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Techniques for Comparing and Ordering Fractions

Date: 07/16/2008 at 22:33:36
From: Ed
Subject: Ordering fractions

Hi, I have a timed test, and have difficulty ordering numerous 
fractions.  It's simple for me to do 2 or 3, but on the test they put 
about 5 to 6 fractions which kill me on time.  What is the fastest 
and easiest way to complete these problems?  Thank you

For example, put 3/8, 14/15, 2/3, 13/16, 8/12, 7/10 in order from
least to greatest. 



Date: 07/18/2008 at 10:08:14
From: Doctor Ali
Subject: Re: Ordering fractions

Hi Ed!

Thanks for writing to Dr. Math.

We have a bunch of fractions and we want to sort them.  This process
has a general method, which is not fast, but it always helps.  I'll
first explain the general method, then we can look at some possible
time-savers.

We know that if you multiply all numbers by a positive constant, the 
order doesn't change.  Do you agree?

I would firstly get rid of the fractions by multiplying them all by 
the least common multiple of the denominators.

You might ask, why the LCM?  Note that since we want to get rid of 
the denominators, we have to multiply the fractions by a number which 
is a multiple of all of the denominators, so that they cancel out and 
form integers.  Why do we choose the least among the common multiples?
This is because if you multiply fractions by a multiple of the LCM of
the denominators, you'll just get greater numbers and it is totally in
vain.

So, we need to find the LCM of the denominators.  This step will need 
the maximum amount of time.  But, if you do this, you'll have some 
integers that you can easily sort.

In your example,

  3/8, 14/15, 2/3, 13/16, 8/12, 7/10

The denominators are,

  8, 15, 3, 16, 12, 10

Factor them,

  2^3, 3 x 5, 3, 2^4, 2^2 x 3, 2 x 5

We know that LCM can be evaluated by writing all of the primes with 
the highest power.  That is,

     LCM = 2^4 x 3 x 5

Or,

     LCM = 240

You can multiply all numbers by 240 and we are sure that you'll get 
only integers.  See:

The numbers will be,

  +-----+-------+-----+-------+------+------+
  | 3/8 | 14/15 | 2/3 | 13/16 | 8/12 | 7/10 | --+
  +-----+-------+-----+-------+------+------+   | x 240
  |  90 |  224  | 160 |  195  |  160 |  168 | <-+
  +-----+-------+-----+-------+------+------+

You can sort them easily this way!

But, as you see, this is not probably the best method for us to do
under time pressure. 

I'll try to suggest some tips to improve the above method, so that it 
can be done by hand much faster.  I won't explain all the steps as an 
algorithm.  I'd rather suggest tips.

Note that we can classify the fractions into three groups in the 
first look.  The groups are,

  1)  Fractions having a value less that 1/2.
  2)  Fractions having a value between 1/2 and 1.
  3)  Fractions having a value greater than 1.

I know that we may have the boundaries among the fractions.  I mean, 
we may have 1/2 itself among the fractions.  I would say that is 
good because you are sure where to put them right away.

Again, in your problem,

  3/8, 14/15, 2/3, 13/16, 8/12, 7/10      

we can save some work by noting that 3/8 is the only fraction less 
than 1/2 (since the numerator is less than half of the denominator),
so it's got to go first,

  3/8, ....

  Left: {14/15, 2/3, 13/16, 8/12, 7/10}

Also note that 14/15 is closest to 1, so it's got to go last, 

  3/8, ....... , 14/15

  Left: {2/3, 13/16, 8/12, 7/10}

Very quickly, we can find that 8/12 equals 2/3 (multiply both the 2
and the 3 by 4 to rewrite 2/3 as 8/12), so we may combine them together,

  3/8, ....... , 14/15

  Left: {[2/3, 8/12] , 13/16, 7/10}
          \_______/
     They'll be together.

So, the problem changes to sorting,

  2/3, 13/16, 7/10

Now we may use the general method we talked about earlier.  Here, 
finding the LCM of three denominators becomes feasible in a short 
period of time.

In order to minimize the amount of arithmetic we do, we may come up 
with some rather strange methods.

One might try to sort the complements (the amount needed to turn the
fraction into 1), instead of the fractions, then order their distance
from 1, instead of their distance from 0.

As an example, we can continue from where we left the previous 
method.  We wanted to sort

  2/3, 13/16, 7/10

The complements are,

  1/3, 3/16, 3/10

Now as you see, we have a pair with equal numerators, 3/16 and 3/10.

You may do the following process in your head to decide about these 
two:

  16 > 10

  1/16 < 1/10

So,

  3/16 < 3/10

Or, you might say that when two positive fractions have a common 
numerator, the one with greater denominator is less in value.

So we have,

  3/16, 3/10

  Left: {1/3}

Now, we should decide about where to put 1/3.  The greater in the 
order is 3/10 or 0.3.  We know that 1/3 ~= 0.33333...  So, 1/3 is 
greater than 3/10.  So, it comes last.  That is,

  3/16, 3/10, 1/3

Since these fractions were the complements of the originals and we 
didn't have them among in the list, we have to reverse the order.  
That is,

From,

  3/16 < 3/10 < 1/3

We can deduce that,

  13/16 > 7/10 > 2/3

Does that make sense?

We already came up with the following order:

  3/8, ....... , 14/15

  Left: {[2/3, 8/12] , 13/16, 7/10}
          \_______/
     They'll be together.

So, the order will be,

  3/8, 2/3 = 8/12, 7/10, 13/16, 14/15

Did you get the idea?

Note that you may not use all of the above tips in a single problem.  
Your talent will tell you which one to use!  Also, be aware that if
you use the tips, it may sometimes take longer than using the general
method from the beginning!

To close the discussion, I might suggest you use these tips to decide 
about where to put as many as numbers as you can in their correct 
order.  Then, you can apply the general method, which is more 
trustworthy, for the remaining numbers.

Please write back if you still have any difficulties.

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



Date: 07/18/2008 at 15:02:40
From: Ed
Subject: Thank you (Ordering fractions)

Thank you so much for the quick response.  Very helpful with the first
method.  I have 25 mins on a timed math test with a few questions like
this, which puts a lot of pressure on time.  Thanks again.
Associated Topics:
Middle School Fractions

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