Learn how to compare fractions

Learn how to compare ordinary fractions. Steps. Examples.

How to compare two fractions?

1. Fractions that have different signs:

  • Any positive fraction is larger than any negative fraction:
  • ie: 4/25 > - 19/2

2. A proper and an improper fraction:

  • Any positive improper fraction is larger than any positive proper fraction:
  • ie: 44/25 > 1 > 19/200
  • Any negative improper fraction is smaller than any negative proper fraction:
  • ie: - 44/25 < -1 < - 19/200

3. Fractions that have both like numerators and denominators:

  • The fractions are equal:
  • ie: 89/50 = 89/50

4. Fractions that have unlike (different) numerators but like (equal) denominators.

  • Positive fractions: compare the numerators, the larger fraction is the one with the larger numerator:
  • ie: 24/25 > 19/25
  • Negative fractions: compare the numerators, the larger fraction is the one with the smaller numerator:
  • ie: - 19/25 < - 17/25

5. Fractions that have unlike (different) denominators but like (equal) numerators.

  • Positive fractions: compare the denominators, the larger fraction is the one with the smaller denominator:
  • ie: 24/25 > 24/26
  • Negative fractions: compare the denominators, the larger fraction is the one with the larger denominator:
  • ie: - 17/25 < - 17/29

6. Fractions that have different denominators and numerators (unlike denominators and numerators).

  • To compare them, fractions should be built up to the same denominator (or if it's easier, to the same numerator).
  • 1) If necessary, start by reducing the fractions to the lowest terms (simplify the fractions to their simplest form equivalents).

    • Factor the numerator and the denominator of each fraction as products of primes, in exponential form.
    • Factor numbers as products of primes, in exponentional form, online
    • Calculate the greatest common factor, GCF, of the numerator and the denominator of each fraction: multiply all their unique common prime factors, by the lower exponents.
    • We will calculate a GCF for each fraction.
    • Each GCF calculated will be used to divide both the numerator and the denominator of each fraction, in order to simplify that fraction.
    • Calculate the greatest common factor, GCF, online
    • Divide both the numerator and the denominator of each fraction by their greatest common factor, GCF.
    • At this moment, the fractions are reduced to their lowest terms (or, if you like, simplified to their simplest form equivalent, irreducible).
    • By reducing a fraction its value does not change, only an equivalent fraction is produced, which is easier to work with.
    • Reduce fractions to their lowest terms, online, with explanations.
  • 2) Calculate the least common multiple, LCM, of all the fractions' denominators (the least common denominator):

    • LCM will be the new denominator of the equivalent fractions to be compared.
    • In this particular case, LCM is also called the least common denominator.
    • Factor all the fractions' denominators as products of primes in exponential form.
    • To calculate LCM multiply all the unique prime factors of the denominators, by the largest exponents.
    • Calculate the least common multiple of two numbers, LCM, online.
  • 3) Build up the fractions to the same denominator, by expanding them.

    • Expanding a fraction = multiply both the numerator and the denominator of a fraction by the same natural number, not zero, called the expanding number, in order to build an equivalent fraction.
    • Calculate each fraction's expanding number: that is a non-zero number we get by dividing the least common multiple, LCM, by each fraction's denominator.
    • Expand each fraction: multiply each fraction's both numerator and denominator by the corresponding expanding number calculated above.
    • The value of a fraction is not changing by expanding it, only an equivalent fraction that is easier to work with is being produced.
  • 4) Compare the numerators of the new equivalent fractions.

    • At this point, the fractions are built up to the same denominator, so it is only a simple task of comparing fractions' numerators.
    • The larger fraction will be the one with the larger numerator if the fractions are positive.
    • If they are negative, the larger fraction will be the one with the smaller numerator.

An example: compare two positive proper fractions that have different denominators and numerators (unlike denominators and numerators), with explanations: 16/24 vs. 45/75

  • 1) Start by reducing the fractions to the lowest terms (simplify the fractions to their simplest form equivalents).

    • Fraction 16/24:
      • Factor both the numerator and the denominator as products of primes in exponential form:
      • 16 = 24;
      • 24 = 23 × 3;
      • Calculate the greatest common factor, GCF, of the numerator and the denominator, multiply all their unique common prime factors, by their lower exponents, if there are:
      • GCF (16; 24) = GCF (24; 23 × 3) = 23;
      • Divide both the numerator and the denominator by GCF:
      • 16/24 = 24 / (23 × 3) = (24 ÷ 23) / ((23 × 3) ÷ 23) = 2/3.
    • Fraction 45/75:
      • Factor both the numerator and the denominator as products of primes in exponential form:
      • 45 = 32 × 5;
      • 75 = 3 × 52;
      • Calculate the greatest common factor, GCF, of the numerator and the denominator, multiply all their unique common prime factors, by their lower exponents, if there are:
      • GCF (45; 75) = GCF (32 × 5; 3 × 52) = 3 × 5;
      • Divide both the numerator and the denominator by GCF:
      • 45/75 = (32 × 5) / (3 × 52) = ((32 × 5) ÷ (3 × 5)) / ((3 × 52) ÷ (3 × 5)) = 3/5.
    • The reduced (simplified) fractions are:
      • 16/24 = 2/3;
      • 45/75 = 3/5.
    • The simplified fractions are equivalent to the original ones, having the same values as the original ones:
      • 16/24 ≈ 0.67; 2/3 ≈ 0.67;
      • 45/75 = 0.6; 3/5 = 0.6;
  • 2) Calculate the least common multiple, LCM, of all the denominators of the reduced fractions (the least common denominator):

    • LCM will be the new denominator of the equivalent fractions to be compared.
    • LCM is also called the least common denominator.
    • To calculate LCM, factor all the denominators as products of primes in exponential notation and then multiply all their unique prime factors by the larger exponents.
      • The denominator of the fraction 2/3 is 3, it's already a prime number, it cannot be broken down to any other primes;
      • The denominator of the fraction 3/5 is 5, it's also a prime number, it also cannot be broken down to other primes.
    • LCM (3, 5) = 3 × 5 = 15.
  • 3) Build up the fractions to the same denominator, by expanding them.

    • Expanding a fraction = multiply both the numerator and the denominator of a fraction by the same natural number, not zero, called the expanding number, in order to build an equivalent fraction.
    • Each fraction's expanding number is calculated by dividing the least common multiple LCM by each fraction's denominator:
      • For the first fraction: 15 ÷ 3 = 5;
      • For the second fraction: 15 ÷ 5 = 3.
    • Expand each fraction by its expanding number:
      • The first fraction: 2/3 = (5 × 2) / (5 × 3) = 10/15;
      • The second fraction: 3/5 = (3 × 3) / (3 × 5) = 9/15.
    • As in the case of simplifying, expanding fractions does not change their values, it only produces some equivalent fractions which are easier to work with:
      • 2/3 ≈ 0.67; 10/15 ≈ 0.67;
      • 3/5 = 0.6; 9/15 = 0.6.
  • 4) Compare the numerators of the new equivalent fractions.

    • Our fractions now have the same denominator, all we have to do is to compare the numerators:
    • 10 > 9 =>
    • 10/15 > 9/15 =>
    • 16/24 > 45/75.

More on ordinary (common) math fractions theory: