# Comparing and ordering fractions

Sometimes we run into problems when we are asked compare y order fractions, that is, that we indicate, for example, which is the largest or the smallest or order them in ascending or descending order.

To compare two fractions we use the symbols “less than” (<) and “greater than (>)”; in the case of more than two fractions we can also order them, either in ascending order (least to greatest) or descending order (greatest to least).  ## Compare fractions

Sometimes we need compare two or more fractions to know which ones are older and which ones are minor. There are two simple ways to compare fractions: (1) if they have the same denominator and (2) a different denominator.

### Same denominator

We consider the numerators of the fractions and compare it; the largest will be the one whose numerator is greater and the smallest, whichever is less.

Example: From the next pair of fractions: 12/25 and 12/23. Which is the largest?

Both fractions have the same denominator, therefore, the largest fraction is 25/12 because it has a greater numerator, that is:

12/25> 12/23

### Different denominator

In this case, we must find fractions equivalent to the given fractions, where they have the same denominator. For this we follow the following steps:

• We find the least common multiple (lcm) of the denominators.
• We multiply the numerator and denominator of the fractions by a number that makes their denominators equal to the lcm.
• Since the fractions have the same denominator, we compare the numerators. The largest will be the one whose numerator is greater and the smallest, whichever is less.

Example: From the next pair of fractions: 7/6 and 16/15. Which is the largest?

The least common multiple of 6 and 15 is 30. We multiply and divide the first fraction by 5:

(7 × 5) / (6 × 5) = 35/30

Now, we multiply and divide the second by 2:

(16 × 2) / (15 × 2) = 32/30

Both fractions have the same denominator, therefore, the largest fraction is 35/30 because it has the largest numerator, that is:

35/30> 32/30

## Order fractions

Sometimes we must order a set of fractions, either ascending or descending. There are three simple ways to compare fractions: (1) if they have the same denominator; (2) same numerator and (3) different denominator.

### Same denominator

We consider the numerators of the fractions; the largest will be the one whose numerator is greater and the smallest, whichever is less. Example: Sort the following set of fractions in descending order:

2/5   ;  7/5    ;   18/5   ;  -3/5    ;    1/5

All fractions have the same denominator; we order from highest to lowest:

5/18> 5/7> 2/5> 1/5> -3/5

### With the same numerator

We consider the denominators of the fractions; the highest will be the one with the highest denominator and the lowest, the lowest. Example: Sort the following set of fractions in descending order:

7/5   ;  7/(-2)    ;   7/3   ;  7/9    ;    7/2

All fractions have the same numerator; we order from highest to lowest:

7/9> 7/5> 7/3> 7/2> 7 / (- 2)

### With different numerators and denominators

In this case, we must find fractions equivalent to the given fractions, where they have the same denominator. For this we follow the following steps:

• We find the least common multiple (lcm) of the denominators.
• We multiply the numerator and denominator of the fractions by a number that makes their denominators equal to the lcm.
• Since the fractions have the same denominator, we consider the numerators. The largest will be the one whose numerator is greater and the smallest, whichever is less.

Example: Sort the following set of fractions in ascending order:

7/5   ;  5/2    ;   3/15   ;  6/3    ;    1/2

The least common multiple of 5, 2,1 5, 3, and 2 is 30. We multiply and divide the first fraction by 6:

(7 × 6) / (5 × 6) = 42/30

We multiply and divide the second by 15:

(5 × 15) / (2 × 15) = 75/30

We multiply and divide the third by 2:

(3 × 2) / (15 × 2) = 6/30

We multiply and divide the fourth by 10:

(6 × 10) / (3 × 10) = 60/30

We multiply and divide the fifth by 15:

(1 × 15) / (2 × 15) = 15/30

All fractions have the same denominator; we order from smallest to largest:

6/30 <15/30 <42/30 <60/30 <75/30