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When to Add or Multiply Denominators?


Date: 04/02/2001 at 01:35:41
From: Laurie
Subject: Differences in adding and multiplying denominators

When you multiply two fractions together you multiply the numerators 
and then multiply the denominators. When you add two fractions you add 
the numerators but don't add the denominators. Why the difference?


Date: 04/02/2001 at 13:00:44
From: Doctor Peterson
Subject: Re: Differences in adding and multiplying denominators

Hi, Laurie.

Basically, they are just different things. Let's look closely at what 
the two operations mean, and see if we can find a simple description 
of what that difference is.

When you multiply by a fraction, you are dividing something up into 
some number of pieces (the denominator, which therefore tells what 
kind of pieces they are, such as thirds), and taking a certain number 
of those (the numerator, which tells how many there are). We can 
diagram this by representing each of the two fractions by a cut in a 
different direction. 

For example, suppose we multiply 3/4 by 2/3. We can start with 3/4 
(the shaded 3 out of 4 bars):

    +-----------------+
    |/////////////////|
    +-----------------+
    |/////////////////|
    +-----------------+
    |/////////////////|
    +-----------------+
    |                 |
    +-----------------+

When we multiply this by 2/3, we cut the whole thing into 3 parts 
vertically, and take 2 of them:

    +-----+-----+-----+
    |XXXXX|XXXXX|/////|
    +-----+-----+-----+
    |XXXXX|XXXXX|/////|
    +-----+-----+-----+
    |XXXXX|XXXXX|/////|
    +-----+-----+-----+
    |     |     |     |
    +-----+-----+-----+

The product is the doubly-shaded (X'ed) part, which consists of 6 
twelfths. Notice where these numbers come from: we divided the 
original whole square into 3*4 = 12 pieces, so the denominator of the 
result is 12 (the product of the denominators); we've selected 2*3 = 6 
of these, so the numerator is 6 (the product of the numerators). We 
multiplied both numerator and denominator because our cuts affected 
both the number and size of the pieces.

How about addition? Let's add 2/3 to 3/4. This time, we have a 
different situation. We aren't cutting each piece up; rather, we're 
putting pieces of two different kinds (thirds and fourths) together. 
Since we can't add apples and oranges, we have to turn both fractions 
into the same kind of piece. We can do this by using a common 
denominator: turning 2/3 into 8/12 and 3/4 into 9/12:

    +-----------------+     +-----+-----+-----+
    |/////////////////|     |\\\\\|\\\\\|     |
    +-----------------+     |\\\\\|\\\\\|     |
    |/////////////////|     |\\\\\|\\\\\|     |
    +-----------------+  +  |\\\\\|\\\\\|     |
    |/////////////////|     |\\\\\|\\\\\|     |
    +-----------------+     |\\\\\|\\\\\|     |
    |                 |     |\\\\\|\\\\\|     |
    +-----------------+     +-----+-----+-----+
            3/4                     2/3

is the same as

    +-----+-----+-----+     +-----+-----+-----+
    |/////|/////|/////|     |\\\\\|\\\\\|     |
    +-----+-----+-----+     +-----+-----+-----+
    |/////|/////|/////|     |\\\\\|\\\\\|     |
    +-----+-----+-----+  +  +-----+-----+-----+
    |/////|/////|/////|     |\\\\\|\\\\\|     |
    +-----+-----+-----+     +-----+-----+-----+
    |     |     |     |     |\\\\\|\\\\\|     |
    +-----+-----+-----+     +-----+-----+-----+
            9/12                    8/12

Now that we are in a position to add the fractions, all we do is add 
the numerators, because the size of the pieces has already been 
determined: we have 17 twelfths. (You may notice that we actually did 
multiply the denominators together to get the common denominator; 
that's not always necessary, because the lowest common denominator may 
be smaller than the product, but you can always use the product as a 
common denominator.) When we did the addition itself, the denominator 
was not affected.

Now, what's the difference between the two problems? 

One way to say it is that multiplication is a two-dimensional activity 
(like finding an area from two lengths, which is what I really did in 
my pictures), while addition is a one-dimensional activity (putting 
together two lengths, or counts). 

In my picture for addition, the second dimension came in only as part 
of the conversion to a common denominator (where I multiplied). Where 
two dimensions are involved, I am cutting across both numerator (the 
chosen part of the fraction) and denominator (the whole) at once; both 
numbers are therefore affected. So the multiplication and the 
conversion multiplied both numerator and denominator. But when I do 
the actual addition, I'm just setting one group of pieces next to 
another and counting; that adds together the numbers of pieces 
(numerators), but doesn't change what they are (the denominator: 
twelfths).

To put it more briefly: addition puts together two groups of the same 
type (denominator) of things, while multiplication changes each of the 
things you multiply.

If you'd like either a deeper (algebraic) or shallower answer, write 
back and let me know what you think.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Fractions
Middle School Fractions

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