Sort the Common Fractions String 234/357, 266/408, 250/378, 234/415, 240/470 in Ascending Order. Online Calculator
Multiple fractions 234/357, 266/408, 250/378, 234/415, 240/470 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:
234/357, 266/408, 250/378, 234/415, 240/470
Analyze the fractions to be compared and ordered, by category:
positive proper fractions: 234/357, 266/408, 250/378, 234/415, 240/470
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: 234/357
- The prime factorizations of the numerator and denominator:
- 234 = 2 × 32 × 13
- 357 = 3 × 7 × 17
- 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 (234; 357) = 3
234/357 = (234 ÷ 3)/(357 ÷ 3) = 78/119
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
234/357 = (2 × 32 × 13)/(3 × 7 × 17) = ((2 × 32 × 13) ÷ 3)/((3 × 7 × 17) ÷ 3) = 78/119
The fraction: 266/408
- 266 = 2 × 7 × 19
- 408 = 23 × 3 × 17
- GCF (266; 408) = 2
266/408 = (266 ÷ 2)/(408 ÷ 2) = 133/204
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
266/408 = (2 × 7 × 19)/(23 × 3 × 17) = ((2 × 7 × 19) ÷ 2)/((23 × 3 × 17) ÷ 2) = 133/204
The fraction: 250/378
- 250 = 2 × 53
- 378 = 2 × 33 × 7
- GCF (250; 378) = 2
250/378 = (250 ÷ 2)/(378 ÷ 2) = 125/189
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
250/378 = (2 × 53)/(2 × 33 × 7) = ((2 × 53) ÷ 2)/((2 × 33 × 7) ÷ 2) = 125/189
The fraction: 234/415
234/415 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 234 = 2 × 32 × 13
- 415 = 5 × 83
- GCF (234; 415) = 1
The fraction: 240/470
- 240 = 24 × 3 × 5
- 470 = 2 × 5 × 47
- GCF (240; 470) = 2 × 5 = 10
240/470 = (240 ÷ 10)/(470 ÷ 10) = 24/47
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
240/470 = (24 × 3 × 5)/(2 × 5 × 47) = ((24 × 3 × 5) ÷ (2 × 5))/((2 × 5 × 47) ÷ (2 × 5)) = 24/47
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:
78 = 2 × 3 × 13
133 = 7 × 19
125 = 53
234 = 2 × 32 × 13
24 = 23 × 3
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 (78, 133, 125, 234, 24) = 23 × 32 × 53 × 7 × 13 × 19 = 15,561,000
Calculate the expanding number of each fraction:
Divide the LCM by the numerator of each fraction.
78/119 ⟶ 15,561,000 ÷ 78 = (23 × 32 × 53 × 7 × 13 × 19) ÷ (2 × 3 × 13) = 199,500
133/204 ⟶ 15,561,000 ÷ 133 = (23 × 32 × 53 × 7 × 13 × 19) ÷ (7 × 19) = 117,000
125/189 ⟶ 15,561,000 ÷ 125 = (23 × 32 × 53 × 7 × 13 × 19) ÷ 53 = 124,488
234/415 ⟶ 15,561,000 ÷ 234 = (23 × 32 × 53 × 7 × 13 × 19) ÷ (2 × 32 × 13) = 66,500
24/47 ⟶ 15,561,000 ÷ 24 = (23 × 32 × 53 × 7 × 13 × 19) ÷ (23 × 3) = 648,375
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:
78/119 = (199,500 × 78)/(199,500 × 119) = 15,561,000/23,740,500
133/204 = (117,000 × 133)/(117,000 × 204) = 15,561,000/23,868,000
125/189 = (124,488 × 125)/(124,488 × 189) = 15,561,000/23,528,232
234/415 = (66,500 × 234)/(66,500 × 415) = 15,561,000/27,597,500
24/47 = (648,375 × 24)/(648,375 × 47) = 15,561,000/30,473,625
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:
15,561,000/30,473,625 < 15,561,000/27,597,500 < 15,561,000/23,868,000 < 15,561,000/23,740,500 < 15,561,000/23,528,232
The initial fractions sorted in ascending order:
240/470 < 234/415 < 266/408 < 234/357 < 250/378
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: