Compare and sort the fractions in ascending order: ^{- 17}/₃₁, ^{- 18}/₃₄, ^{- 28}/₁₉. Common ordinary fractions compared and sorted in ascending order, result explained below

To compare and sort multiple fractions, they should either have the same denominator or the same numerator.

1 negative improper fraction: - ²⁸/₁₉

negative proper fractions: - ¹⁷/₃₁, - ¹⁸/₃₄

- any negative improper fraction is smaller than...

- any negative proper fraction.

- ¹⁷/₃₁ is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
17 is a prime number.
31 is a prime number.

- ¹⁸/₃₄ = - ^{(2 × 32)}/_{(2 × 17)} = - ^{((2 × 32) ÷ 2)}/_{((2 × 17) ÷ 2)} = - ⁹/₁₇

Internal link > Reduce (simplify) common (ordinary) fractions to the lowest terms (to their simplest form equivalent), online calculator

1) calculate their common numerator

2) then calculate the expanding number of each fraction

3) then build up their numerators the same by expanding the fractions to equivalent forms, which all have equal numerators

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:

17 is a prime number.

9 = 3²

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 (17, 9) = 3² × 17 = 153

External link > Calculate LCM, the least common multiple of numbers, online calculator

Divide the LCM by the numerator of each fraction.

- ¹⁷/₃₁ : 153 ÷ 17 = (3² × 17) ÷ 17 = 9

- ⁹/₁₇ : 153 ÷ 9 = (3² × 17) ÷ 3² = 17

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:

- ¹⁷/₃₁ = - ^{(9 × 17)}/_{(9 × 31)} = - ¹⁵³/₂₇₉

- ⁹/₁₇ = - ^{(17 × 9)}/_{(17 × 17)} = - ¹⁵³/₂₈₉

The larger the denominator the larger the negative fraction.

The larger the denominator the smaller the positive fraction.

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.