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### 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

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

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