Sort the Common Fractions String 55/116, 135/87, 129/95, 74/129, 80/133 in Ascending Order. Online Calculator
Multiple fractions 55/116, 135/87, 129/95, 74/129, 80/133 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:
55/116, 135/87, 129/95, 74/129, 80/133
Analyze the fractions to be compared and ordered, by category:
positive proper fractions: 55/116, 74/129, 80/133
positive improper fractions: 135/87, 129/95
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:
55/116, 74/129, 80/133
Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:
- To reduce a fraction to the lowest terms equivalent: divide both the numerator and denominator by their greatest common factor, GCF.
The fraction: 55/116
55/116 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 55 = 5 × 11
- 116 = 22 × 29
- GCF (55; 116) = 1
The fraction: 74/129
74/129 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 74 = 2 × 37
- 129 = 3 × 43
- GCF (74; 129) = 1
The fraction: 80/133
80/133 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 80 = 24 × 5
- 133 = 7 × 19
- GCF (80; 133) = 1
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:
55 = 5 × 11
74 = 2 × 37
80 = 24 × 5
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).
LCM (55, 74, 80) = 24 × 5 × 11 × 37 = 32,560
Calculate the expanding number of each fraction:
Divide the LCM by the numerator of each fraction.
55/116 ⟶ 32,560 ÷ 55 = (24 × 5 × 11 × 37) ÷ (5 × 11) = 592
74/129 ⟶ 32,560 ÷ 74 = (24 × 5 × 11 × 37) ÷ (2 × 37) = 440
80/133 ⟶ 32,560 ÷ 80 = (24 × 5 × 11 × 37) ÷ (24 × 5) = 407
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:
55/116 = (592 × 55)/(592 × 116) = 32,560/68,672
74/129 = (440 × 74)/(440 × 129) = 32,560/56,760
80/133 = (407 × 80)/(407 × 133) = 32,560/54,131
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:
32,560/68,672 < 32,560/56,760 < 32,560/54,131
The initial fractions sorted in ascending order:
55/116 < 74/129 < 80/133
Sort the positive improper fractions in ascending order:
135/87 and 129/95
Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:
- To reduce a fraction to the lowest terms equivalent: divide both the numerator and denominator by their greatest common factor, GCF.
The fraction: 135/87
- The prime factorizations of the numerator and denominator:
- 135 = 33 × 5
- 87 = 3 × 29
- 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 (135; 87) = 3
135/87 = (135 ÷ 3)/(87 ÷ 3) = 45/29
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
135/87 = (33 × 5)/(3 × 29) = ((33 × 5) ÷ 3)/((3 × 29) ÷ 3) = 45/29
The fraction: 129/95
129/95 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 129 = 3 × 43
- 95 = 5 × 19
- GCF (129; 95) = 1
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:
45 = 32 × 5
129 = 3 × 43
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).
LCM (45, 129) = 32 × 5 × 43 = 1,935
Calculate the expanding number of each fraction:
Divide the LCM by the numerator of each fraction.
45/29 ⟶ 1,935 ÷ 45 = (32 × 5 × 43) ÷ (32 × 5) = 43
129/95 ⟶ 1,935 ÷ 129 = (32 × 5 × 43) ÷ (3 × 43) = 15
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:
45/29 = (43 × 45)/(43 × 29) = 1,935/1,247
129/95 = (15 × 129)/(15 × 95) = 1,935/1,425
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:
1,935/1,425 < 1,935/1,247
The initial fractions sorted in ascending order:
129/95 < 135/87
::: The operation of comparing fractions :::
The final answer:
Sort the positive proper fractions in ascending order:
55/116 < 74/129 < 80/133
Sort the positive improper fractions in ascending order:
129/95 < 135/87
All the fractions sorted in ascending order:
55/116 < 74/129 < 80/133 < 129/95 < 135/87
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.
Compare and sort common fractions, online calculator: