Compare and sort in ascending order the two common ordinary fractions, which one is larger: ^{- 71}/₄₀ and ^{- 80}/₄₅. 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.

- ⁷¹/₄₀ is already reduced to the lowest terms.
The numerator and denominator have no common prime factors:
71 is a prime number.
40 = 2³ × 5

- ⁸⁰/₄₅ = - ^{(24 × 5)}/_{(3² × 5)} = - ^{((24 × 5) ÷ 5)}/_{((3² × 5) ÷ 5)} = - ¹⁶/₉

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

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

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:

40 = 2³ × 5

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 (40, 9) = 2³ × 3² × 5 = 360

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

Divide the LCM by the denominator of each fraction.

- ⁷¹/₄₀ : 360 ÷ 40 = (2³ × 3² × 5) ÷ (2³ × 5) = 9

- ¹⁶/₉ : 360 ÷ 9 = (2³ × 3² × 5) ÷ 3² = 40

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:

- ⁷¹/₄₀ = - ^{(9 × 71)}/_{(9 × 40)} = - ⁶³⁹/₃₆₀

- ¹⁶/₉ = - ^{(40 × 16)}/_{(40 × 9)} = - ⁶⁴⁰/₃₆₀

The larger the numerator the smaller the negative fraction.

The larger the numerator the larger 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.