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:

(1) What is a fraction? Fractions types. How do they compare?


(2) Fractions changing form, expand and reduce (simplify) fractions


(3) Reducing fractions. The greatest common factor, GCF


(4) How to, comparing two fractions with unlike (different) numerators and denominators


(5) Sorting fractions in ascending order


(6) Adding ordinary (common, simple) fractions


(7) Subtracting ordinary (common, simple) fractions


(8) Multiplying ordinary (common, simple) fractions


(9) Fractions, theory: rational numbers