Ordering fractions

Before learning about ordering fractions, please read the primer on fractions.


How do you order fractions?


You may be asked to order a list which includes proper and improper fractions and mixed fractions (mixed numbers).


This can easily be achieved by converting any mixed numbers into improper fractions (see also here) and then comparing fractions with like (the same) numerators, like (the same) denominators or unlike (different) denominators.

Let's begin with a simple example

Ordering fractions 3/4, 7/4, 1/4, 4/4 from least to greatest value


No mixed fractions to worry about here! Just two proper fractions (3/4 and 1/4) (numerator less than denominator) and two improper fractions (7/4 and 4/4) (numerator greater than or equal to denominator).


The denominator’s the same in each fraction (4). Therefore, the fraction with the lowest numerator has the least value – 1/4 (numerator 1) in this case. The next lowest value fraction is 3/4 (numerator 3), followed by 4/4 (numerator 4), followed by 7/4 (numerator 7).


So the correct order least to greatest value is

1/4          3/4          4/4          7/4


You may also be asked to order fractions from greatest to least value. For fractions with like denominators, the fraction with the highest numerator has the greatest value – 7/4 (numerator 7) in this example. The next greatest value fraction is 4/4 (numerator 4), followed by 3/4 (numerator 3), followed by 1/4 (numerator 1).


Therefore, the correct order greatest to least value is

7/4          4/4          3/4          1/4


Next try ordering fractions

16/15          4/3         3/5

from least to greatest value.

Here you are being asked to order two improper fractions (16/15 and 4/3) and one proper fraction (3/5).

Notice how the denominators are different. You need to find the least common denominator - the lowest number into which the denominators 15, 3 and 5 will divide. The answer of course is 15.

Leaving 16/15 alone (its denominator’s already 15), convert 4/3 to the equivalent fraction 20/15 by multiplying numerator and denominator by 5

Also convert 3/5 to the equivalent fraction 9/15 by multiplying numerator and denominator by 3

Now you have three fractions with like denominators – 16/15, 20/15 and 9/15. The fraction with the lowest numerator will be of the least value. 9/15’s numerator (9) is less than 16/15’s numerator (16) which in turn is less than 20/15's numerator (20). So 9/15 is of less value than 16/15 which in turn is of less value than 20/15.


The fraction order from least to greatest value is therefore

9/15          16/15          20/15

and because  9/15 = 3/5  and  20/15 = 4/3  the order from least to greatest value can also be written

3/5          16/15          4/3


i.e. 3/5 is of lower value than 16/15 which in turn is of lower value than 4/3.


What about if there's a mixture of proper, improper and mixed fractions?


For instance, order the following fractions from least to greatest value

8/8          11/2         8/16          1  1/4


Don’t panic! Let's go through this logically.


Here we have examples of improper fractions (8/8 and 11/2), a proper fraction (8/16) and a mixed fraction or mixed number (1 1/4).


You could begin by converting 1 1/4 to an improper fraction

1  1/4   =   1 + 1/4   =   4/4 + 1/4   =   5/4


So the list of fractions requiring ordering now looks like this

8/8          11/2          8/16          5/4


There are two fractions with like numerators - 8/8 and 8/16. The fraction with the higher denominator will be of lesser value. 8/16’s denominator (16) is higher than 8/8’s denominator (8), so the order of these two fractions from least to greatest value is

8/16          8/8


Now take a look at 5/4 – multiplying numerator and denominator by 2 gives the equivalent fraction 10/8.

Notice how 10/8 has the same denominator as 8/8 – the fraction with the lower numerator will have the lesser value (i.e. 8/8 is of lower value than 10/8). You can now order three of the fractions in the original list from least to greatest value

8/16          8/8          10/8 (= 5/4 = 1 1/4)


What about 11/2? You could multiply numerator and denominator by 4 to give the equivalent fraction 44/8

44/8 is obviously of greater value than 10/8 because 44/8’s numerator is higher (44 compared to 10) so the fraction list from least to greatest value becomes

8/16          8/8          10/8 (= 5/4 = 1 1/4)          44/8 (= 11/2)


8/16          8/8          1  1/4          11/2


Try ordering some more fractions!

Obviously there may be several different ways of correctly ordering fractions from a given list.

Experiment with different lists - you can easily make up your own. Just remember to start simple, perhaps with some proper and improper fractions, then add in some mixed fractions to spice things up!

Like any math skill, ordering fractions becomes easier with ... practice, practice, practice!

For more on fractions click:

What is a fraction?

Equivalent fractions

Simplifying fractions

Adding fractions

How to subtract fractions

Multiplying fractions

How to divide fractions

Improper fractions

Mixed fractions (mixed numbers)

Comparing fractions

Fraction to decimal

Fraction to decimal chart

Decimal to fraction

Fraction to percent

Percent to fraction

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