Sort the Common Fractions String 163/17, 153/36, 94/28, 113/31, 67/119, 66/134 in Ascending Order. Online Calculator
Multiple fractions 163/17, 153/36, 94/28, 113/31, 67/119, 66/134 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:
163/17, 153/36, 94/28, 113/31, 67/119, 66/134
Analyze the fractions to be compared and ordered, by category:
positive proper fractions: 67/119, 66/134
positive improper fractions: 163/17, 153/36, 94/28, 113/31
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:
67/119 and 66/134
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: 67/119
67/119 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 67 is a prime number.
- 119 = 7 × 17
- GCF (67; 119) = 1
The fraction: 66/134
- The prime factorizations of the numerator and denominator:
- 66 = 2 × 3 × 11
- 134 = 2 × 67
- 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 (66; 134) = 2
66/134 = (66 ÷ 2)/(134 ÷ 2) = 33/67
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
66/134 = (2 × 3 × 11)/(2 × 67) = ((2 × 3 × 11) ÷ 2)/((2 × 67) ÷ 2) = 33/67
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:
67 is a prime number.
33 = 3 × 11
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 (67, 33) = 3 × 11 × 67 = 2,211
Calculate the expanding number of each fraction:
Divide the LCM by the numerator of each fraction.
67/119 ⟶ 2,211 ÷ 67 = (3 × 11 × 67) ÷ 67 = 33
33/67 ⟶ 2,211 ÷ 33 = (3 × 11 × 67) ÷ (3 × 11) = 67
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:
67/119 = (33 × 67)/(33 × 119) = 2,211/3,927
33/67 = (67 × 33)/(67 × 67) = 2,211/4,489
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:
2,211/4,489 < 2,211/3,927
The initial fractions sorted in ascending order:
66/134 < 67/119
Sort the positive improper fractions in ascending order:
163/17, 153/36, 94/28, 113/31
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: 163/17
163/17 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 163 is a prime number.
- 17 is a prime number.
- GCF (163; 17) = 1
The fraction: 153/36
- The prime factorizations of the numerator and denominator:
- 153 = 32 × 17
- 36 = 22 × 32
- 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 (153; 36) = 32 = 9
153/36 = (153 ÷ 9)/(36 ÷ 9) = 17/4
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
153/36 = (32 × 17)/(22 × 32) = ((32 × 17) ÷ 32)/((22 × 32) ÷ 32) = 17/4
The fraction: 94/28
- 94 = 2 × 47
- 28 = 22 × 7
- GCF (94; 28) = 2
94/28 = (94 ÷ 2)/(28 ÷ 2) = 47/14
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
94/28 = (2 × 47)/(22 × 7) = ((2 × 47) ÷ 2)/((22 × 7) ÷ 2) = 47/14
The fraction: 113/31
113/31 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 113 is a prime number.
- 31 is a prime number.
- GCF (113; 31) = 1
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) expand the fractions to equivalent forms having the same denominator
Calculate the common denominator
The common denominator is nothing else than the least common multiple (LCM) of the denominators of the fractions.
To calculate the LCM, we need the prime factorization of the denominators:
17 is a prime number.
4 = 22
14 = 2 × 7
31 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).
LCM (17, 4, 14, 31) = 22 × 7 × 17 × 31 = 14,756
Calculate the expanding number of each fraction:
Divide the LCM by the denominator of each fraction.
163/17 ⟶ 14,756 ÷ 17 = (22 × 7 × 17 × 31) ÷ 17 = 868
17/4 ⟶ 14,756 ÷ 4 = (22 × 7 × 17 × 31) ÷ 22 = 3,689
47/14 ⟶ 14,756 ÷ 14 = (22 × 7 × 17 × 31) ÷ (2 × 7) = 1,054
113/31 ⟶ 14,756 ÷ 31 = (22 × 7 × 17 × 31) ÷ 31 = 476
Make the denominators 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 denominator:
163/17 = (868 × 163)/(868 × 17) = 141,484/14,756
17/4 = (3,689 × 17)/(3,689 × 4) = 62,713/14,756
47/14 = (1,054 × 47)/(1,054 × 14) = 49,538/14,756
113/31 = (476 × 113)/(476 × 31) = 53,788/14,756
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:
49,538/14,756 < 53,788/14,756 < 62,713/14,756 < 141,484/14,756
The initial fractions sorted in ascending order:
94/28 < 113/31 < 153/36 < 163/17
::: The operation of comparing fractions :::
The final answer:
Sort the positive proper fractions in ascending order:
66/134 < 67/119
Sort the positive improper fractions in ascending order:
94/28 < 113/31 < 153/36 < 163/17
All the fractions sorted in ascending order:
66/134 < 67/119 < 94/28 < 113/31 < 153/36 < 163/17
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.
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