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### Comparing Fractions

```
Date: 08/04/99 at 19:47:15
From: Jennifer
Subject: Fractions

Dr. Math,

To start us off this year in pre-calculus our teacher made us place
fractions in descending and ascending order. I don't quite understand
how you can tell just by looking at a fraction whether it's larger or

Sincerely,
Jennifer
```

```
Date: 08/09/99 at 07:55:16
From: Doctor Andrewg
Subject: Re: Fractions

Hi Jennifer!

Fortunately, your question isn't too difficult and I'm sure that you
will soon understand how to order fractions by size.

First of all, remember that fractions are really just numbers. So the
fraction

7
---
2

is really just 7 divided by 2.  This is equal to 3.5 (check with your
calculator if you like). So one way to compare fractions would be to
convert them all to decimals first. So if we wanted to know which of
the following two fractions was the largest

1           2
---   and   ---
3           5

we could work out that 1 divided by 3 is 0.333... (the 3s continue on
forever), and that 2 divided by 5 is 0.4. Can you see that the second
fraction is bigger than the first one now?

You could of course use your calculator to work out the decimals if
you wanted all the time, but it would be faster if you could learn to
do the calculations (at least some of the time) in your head or on
paper when the numbers are small. Your brain can work much faster than
a calculator for arithmetic, especially with a little practice.

Another method, useful if you don't have a calculator or can't convert
the fractions into decimals, is to convert the fractions to have the
same denominator (the value underneath the fraction sign, the top is
called the numerator - you can remember this as 'the _d_enominator
goes on the _d_own side').

For example, if we have two fractions that are 2 divided by some
positive value (call it 'x'), and 3 divided by x, then the first one
must be smaller than the second. Do you see why?

2     3
--- < ---   when x > 0
x     x

This is really the same as

1           1
2 * ---  <  3 * ---
x           x

and since we have more things (where the things are 1's over x's) on
the right-hand side, it is bigger (2 things on the left-hand side, 3
things on the right-hand side). Okay?  Remember though, that the
things have to be the same for this to work.

So we can only use this method if the fractions have the same
denominator. Can you guess what we need to do first? Convert the
fractions to have the same denominator.

The fastest way to convert fractions with different denominators to be
the same is to multiply both the numerator and denominator of each
fraction by the other's denominator. This gets a bit messy with more
than two fractions, since you have to multiply each fraction by all
the other fractions' denominators, so we'll stick to the two-fraction
case here.

Given the two fractions:

1           2
---   and   ---
3           5

We would multiply the first fraction by 5 over 5 (which as a fraction
is really equal to 1, so we're not actually changing the value, just
the representation of it), and the second fraction by 3 over 3. The 5
comes from the other fraction's denominator. We get the 3 in the same
way.

1     5      5
--- * --- = ----
3     5     15

2     3      6
--- * --- = ----
5     3     15

So 5 over 15 and 1 over 3 are really the same fraction. Do you see
this?

Both fractions have the same denominator now.

Now 5 over 15 must be smaller than 6 over 15, so the second fraction
is larger. This is the same conclusion that we reached above with
converting the fractions to decimals. In fact, 5 over 15 still equals
0.33333 (the 3 keeps on repeating) and 6 over 15 still equals 0.4.

Remember that there are two easy ways to compare fractions, by
converting to decimal or by converting the fractions to have the same
denominator. Good luck with your class, and remember that if you have
any problems you can always ask us another question (or even the same
one again if you would like a bit more help).

Take care,

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

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