Compare and sort the fractions in ascending order: 21/39, 9/25, 19/24, 16/25, 19/29, 28/27, 16/12, 33/21. Common ordinary fractions compared and sorted in ascending order, result explained below

Sort: 21/39, 9/25, 19/24, 16/25, 19/29, 28/27, 16/12, 33/21

To compare and sort multiple fractions, they should either have the same denominator or the same numerator.

The operation of sorting fractions in ascending order:
21/39, 9/25, 19/24, 16/25, 19/29, 28/27, 16/12, 33/21

Analyze the fractions to be compared and ordered, by category:

positive proper fractions: 21/39, 9/25, 19/24, 16/25, 19/29


positive improper fractions: 28/27, 16/12, 33/21

How to compare and sort the fractions in ascending order, by categories:

- any positive proper fraction is smaller than...


- any positive improper fraction.



How do we compare and sort all the fractions?

It is clear that there is no point in comparing fractions from different categories.


We will compare and sort the fractions in each of the above categories, separately.


Sort the positive proper fractions in ascending order:
21/39, 9/25, 19/24, 16/25, 19/29

Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:

By reducing the values of the numerators and denominators of the fractions, further calculations with these fractions become easier to do.


To reduce a fraction to the lowest terms equivalent: divide both the numerator and denominator by their greatest common factor, GCF.


Internal link » Reduce (simplify) common (ordinary) fractions to the lowest terms (to their simplest form equivalent), online calculator



21/39 = (3 × 7)/(3 × 13) = ((3 × 7) ÷ 3)/((3 × 13) ÷ 3) = 7/13

9/25 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
9 = 32
25 = 52

19/24 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
19 is a prime number.
24 = 23 × 3

16/25 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
16 = 24
25 = 52

19/29 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
19 is a prime number.
29 is a prime number.

To compare and sort the fractions, make their numerators the same.

To make the fractions' numerators the same - we have to:

1) calculate their common numerator


2) then calculate the expanding number of each fraction


3) then make their numerators the same by expanding the fractions to equivalent forms, which all have equal numerators

Calculate the common numerator

The common numerator is nothing else than the least common multiple (LCM) of the numerators of the fractions.


The LCM will be the common numerator of the compared fractions.


To calculate the LCM, we need the prime factorization of the numerators:


7 is a prime number.


9 = 32


19 is a prime number.


16 = 24


Multiply all the unique prime factors: if there are repeating prime factors we only take them once, and only the ones having the highest exponent (the highest powers).


External link » Calculate LCM, the least common multiple of numbers, online calculator


LCM (7, 9, 19, 16) = 24 × 32 × 7 × 19 = 19,152


Calculate the expanding number of each fraction:

Divide the LCM by the numerator of each fraction.


7/13 : 19,152 ÷ 7 = (24 × 32 × 7 × 19) ÷ 7 = 2,736


9/25 : 19,152 ÷ 9 = (24 × 32 × 7 × 19) ÷ 32 = 2,128


19/24 : 19,152 ÷ 19 = (24 × 32 × 7 × 19) ÷ 19 = 1,008


16/25 : 19,152 ÷ 16 = (24 × 32 × 7 × 19) ÷ 24 = 1,197


19/29 : 19,152 ÷ 19 = (24 × 32 × 7 × 19) ÷ 19 = 1,008



Make the fractions' numerators the same:

Expand each fraction: multiply both its numerator and denominator by its corresponding expanding number, calculated at the step 2, above.


This way all the fractions will have the same numerator:



7/13 = (2,736 × 7)/(2,736 × 13) = 19,152/35,568


9/25 = (2,128 × 9)/(2,128 × 25) = 19,152/53,200


19/24 = (1,008 × 19)/(1,008 × 24) = 19,152/24,192


16/25 = (1,197 × 16)/(1,197 × 25) = 19,152/29,925


19/29 = (1,008 × 19)/(1,008 × 29) = 19,152/29,232


The fractions have the same numerator, compare their denominators.

The larger the denominator the smaller the positive fraction.


The larger the denominator the larger the negative fraction.


The fractions sorted in ascending order:
19,152/53,200 < 19,152/35,568 < 19,152/29,925 < 19,152/29,232 < 19,152/24,192

The initial fractions sorted in ascending order:
9/25 < 21/39 < 16/25 < 19/29 < 19/24


Sort the positive improper fractions in ascending order:
28/27, 16/12, 33/21

Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:

By reducing the values of the numerators and denominators of the fractions, further calculations with these fractions become easier to do.


To reduce a fraction to the lowest terms equivalent: divide both the numerator and denominator by their greatest common factor, GCF.


Internal link » Reduce (simplify) common (ordinary) fractions to the lowest terms (to their simplest form equivalent), online calculator


28/27 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
28 = 22 × 7
27 = 33


16/12 = 24/(22 × 3) = (24 ÷ 22)/((22 × 3) ÷ 22) = 4/3


33/21 = (3 × 11)/(3 × 7) = ((3 × 11) ÷ 3)/((3 × 7) ÷ 3) = 11/7

To compare and sort the fractions, make their denominators the same.

To make the fractions' denominators the same - we have to:

1) calculate their common denominator


2) then calculate the expanding number of each fraction


3) then make their denominators the same by expanding the fractions to equivalent forms, which all have equal denominators

Calculate the common denominator

The common denominator is nothing else than the least common multiple (LCM) of the denominators of the fractions.


The LCM will be the common denominator of the compared fractions.


To calculate the LCM, we need the prime factorization of the denominators:


27 = 33


3 is a prime number.


7 is a prime number.


Multiply all the unique prime factors: if there are repeating prime factors we only take them once, and only the ones having the highest exponent (the highest powers).


External link » Calculate LCM, the least common multiple of numbers, online calculator


LCM (27, 3, 7) = 33 × 7 = 189


Calculate the expanding number of each fraction:

Divide the LCM by the denominator of each fraction.


28/27 : 189 ÷ 27 = (33 × 7) ÷ 33 = 7


4/3 : 189 ÷ 3 = (33 × 7) ÷ 3 = 63


11/7 : 189 ÷ 7 = (33 × 7) ÷ 7 = 27



Make the fractions' denominators the same:

Expand each fraction: multiply both its numerator and denominator by its corresponding expanding number, calculated at the step 2, above.


This way all the fractions will have the same denominator:



28/27 = (7 × 28)/(7 × 27) = 196/189


4/3 = (63 × 4)/(63 × 3) = 252/189


11/7 = (27 × 11)/(27 × 7) = 297/189


The fractions have the same denominator, compare their numerators.

The larger the numerator the larger the positive fraction.


The larger the numerator the smaller the negative fraction.


The fractions sorted in ascending order:
196/189 < 252/189 < 297/189

The initial fractions sorted in ascending order:
28/27 < 16/12 < 33/21


::: The operation of comparing fractions :::
The final answer:

Sort the positive proper fractions in ascending order:
9/25 < 21/39 < 16/25 < 19/29 < 19/24

Sort the positive improper fractions in ascending order:
28/27 < 16/12 < 33/21

All the fractions sorted in ascending order:
9/25 < 21/39 < 16/25 < 19/29 < 19/24 < 28/27 < 16/12 < 33/21

How are the numbers written: comma ',' used as a thousands separator; point '.' as a decimal separator; numbers rounded off to max. 12 decimals (if the case). Used symbols: '/' the fraction bar; ÷ dividing; × multiplying; + plus (adding); - minus (subtracting); = equal; ≈ approximately equal.

Other similar operations

Compare and sort the fractions in ascending order:
27/49, 13/33, 28/32, 25/35, 23/35, 33/34, 26/16, 41/25

Compare and sort common ordinary fractions, online calculator:

The latest common ordinary fractions compared and sorted in ascending order

Tutoring: Comparing ordinary fractions

How to compare two fractions?

1. Fractions that have different signs:

  • Any positive fraction is larger than any negative fraction:
  • ie: 4/25 > - 19/2

2. A proper and an improper fraction:

  • Any positive improper fraction is larger than any positive proper fraction:
  • ie: 44/25 > 1 > 19/200
  • Any negative improper fraction is smaller than any negative proper fraction:
  • ie: - 44/25 < -1 < - 19/200

3. Fractions that have both like numerators and denominators:

  • The fractions are equal:
  • ie: 89/50 = 89/50

4. Fractions that have unlike (different) numerators but like (equal) denominators.

  • Positive fractions: compare the numerators, the larger fraction is the one with the larger numerator:
  • ie: 24/25 > 19/25
  • Negative fractions: compare the numerators, the larger fraction is the one with the smaller numerator:
  • ie: - 19/25 < - 17/25

5. Fractions that have unlike (different) denominators but like (equal) numerators.

  • Positive fractions: compare the denominators, the larger fraction is the one with the smaller denominator:
  • ie: 24/25 > 24/26
  • Negative fractions: compare the denominators, the larger fraction is the one with the larger denominator:
  • ie: - 17/25 < - 17/29

6. Fractions that have different denominators and numerators (unlike denominators and numerators).

More on ordinary (common) fractions / theory: