To compare and sort multiple fractions, they should either have the same denominator or the same numerator.
The operation of comparing fractions:
^{- 202}/_{122} and ^{- 205}/_{126}
Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:
- ^{202}/_{122} = - ^{(2 × 101)}/_{(2 × 61)} = - ^{((2 × 101) ÷ 2)}/_{((2 × 61) ÷ 2)} = - ^{101}/_{61}
- ^{205}/_{126} is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
205 = 5 × 41
126 = 2 × 3^{2} × 7
To compare and sort the fractions, build them up to the same denominator.
To build the fractions up to the same denominator we have to:
1) calculate their common denominator
2) then calculate the expanding number of each fraction
3) then build up 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:
61 is a prime number.
126 = 2 × 3^{2} × 7
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 (61, 126) = 2 × 3^{2} × 7 × 61 = 7,686
Calculate the expanding number of each fraction:
Divide the LCM by the denominator of each fraction.
- ^{101}/_{61} : 7,686 ÷ 61 = (2 × 3^{2} × 7 × 61) ÷ 61 = 126
- ^{205}/_{126} : 7,686 ÷ 126 = (2 × 3^{2} × 7 × 61) ÷ (2 × 3^{2} × 7) = 61
Build up the fractions to the same common denominator:
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:
- ^{101}/_{61} = - ^{(126 × 101)}/_{(126 × 61)} = - ^{12,726}/_{7,686}
- ^{205}/_{126} = - ^{(61 × 205)}/_{(61 × 126)} = - ^{12,505}/_{7,686}
The fractions have the same denominator, compare their numerators.
The larger the numerator the smaller the negative fraction.
The larger the numerator the larger the positive fraction.
::: The operation of comparing fractions :::
The final answer:
The fractions sorted in ascending order:
- ^{12,726}/_{7,686} < - ^{12,505}/_{7,686}
The initial fractions sorted in ascending order:
^{- 202}/_{122} < ^{- 205}/_{126}
How are the numbers being written on our website: comma ',' is used as a thousands separator; point '.' used as a decimal separator; numbers rounded off to max. 12 decimals (if the case). The set of the used symbols on our website: '/' the fraction bar; ÷ dividing; × multiplying; + plus (adding); - minus (subtracting); = equal; ≈ approximately equal.