Comparing Fractions

Date: 03/11/99 at 21:12:40
From: ERIC
Subject: Comparison of Fractions

What is the easiest way of telling that one fraction is bigger than 
another (e.g. 5/9 and 5/6)? 

Date: 03/12/99 at 12:22:32
From: Doctor Peterson
Subject: Re: Comparison of Fractions

For the particular problem you gave, there is a very easy way! But let 
us look at three different kinds of problems, each of which can be done 
a different way.

First, suppose the denominators are the same, like 5/9 and 7/9. Then 
it is pretty easy: just compare the numerators, which tell how many 
pieces (ninths, in this case) there are. Since the "pieces" are the 
same size in each fraction, the more pieces, the bigger the fraction. 
So the rule is:

     a     b
    --- > ---  if  a > b
     n     n

Since 5 < 7, 5/9 < 7/9. They are in the same order as their numerators.

Second, suppose the numerators are the same, as in your problem, 5/9 
and 5/6. Then it works the opposite way: since you have the same number 
of pieces, you compare the denominators, which tell how big the pieces 
are. The bigger the denominator, the smaller the fraction. So the rule 
is:
     n     n
    --- > ---  if  a < b
     a     b

Since 9 > 6, 5/9 < 5/6. They are in the reverse order of their 
denominators.

Finally we have the general case, which is harder than the other two 
ways, but works for any fractions at all. You simply convert the 
fractions to the first case, by giving them a common denominator. You 
do not really have to worry about finding the least common 
denominator, though sometimes that will save a lot of work. Let us 
compare 5/9 and 4/7. Since we do not see any common factors immediately 
(and in fact there are not any), we can just multiply the denominators 
to get a common denominator, 63. To convert 5/9 to 63rds, we multiply 
by 7; to convert 4/7 to 63rds, we multiply by 9:

     5     4
    --- ? ---
     9     7

    5*7   4*9
    --- ? ---
    9*7   7*9

     35 < 36, so

     5     4
    --- < ---
     9     7

You may notice that I did not have to bother calculating the 63. Since 
I know the denominators are the same, I only need the numerators. What 
I have really done is just to multiply the 5 by 7 and the 4 by 9 and 
compare the results. Some people call this "cross multiplication":

    5*7=35          4*9=36; since this is bigger, 4/7 is bigger
          \        /
           5     4
          --- ? ---
           9     7

This is what I DO, but what I THINK is what I described above - 
otherwise I would never remember which product has to be bigger to make 
which fraction bigger! And this also lets me think in terms of an LCD 
if that helps. For instance, which is bigger, 5/9 or 44/81? I see that 
81 is a multiple of 9, so I do not have to go to the trouble of 
multiplying 5 by 81 and 44 by 9; I just multiply 5 by 9 and compare to 
44:
     5    44
    --- ? --
     9    81

    5*9   44
    --- ? --
    9*9   81

     45 > 44 so

     5    44
    --- > --
     9    81

A little thinking saved a lot of work. And that is what math is all 
about!

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