Sort the Common Fractions String 109/165, 106/169, 110/186, 99/153, 118/172, 186/109 in Ascending Order. Online Calculator

Multiple fractions 109/165, 106/169, 110/186, 99/153, 118/172, 186/109 compared and then sorted in ascending order

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:
109/165, 106/169, 110/186, 99/153, 118/172, 186/109

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

positive proper fractions: 109/165, 106/169, 110/186, 99/153, 118/172

1 positive improper fraction: 186/109

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

- any positive proper fraction is smaller than...

- any positive improper fraction.


Sort the positive proper fractions in ascending order:
109/165, 106/169, 110/186, 99/153, 118/172

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


The fraction: 109/165

109/165 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 109 is a prime number.
  • 165 = 3 × 5 × 11
  • GCF (109; 165) = 1


The fraction: 106/169

106/169 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors:


  • 106 = 2 × 53
  • 169 = 132
  • GCF (106; 169) = 1


The fraction: 110/186

  • The prime factorizations of the numerator and denominator:
  • 110 = 2 × 5 × 11
  • 186 = 2 × 3 × 31
  • Multiply all the common prime factors: if there are repeating prime factors we only take them once, and only the ones having the lowest exponent (the lowest powers).
  • GCF (110; 186) = 2

110/186 = (110 ÷ 2)/(186 ÷ 2) = 55/93


The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:


110/186 = (2 × 5 × 11)/(2 × 3 × 31) = ((2 × 5 × 11) ÷ 2)/((2 × 3 × 31) ÷ 2) = 55/93



The fraction: 99/153

  • 99 = 32 × 11
  • 153 = 32 × 17
  • GCF (99; 153) = 32 = 9

99/153 = (99 ÷ 9)/(153 ÷ 9) = 11/17


The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:


99/153 = (32 × 11)/(32 × 17) = ((32 × 11) ÷ 32)/((32 × 17) ÷ 32) = 11/17



The fraction: 118/172

  • 118 = 2 × 59
  • 172 = 22 × 43
  • GCF (118; 172) = 2

118/172 = (118 ÷ 2)/(172 ÷ 2) = 59/86


The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:


118/172 = (2 × 59)/(22 × 43) = ((2 × 59) ÷ 2)/((22 × 43) ÷ 2) = 59/86




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) calculate the expanding number of each fraction
  • 3) expand the fractions to equivalent forms having equal numerators

Calculate the common numerator

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


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

109 is a prime number.

106 = 2 × 53

55 = 5 × 11

11 is a prime number.

59 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 (109, 106, 55, 11, 59) = 2 × 5 × 11 × 53 × 59 × 109 = 37,492,730



Calculate the expanding number of each fraction:

Divide the LCM by the numerator of each fraction.

109/165 ⟶ 37,492,730 ÷ 109 = (2 × 5 × 11 × 53 × 59 × 109) ÷ 109 = 343,970

106/169 ⟶ 37,492,730 ÷ 106 = (2 × 5 × 11 × 53 × 59 × 109) ÷ (2 × 53) = 353,705

55/93 ⟶ 37,492,730 ÷ 55 = (2 × 5 × 11 × 53 × 59 × 109) ÷ (5 × 11) = 681,686

11/17 ⟶ 37,492,730 ÷ 11 = (2 × 5 × 11 × 53 × 59 × 109) ÷ 11 = 3,408,430

59/86 ⟶ 37,492,730 ÷ 59 = (2 × 5 × 11 × 53 × 59 × 109) ÷ 59 = 635,470



Make the numerators of the fractions the same:

  • Expand each fraction: multiply both its numerator and denominator by its corresponding expanding number, calculated above.
  • This way all the fractions will have the same numerator:

109/165 = (343,970 × 109)/(343,970 × 165) = 37,492,730/56,755,050

106/169 = (353,705 × 106)/(353,705 × 169) = 37,492,730/59,776,145

55/93 = (681,686 × 55)/(681,686 × 93) = 37,492,730/63,396,798

11/17 = (3,408,430 × 11)/(3,408,430 × 17) = 37,492,730/57,943,310

59/86 = (635,470 × 59)/(635,470 × 86) = 37,492,730/54,650,420



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:
37,492,730/63,396,798 < 37,492,730/59,776,145 < 37,492,730/57,943,310 < 37,492,730/56,755,050 < 37,492,730/54,650,420

The initial fractions sorted in ascending order:
110/186 < 106/169 < 99/153 < 109/165 < 118/172

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

Sort the positive proper fractions in ascending order:
110/186 < 106/169 < 99/153 < 109/165 < 118/172

All the fractions sorted in ascending order:
110/186 < 106/169 < 99/153 < 109/165 < 118/172 < 186/109

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:
115/172, 112/180, 118/197, 108/158, 120/178, 197/111

» New. All the Calculations Performed by Our Visitors: Common Fractions Compared and Sorted. Data organized on a Monthly Basis

Compare and sort common fractions, online calculator:

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: