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### Order of Operations and Fractions

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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??
```

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

Hi Marlene,

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.
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Associated Topics:
Elementary Fractions
Elementary Multiplication