on here, I suspect, is that when you look at 

  105 / ab

what you _see_ is 

  105 / (ab)

but that's not what it _says_.  What it says is

  (105 / a)b

because by convention, multiplications and divisions _are_ performed 
left to right, in the order in which they occur. So in fact, when a=3 
and b=5, 

  105 / ab = (105 / a)b 

           = (105 / 3)5

           = (35)5

           = 175

One way to think about what's happening is this: Spaces are NOT 
operators. So long as you don't stick them into the middle of 
identifiers (i.e., variables with names that are more than 1 character 
long, like 'width' or 'height' - if all your variables are one letter, 
this isn't an issue), adding or removing spaces makes no difference at 
all in the meaning of an expression. So 

  105 / ab = 105/ab = 105/a b = 105      /     a         b

Also, when you cram two identifiers together, there is an implied 
multiplication, so 

  105 / ab = 105 / a*b = 105 / a * b = 105/a   *   b

Now, you'd probably agree that the expression on the far right is the 
same as

  (105/a)  *  b

right? Well, that means it's also the same as the one on the far left. 

Of course, the _point_ of the question was to bring up exactly this 
issue: grouping with white space is _not_ the same as grouping with 
parentheses, or with a horizontal bar:

                105 
  105 / (ab) = ------
                 ab

In a sense, your teacher is serving notice that - quite correctly - 
any student who writes 

  105 / ab

when he means 

  105 / (ab)

is going to lose points. Dropping the '*' and cramming two variables 
together is _not_ the same as putting parentheses around them, even 
though it's somewhat natural for a human brain to interpret it that 
way in some contexts.  :^D

Does this help? 

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

Date: 02/23/2002 at 16:45:53
From: Marlene Pealer
Subject: Not clear in order of operations

I am very grateful for the clarity of the answer.  Please, one 
follow-up.  Why does it say in the order of operations "..in a 
fraction, treat each side as if it were in parentheses even if the 
parentheses are not there" (I found this in the archives) And couldn't 
I consider "105 divided by ab" as a fraction, just as I rewrite 10 
divided by 2 as 10/2 in fraction format?  

WHEN do I treat each side as if it were in parentheses. Clearly, if I 
re-write it in numerator = 105/denominator = ab format, the answer is 
7. Would you KNOW beyond doubt to rewrite it in fraction format as 
105/a times b?  What is confusing me is the use of the division symbol 
and why it says to treat the number(s) as if it were in parentheses.

Date: 02/23/2002 at 18:19:35
From: Doctor Ian
Subject: Re: Not clear in order of operations

Hi Marlene, 

To write '105 divided by (ab)' as a fraction without parentheses, 
you'd have to use a horizontal divider as a grouping symbol:

    105
  -------
     ab

When you use this symbol, parentheses around the entire numerator, and 
around the entire denominator, are implied, e.g., 

  1 + 2 + 3 + 4
  ------------- = (1 + 2 + 3 + 4) / (4 + 3 + 2 + 1) 
  4 + 3 + 2 + 1

                = (10) / (10)

                = 1

but

  1 + 2 + 3 + 4 / 4 + 3 + 2 + 1 = 1 + 2 + 3 + (4/4) + 3 + 2 + 1

                                = 1 + 2 + 3 + 1 + 3 + 2 + 1

                                = 11

In essence, when you write

   something
  ----------------
   something_else

you can simplify the top and bottom expressions independently of each 
other - which is what you do with 

    105      105
  ------- = ----- = 7
   3 * 5      15

It's important to keep in mind that _all_ of this is about saving 
effort in writing.  All ambiguity would be removed if parentheses were 
always used; but that's a lot of extra writing, and when you're 
manipulating equations - which often requires writing the same things 
over and over again, line after line - you want to streamline things 
as much as possible.  

And so, while it would be completely unambiguous to write

  1 + ((3 + ((6 * 4) / 8)) + 2) - (3 * 2)

it's much easier to write

  1 + 3 + 6 * 4 / 8 + 2 - 3 * 2

and we can do this if we _all_ agree that we'll do multiplications and 
divisions first, in left to right order,

  1 + 3 + 6 * 4 / 8 + 2 - 3 * 2
          -----

  1 + 3 +    24 / 8 + 2 - 3 * 2
             ------

  1 + 3 +         3 + 2 - 3 * 2
                          -----

  1 + 3 +         3 + 2 - 6

and _then_ do additions and subtractions, again from left to right, 

  1 + 3 +         3 + 2 - 6
  -----

      4 +         3 + 2 - 6
      -------------
 
                  7 + 2 - 6
                  -----

                      9 - 6
                      -----
   
                          3

The use of  a vinculum ('---------') to represent fractions is another 
convention, which allows us to get away without parenthesizing the 
numerator and denominator of a complicated fraction. 

But by design, the vinculum shares an important property with 
parentheses:  it can indicate the extent of an enclosure.  This is 
_not_ true of '+', '-', '*', or '/'. 

And so this is yet another way that you can keep these conventions 
straight. If you assume that the 'reach' of each operator is exactly 
one operand, a lot of the mystery should disappear. If you want to 
extend the 'reach' of an operator, you have to use parentheses, a 
vinculum, a subscript or superscript (which you'll encounter when you 
get to exponents), or some equivalent notation. E.g.,

     3 + 2    5
    2      = 2  = 32

In the case of 105/ab, if you want the '/' to reach both the a and the 
b, you need to do something to extend its normal reach, e.g., 

  105/(ab)
     ^
     |
     Now the 'next operand' is the whole parenthesized expression

or 

  105
  ---  <------ The operands of the division are the entire numerator
   ab          and the entire denominator. 

Does this help?  Keep writing back if it's still not making sense. 

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

Date: 02/23/2002 at 21:24:20
From: Marlene Pealer
Subject: Not clear in order of operations

Dr. Ian, my deep thanks. I understand you do this as a volunteer and I 
hope your needs in it are met as well as mine were. I get it! And I'm 
sorry about the things I said about my son's math teacher. Now I must 
teach my son to be gracious when you screw up.