Compare and sort the fractions in ascending order: 48/83, 46/76, 47/39, 61/35, 58/27, 82/33. Common ordinary fractions compared and sorted in ascending order, result explained below
Sort: 48/83, 46/76, 47/39, 61/35, 58/27, 82/33
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:
48/83, 46/76, 47/39, 61/35, 58/27, 82/33
Analyze the fractions to be compared and ordered, by category:
positive proper fractions: 48/83, 46/76
positive improper fractions: 47/39, 61/35, 58/27, 82/33
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:
48/83 and 46/76
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.
48/83 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
48 = 24 × 3
83 is a prime number.
46/76 = (2 × 23)/(22 × 19) = ((2 × 23) ÷ 2)/((22 × 19) ÷ 2) = 23/38
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:
48 = 24 × 3
23 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 (48, 23) = 24 × 3 × 23 = 1,104
Calculate the expanding number of each fraction:
Divide the LCM by the numerator of each fraction.
48/83 : 1,104 ÷ 48 = (24 × 3 × 23) ÷ (24 × 3) = 23
23/38 : 1,104 ÷ 23 = (24 × 3 × 23) ÷ 23 = 48
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:
48/83 = (23 × 48)/(23 × 83) = 1,104/1,909
23/38 = (48 × 23)/(48 × 38) = 1,104/1,824
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:
1,104/1,909 < 1,104/1,824
The initial fractions sorted in ascending order:
48/83 < 46/76
Sort the positive improper fractions in ascending order:
47/39, 61/35, 58/27, 82/33
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.
47/39 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
47 is a prime number.
39 = 3 × 13
61/35 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
61 is a prime number.
35 = 5 × 7
58/27 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
58 = 2 × 29
27 = 33
82/33 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
82 = 2 × 41
33 = 3 × 11
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:
39 = 3 × 13
35 = 5 × 7
27 = 33
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 (39, 35, 27, 33) = 33 × 5 × 7 × 11 × 13 = 135,135
Calculate the expanding number of each fraction:
Divide the LCM by the denominator of each fraction.
47/39 : 135,135 ÷ 39 = (33 × 5 × 7 × 11 × 13) ÷ (3 × 13) = 3,465
61/35 : 135,135 ÷ 35 = (33 × 5 × 7 × 11 × 13) ÷ (5 × 7) = 3,861
58/27 : 135,135 ÷ 27 = (33 × 5 × 7 × 11 × 13) ÷ 33 = 5,005
82/33 : 135,135 ÷ 33 = (33 × 5 × 7 × 11 × 13) ÷ (3 × 11) = 4,095
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:
47/39 = (3,465 × 47)/(3,465 × 39) = 162,855/135,135
61/35 = (3,861 × 61)/(3,861 × 35) = 235,521/135,135
58/27 = (5,005 × 58)/(5,005 × 27) = 290,290/135,135
82/33 = (4,095 × 82)/(4,095 × 33) = 335,790/135,135
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:
162,855/135,135 < 235,521/135,135 < 290,290/135,135 < 335,790/135,135
The initial fractions sorted in ascending order:
47/39 < 61/35 < 58/27 < 82/33
::: The operation of comparing fractions :::
The final answer:
Sort the positive proper fractions in ascending order:
48/83 < 46/76
Sort the positive improper fractions in ascending order:
47/39 < 61/35 < 58/27 < 82/33
All the fractions sorted in ascending order:
48/83 < 46/76 < 47/39 < 61/35 < 58/27 < 82/33
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 common ordinary fractions, online calculator: