Compare the Two Common Fractions 814/854 and 819/859, Which One is Larger? Online Calculator
Fractions 814/854 and 819/859 are compared by building equivalent fractions, which have either equal denominators or equal numerators
To compare and sort multiple fractions, they should either have the same denominator or the same numerator.
The operation of comparing fractions:
814/854 and 819/859
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: 814/854
- The prime factorizations of the numerator and denominator:
- 814 = 2 × 11 × 37
- 854 = 2 × 7 × 61
- 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 (814; 854) = 2
814/854 = (814 ÷ 2)/(854 ÷ 2) = 407/427
The fraction can also be reduced without calculating GCF; factor the numerator and denominator and cross out all the common prime factors:
814/854 = (2 × 11 × 37)/(2 × 7 × 61) = ((2 × 11 × 37) ÷ 2)/((2 × 7 × 61) ÷ 2) = 407/427
The fraction: 819/859
819/859 is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
- 819 = 32 × 7 × 13
- 859 is a prime number.
- GCF (819; 859) = 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:
407 = 11 × 37
819 = 32 × 7 × 13
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 (407, 819) = 32 × 7 × 11 × 13 × 37 = 333,333
Calculate the expanding number of each fraction:
Divide the LCM by the numerator of each fraction.
407/427 ⟶ 333,333 ÷ 407 = (32 × 7 × 11 × 13 × 37) ÷ (11 × 37) = 819
819/859 ⟶ 333,333 ÷ 819 = (32 × 7 × 11 × 13 × 37) ÷ (32 × 7 × 13) = 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:
407/427 = (819 × 407)/(819 × 427) = 333,333/349,713
819/859 = (407 × 819)/(407 × 859) = 333,333/349,613
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:
333,333/349,713 < 333,333/349,613
The initial fractions sorted in ascending order:
814/854 < 819/859
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: