Compare and sort the fractions in ascending order: 87/142, 87/133, 91/127, 80/114. Common ordinary fractions compared and sorted in ascending order, result explained below
Sort: 87/142, 87/133, 91/127, 80/114
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:
87/142, 87/133, 91/127, 80/114
Analyze the fractions to be compared and ordered, by category:
positive proper fractions: 87/142, 87/133, 91/127, 80/114
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.
87/142 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
87 = 3 × 29
142 = 2 × 71
87/133 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
87 = 3 × 29
133 = 7 × 19
91/127 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
91 = 7 × 13
127 is a prime number.
80/114 = (24 × 5)/(2 × 3 × 19) = ((24 × 5) ÷ 2)/((2 × 3 × 19) ÷ 2) = 40/57
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:
87 = 3 × 29
91 = 7 × 13
40 = 23 × 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 (87, 91, 40) = 23 × 3 × 5 × 7 × 13 × 29 = 316,680
Calculate the expanding number of each fraction:
Divide the LCM by the numerator of each fraction.
87/142 : 316,680 ÷ 87 = (23 × 3 × 5 × 7 × 13 × 29) ÷ (3 × 29) = 3,640
87/133 : 316,680 ÷ 87 = (23 × 3 × 5 × 7 × 13 × 29) ÷ (3 × 29) = 3,640
91/127 : 316,680 ÷ 91 = (23 × 3 × 5 × 7 × 13 × 29) ÷ (7 × 13) = 3,480
40/57 : 316,680 ÷ 40 = (23 × 3 × 5 × 7 × 13 × 29) ÷ (23 × 5) = 7,917
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:
87/142 = (3,640 × 87)/(3,640 × 142) = 316,680/516,880
87/133 = (3,640 × 87)/(3,640 × 133) = 316,680/484,120
91/127 = (3,480 × 91)/(3,480 × 127) = 316,680/441,960
40/57 = (7,917 × 40)/(7,917 × 57) = 316,680/451,269
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 operation of comparing fractions :::
The final answer:
The fractions sorted in ascending order:
316,680/516,880 < 316,680/484,120 < 316,680/451,269 < 316,680/441,960
The initial fractions sorted in ascending order:
87/142 < 87/133 < 80/114 < 91/127
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: