Putting Rational Numbers in Order

Date: 09/22/2002 at 18:08:16
From: Eric
Subject: Ordering rational numbers

Hello, 

I am stuck on a problem and was wondering if you could walk me through 
it. I have to order 4/15, 6/17, and 3/16 from least to greatest. I 
can't use a calculator so it's hard to find a common denominator for 
all three of those fractions. How would I solve this and other similar 
problems?

Date: 09/22/2002 at 19:26:58
From: Doctor Sarah
Subject: Re: Ordering rational numbers

Hi Eric - thanks for writing to Dr. Math.

If you can't use a calculator, you could multiply together the 
denominators (15 * 16 * 17) and use that denominator:

  15 * 16 * 17 = 4080

  4080/15 = 272  1/15 = 408/4080   4/15 =1088/4080

  4080/16 = 255  1/16 = 255/4080   3/16 =  765/4080

  4080/17 = 240  1/17 = 240/4080   6/17 = 1440/4080

If that looks like too much calculating, try division:
 
      __.187_
   16 )3.000      3/16 = .187
       
      __.266_
   15 )4.000      4/15 = .266

      __.352_
   17 )6.000      6/17 = .352

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

Date: 09/22/2002 at 23:51:42
From: Doctor Greenie
Subject: Re: Ordering rational numbers

Hi, Eric -

Here is another method which can sometimes be used (it works very 
nicely on this problem).

Let's think of each fraction as representing the record of some sports 
team. For example, the fraction 7/10 would represent the record of a 
team which has won 7 out the 10 games it has played.

Now let's compare the fractions 3/16 and 4/15.  A team whose record 
is represented by the fraction 3/16 has played one more game than the 
team whose record is represented by the fraction 4/15, but they have 
won one game less. If they have played more total games but have won 
fewer, then clearly their record is worse - that is, their winning 
percentage is lower. So we know that 3/16 is less than 4/15 without 
doing any difficult arithmetic.

Now let's compare the fractions 4/15 and 6/17 in a slightly different 
but still very similar manner. This time we can think of these two 
fractions as representing the record of the same team, but a couple of 
games apart. So the fraction 4/15 represents the fact that the team at 
one point had played 15 games and won 4; the fraction 6/17 represents 
the fact that, after the team had played 17 games, it had won 6 of 
them. In going from its earlier record of winning 4 out of 15 games to 
its later record of winning 6 out of 17 games, it must have won 2 out 
of 2 games. The winning percentage for winning 2 out of 2 games is 
much higher than the winning percentage for winning 4 out of 15 games; 
so winning those 2 out of 2 games must have raised the team's winning 
percentage. That is, the fraction 6/17 must be bigger than the 
fraction 4/15.

I hope you find this method of comparing fractions useful in at least 
some problems of this type.

Please write back if you have any questions about this method.

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