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

Tutoring: Comparing ordinary fractions

How to compare two fractions?

1. EQUAL DENOMINATORS but unlike numerators fractions

  • a) To compare two positive fractions that have EQUAL DENOMINATORS (like denominators) but different numerators (unlike numerators), simply compare the numerators: the larger fraction is the one with the larger numerator, ie: 24/25 > 19/25
  • b) To compare two negative fractions that have EQUAL DENOMINATORS (like denominators) but different numerators (unlike numerators), simply compare the numerators: the larger fraction is the one with the smaller numerator, ie: -19/25 < -17/25
  • c) To compare two fractions of different signs (one positive and one negative) that have EQUAL DENOMINATORS (like denominators) but different numerators (unlike numerators), the rule is that any positive fraction is larger than any negative fraction, ie: 2/25 > -1/25

2. EQUAL NUMERATORS but unlike denominators fractions

  • a) To compare two positive fractions that have EQUAL NUMERATORS (like numerators) but different denominators (unlike denominators), simply compare the denominators: the larger fraction is the one with the smaller denominator, ie: 24/25 > 24/26
  • b) To compare two negative fractions that have EQUAL NUMERATORS (like numerators) but different denominators (unlike denominators), simply compare the denominators: the larger fraction is the one with the larger denominator, ie: -17/25 < -17/29
  • c) To compare two fractions of different signs (one positive and one negative) that have EQUAL NUMERATORS (like numerators) but different denominators (unlike denominators), the rule is that any negative fraction is smaller than any positive fraction, ie: -1/25 < 1/200

3. Different denominators and numerators (unlike denominators and unlike numerators) fractions

  • a) To compare two fractions of the same sign (both positive or both negative) that have different denominators and numerators (unlike denominators and unlike numerators), fractions should be brought to the same denominator (or if it's easier, to the same numerators). Please see the next paragraph, 3.a)
  • b) To compare two fractions of different signs (one positive and one negative) that have different denominators and numerators (unlike denominators and unlike numerators), the rule is that any negative fraction is smaller than any positive fraction, ie: -11/24 < 10/13

3.a) How to compare two fractions of the same sign (both positive or both negative) that have unlike denominators and unlike numerators? Below, we will bring the fractions to the same denominator, but it's also possible to compare fractions by bringing them to the same numerator.

If appropriate, you should start by reducing the fractions to the lowest terms (simplifying the fractions).

Calculate the lowest common multiple, LCM, of fractions' denominators

  • Factor the fractions' denominators down to their prime factors
  • The fractions' denominators lowest common multiple, LCM, will contain all their unique prime factors, by 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: two numbers' lowest common multiple, LCM.

Bring the fractions to the same denominator.

  • Calculate each fraction's expanding number: that is a non-zero number obtained by dividing the lowest common multiple LCM calculated above by each fraction's denominator
  • Expand each fraction: multiply each fraction's both numerator and denominator by the corresponding expanding 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 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 of comparing two fractions of the same sign that have different denominators and numerators (unlike denominators and numerators), with explanations: 16/24 vs. 45/75

Reduce (simplify) each fraction to lowest terms:

  • Factor both the numerator and denominator of each fraction down to prime factors
  • Divide each fraction's both numerator and denominator by their greatest common factor GCF (greatest common divisor GCD)
  • If you don't know how to calculate the greatest common divisor, go to numere-prime.ro: greatest common factor (GCF).
  • Reduce (simplify) the fraction 16/24 = 24 / (23 * 3) = (24 ÷ 23) / ((23 * 3) ÷ 23) = 2/3
  • Reduce (simplify) the fraction 45/75 = (32 * 5) / (3 * 52) = ((32 * 5) ÷ (3 * 5)) / ((3 * 52) ÷ (3 * 5)) = 3/5
  • At this point, the two fractions are reduced to the lowest terms (simplified): 16/24 = 2/3 și 45/75 = 3/5

Calculate the fractions' denominators lowest common multiple, LCM:

  • Factor fractions' denominators down to prime factors and then take all the unique prime factors, by the highest powers
  • First fraction's denominator factored down to prime factors: denominator is 3, it's already a prime number, it cannot be prime factorized
  • Second fraction's denominator factored down to prime factors: denominator is 5, it's a prime number, it cannot be prime factorized
  • The fractions' denominators lowest common multiple LCM must contain all their unique prime factors, by the highest powers: LCM (3, 5) = 3 * 5 = 15.

Bring fractions to the same denominator:

  • Each fraction's expanding number is calculated for each fraction by dividing the lowest common multiple LCM by each fraction's denominator:
  • first fraction's expanding number is: 15 ÷ 3 = 5
  • second fraction's expanding number ÷ 15 ÷ 5 = 3
  • To bring fractions to the same denominator, expand fractions: multiply each fraction's both numerator and denominator by their corresponding expanding number:
  • the first fraction is expanding as 2/3 = (5 * 2) / (5 * 3) = 10/15
  • the second fraction is expanded as 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: