Fractions, theory: how to compare two fractions with both unlike (different) nominators and denominators?

Tutoring: Comparing ordinary fractions

How to compare two fractions?

To compare two fractions with equal denominators (like denominators), but different numerators (unlike numerators), the larger (bigger, greater) fraction is the one with the larger numerator.

To compare two fractions with equal numerators (like numerators), but different denominators (unlike denominators), the larger (bigger, greater) fraction is the one with the smaller denominator.

To compare two fractions with different denominators and numerators (unlike denominators, unlike numerators), fractions should be first brought to the same denominator (or if it's easier, to the same numerators):

  • Where appropriate, you should start by reducing the fractions to the lowest terms (simplifying the fractions). If you don't know how, or if you'd like to practice the whole process, with examples, go to this page www.fractii.ro: reducing to lowest terms (simplify) ordinary math fractions online, with explanations.
  • To bring all the fractions to the same denominator, you must calculate the lowest common multiple LCM of fractions denominators:
    • Fractions denominators are factorized into prime factors
    • .
    • Lowest common multiple LCM will contain all of the fractions denominators factors, at the highest powers.
    • .
    • If you don't know how, or if you'd like to practice the process, go to this page on numere-prime.ro website: lowest common multiple LCM of two numbers.
  • Multiplying number of each fraction's both numerator and denominator must be calculated: divide the lowest common multiple LCM calculated above, by the denominator of each fraction, obtaining the corresponding multiplying number for each fraction
  • Multiply each fraction's both numerator and denominator by the corresponding multiplying number calculated above.
  • At this point, fractions are brought to the same denominator, so it's now only a simple task of comparing fractions' numerators.
  • The larger (bigger, greater) fraction will be the one with the larger numerator.

An example of comparing two fractions with different denominators and numerators (unlike denominators and numerators)

  • Let's compare these fractions: 16/24 and 45/75.
  • Reduce each fraction to lowest terms (simplify it):
    • Factorize into prime factors both the numerator and denominator of each fraction
    • Divide each fraction's both the numerator and denominator by the number containing only their common factors, at lowest powers - this is the greatest common factor GCF (greatest common divisor GCD) of each fraction's numerator and denominator
    • If you don't know how to calculate the greatest common divisor, go to numere-prime.ro: greatest common factor (GCF).
    • Simplify the fraction 16/24 = 24 / (23 * 3) = (24 : 23) / ((23 * 3) : 23) = 2/3
    • Simplify the fraction 45/75 = (32 * 5) / (3 * 52) = ((32 * 5) : (3 * 5)) / ((3 * 25) : (3 * 5)) = 3/5
  • At this point, fractions are reduced to the lowest terms (simplified): 16/24 = 2/3 and 45/75 = 3/5
  • Next, we calculate the lowest common multiple LCM of fractions' denominators.
    • Factorize fractions denominators into prime factors and take all the factors contained, at the highest powers
    • First fraction's denominator factorized into prime factors: denominator is 3, it's already a prime number, it cannot be factorized furthermore into prime factors
    • Second fraction's denominator factorized into prime factors: denominator is 5, it's a prime number, it cannot be factorized furthermore into prime factors
    • The lowest common multiple LCM must contain all the factors from fractions' denominators factorized into prime numbers, at the highest powers: LCM (3, 5) = 3 * 5 = 15.
  • Multiplying number of each fraction's both numerator and denominator is calculated for each fraction by dividing the lowest common multiple LCM to each fraction denominator:
    • multiplying number for the first fraction is: 15:3 = 5
    • multiplying number for the second fraction: 15:5 = 3
  • To bring fractions to the same denominator, multiply each fraction's both numerator and denominator by their corresponding multiplying number:
    • the first fraction yields 2/3 = (5 * 2) / (5 * 3) = 10/15
    • the second fraction yields 3/5 = (3 * 3) / (3 * 5) = 9/15
  • Obviously, the larger fraction it's the one with the larger numerator, 10/15 > 9/15, which means that the initial fraction 16/24 is larger than the initial fraction 45/75.

More on ordinary math fractions theory:

Fractions operations that can be run automatically, with explanations: