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 
smaller than another one. Can you please help me?

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/