Do Fractions Have to be Equal?

Date: 05/09/2003 at 11:13:16
From: Phil Collins
Subject: Fractions

My grandson's textbook defines fractions as "one or more of the equal 
parts of a whole." Is this true? Do the parts of the whole have to be 
equal?

Date: 05/09/2003 at 15:07:46
From: Doctor Peterson
Subject: Re: Fractions

Hi, Phil.

The book is correct, though there can be some misunderstandings about 
what it is saying.

When we name a fraction such as 1/3, we mean that you COULD obtain 
this amount by cutting three EQUAL pieces and taking one of them. We 
don't necessarily mean that you DID cut it just that way. But we do 
mean that if you cut a pie into three DIFFERENT pieces, they can't 
all be called thirds.

Here is a pie cut into thirds (as well as I can do so in text!):

                      ooooooooo
                oooooo.........oooooo
             ooo.................../ ooo
           oo...................../     oo
         oo....................../        oo
        o......................./           o
       o......................./             o
      o......................./               o
     o......................./                 o
     o....................../                  o
    o....................../                    o
    o---------------------.                     o
    o                      \                    o
     o                      \                  o
     o                       \                 o
      o                       \               o
       o                       \             o
        o                       \           o
         oo                      \        oo
           oo                     \     oo
             ooo                   \ ooo
                oooooo         oooooo
                      ooooooooo

All three pieces are equal, and I have chosen one of them, so that 
piece is 1/3 of the pie.

Here is a pie cut into three pieces that are not equal, but the chosen 
piece is still 1/3, because I COULD have made it the same way as 
before:

                      ooooooooo
                oooooo.........oooooo
             ooo.................../ ooo
           oo...................../     oo
         oo....................../        oo
        o......................./           o
       o......................./             o
      o......................./               o
     o......................./                 o
     o....................../                  o
    o....................../                    o
    o---------------------+---------------------o
    o                                           o
     o                                         o
     o                                         o
      o                                       o
       o                                     o
        o                                   o
         oo                               oo
           oo                           oo
             ooo                     ooo
                oooooo         oooooo
                      ooooooooo

Only the shaded piece is actually 1/3; the others happen to be 1/6 
and 1/2. Because they are not equal, we can't call them thirds. But 
the shaded piece is the same size as the third I made before, so it 
can be called 1/3.

Here is another pie, which I cut into six pieces, but I chose two of 
them; the shaded area is still 1/3, because again I could have cut it 
into three equal pieces to get the same amount:

                      ooooooooo
                oooooo.........oooooo
             ooo.\................./ ooo
           oo.....\.............../     oo
         oo........\............./        oo
        o...........\.........../           o
       o.............\........./             o
      o...............\......./               o
     o.................\...../                 o
     o..................\.../                  o
    o....................\./                    o
    o---------------------+---------------------o
    o                    / \                    o
     o                  /   \                  o
     o                 /     \                 o
      o               /       \               o
       o             /         \             o
        o           /           \           o
         oo        /             \        oo
           oo     /               \     oo
             ooo /                 \ ooo
                oooooo         oooooo
                      ooooooooo

(Of course, this can also be called 2/6, which is equivalent to 1/3.)

And here is a pie that I cut into three unequal pieces, none of which 
is a third:

                      ooooooooo
                oooooo....|    oooooo
             ooo..........|          ooo
           oo.............|             oo
         oo...............|               oo
        o.................|                 o
       o..................|                  o
      o...................|                   o
     o....................|                    o
     o....................|                    o
    o.....................|                     o
    o---------------------*                     o
    o                      \                    o
     o                     \                   o
     o                      \                  o
      o                     \                 o
       o                     \               o
        o                    \              o
         oo                   \           oo
           oo                 \         oo
             ooo               \     ooo
                oooooo         oooooo
                      ooooooooo

The point is that a third is not defined based on how you make it, but 
based on how big it is. If three identical pieces put together make up 
a whole, then those pieces are thirds. Three different pieces, or 
three pieces that don't add up to a whole, are not necessarily thirds.

Note also that this defines "fraction" in this specific mathematical 
sense. In everyday language we can use the word in other ways, such 
as "only a fraction of the population understands math well." We can 
even talk about one child getting the "bigger half" of a pie. But 
when we are writing a fraction using a numerator and denominator, the 
denominator has to indicate a number of equal pieces into which a 
whole can be cut, and the numerator is the number of those pieces 
chosen.

Does that help to clarify things?

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

Date: 05/09/2003 at 20:54:04
From: Doctor Dotty
Subject: Re: Fractions

Hi Phil,

Thanks for the question.

The textbook definition is correct, for this reason:

Here is a pie split up into bits of different sizes:

                * | *
            *     |   1 *
                  |   -   
         *        |   4    *
              11  |_ _ _ _ _
        *     --   \    1   *
              20    \   -   
         *           \  5  *
                      \   
            *           *
                *   *

Each piece is a fraction of the whole.

11/20 is "11 equal parts of a whole"
1/5 is "1 part of a whole"
1/4 is "1 part of a whole"

Together, if you add them all up, you do indeed get a whole.

The definition was slightly confusing though, I agree, as it could 
be taken (as I suspect you did) as meaning that all fractions 
regarding the same whole have to have the same denominator - rather 
than you can't have two different denominators in the same fraction.

A better way of defining a fraction could be:

"A number written in the form a/b where both a and b are integers."

Does that make sense? 

If I can help any more with this problem or any other, please write 
back.

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