Reduce Fractions Step by Step

Detailed Examples, Explanations, and Practice Questions

We present examples on how to reduce fractions to lowest terms. Detailed solutions and explanations are also included. More questions and their answers are included below. Reducing fractions including variables is also covered.

What is a Reduced Fraction to its Lowest Terms?

A fraction is reduced to the lowest terms if the numerator and the denominator have no common factors except 1.

Example 1

a) The following fractions are in reduced form:

\[ \dfrac{2}{5}, \quad \dfrac{10}{13}, \quad \dfrac{25}{28} \]

b) The following fractions are NOT in reduced form:

  • \( \dfrac{2}{6} \): because the numerator \( 2 = 2 \times 1 \) and the denominator \( 6 = 3 \times 2 \), and they therefore have a common factor equal to 2.
  • \( \dfrac{12}{16} \): because the numerator \( 12 = 3 \times 4 \) and the denominator \( 16 = 4 \times 4 \), and they therefore have a common factor equal to 4.
  • \( \dfrac{10}{50} \): because the numerator \( 10 = 10 \times 1 \) and the denominator \( 50 = 5 \times 10 \), and they therefore have a common factor equal to 10.

There are two ways to write fractions in reduced form. In the first method, you first need to find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both numerator and denominator by the GCF to find an equivalent fraction in lowest terms.

Reduce Fractions Using The Greatest Common Factor (GCF)

Example 2

Reduce the fraction \( \dfrac{12}{28} \) to lowest terms.

Solution

Solution to Example 2
The GCF of the numerator \( 12 \) and denominator \( 28 \) is \( 4 \). Hence we obtain an equivalent fraction in reduced form by dividing the numerator and denominator by the GCF:

\[ \dfrac{12}{28} = \dfrac{12 \color{red}{\div 4}}{28 \color{red}{\div 4}} = \dfrac{3}{7} \]

The numerator \( 3 \) and denominator \( 7 \) have no common factor except \( 1 \) and therefore the fraction \( \dfrac{3}{7} \) is the reduced form of the given fraction.

Example 3

Reduce the fraction \( \dfrac{240}{360} \) to lowest terms.

Solution

Solution to Example 3
The numerator and denominator are multiples of ten; we can therefore start by dividing the numerator and denominator by \( 10 \) to obtain an equivalent fraction:

\[ \dfrac{240}{360} = \dfrac{240 \color{red}{\div 10}}{360 \color{red}{\div 10}} = \dfrac{24}{36} \]

The GCF of the numerator \( 24 \) and denominator \( 36 \) is \( 12 \). An equivalent fraction in reduced form is obtained by dividing the numerator and denominator by the GCF:

\[ \dfrac{24}{36} = \dfrac{24 \color{red}{\div 12}}{36 \color{red}{\div 12}} = \dfrac{2}{3} \]

The fraction \( \dfrac{2}{3} \) is the reduced form of the given fraction.

Reduce Fractions by Successive Divisions

In this method, we divide the numerator and denominator by prime numbers: \( 2, 3, 5, 7, 11, \dots \) until the numerator and denominator have no common factor except 1.

Example 4

Reduce the fraction \( \dfrac{36}{168} \) to lowest terms.

Solution

Solution to Example 4
The numerator and denominator are even integers and therefore divisible by 2. Hence we divide numerator and denominator by 2:

\[ \dfrac{36}{168} = \dfrac{36 \div 2}{168 \div 2} = \dfrac{18}{84} \]

The numerator and denominator of the fraction obtained above are even integers, so we divide numerator and denominator by 2 one more time:

\[ \dfrac{18}{84} = \dfrac{18 \div 2}{84 \div 2} = \dfrac{9}{42} \]

The numerator and denominator of the fraction obtained above are divisible by 3, hence:

\[ \dfrac{9}{42} = \dfrac{9 \div 3}{42 \div 3} = \dfrac{3}{14} \]

\( \dfrac{3}{14} \) is the reduced form of the given fraction.

Reduce Fractions Including Variables

We reduce the numerical part and the algebraic part.

Example 5

Reduce the fraction \( \dfrac{21x}{36x^2} \) to lowest terms.

Solution

Solution to Example 5
The given fraction may be written as the product of two fractions: one in numerical form and the other in algebraic form:

\[ \dfrac{21x}{36x^2} = \dfrac{21}{36} \times \dfrac{x}{x^2} \]

Reduce the fraction in numerical form. The GCF of 21 and 36 is equal to 3:

\[ \dfrac{21}{36} = \dfrac{21 \div 3}{36 \div 3} = \dfrac{7}{12} \]

Reduce the fraction in algebraic form. The common factor to \( x \) and \( x^2 \) is \( x \). We divide the numerator and the denominator by \( x \):

\[ \dfrac{x}{x^2} = \dfrac{x \div x}{x^2 \div x} = \dfrac{1}{x} \]

We now rewrite the given fraction in reduced form:

\[ \dfrac{21x}{36x^2} = \dfrac{7}{12} \times \dfrac{1}{x} = \dfrac{7}{12x} \]

\( \dfrac{7}{12x} \) is the reduced form of the given fraction.

Questions and Solutions

  1. Question 1: Which of the following fractions are NOT in reduced form?

    1. \( \dfrac{20}{155} \)
    2. \( \dfrac{3}{8} \)
    3. \( \dfrac{246}{1246} \)
    4. \( \dfrac{5}{13} \)
    5. \( \dfrac{9}{27} \)
    6. \( \dfrac{2x}{3x} \)
    7. \( \dfrac{7xy}{11y} \)
    2. Solution

    The following fractions are NOT in reduced form:

    1. \( \dfrac{20}{155} \): numerator and denominator are divisible by 5 and therefore have a common factor of 5.
    2. \( \dfrac{246}{1246} \): numerator and denominator are even numbers and therefore divisible by 2; hence they have a common factor of 2.
    3. \( \dfrac{9}{27} \): numerator and denominator are divisible by 3 and therefore have a common factor of 3.
    4. \( \dfrac{2x}{3x} \): numerator and denominator have a common factor of \( x \).
    5. \( \dfrac{7xy}{11y} \): numerator and denominator have a common factor of \( y \).

  2. Question 2: Write the following fractions in reduced form:

    1. \( \dfrac{18}{48} \)
    2. \( \dfrac{14}{63} \)
    3. \( \dfrac{35}{165} \)
    4. \( \dfrac{1230}{2340} \)
    5. \( \dfrac{33x^2}{44x} \)
    6. \( \dfrac{15xy}{45xy^2} \)
    7. \( \dfrac{12(x-1)}{18(x-1)^2} \)
    3. Solution

    1. \( \dfrac{18}{48} \)
      GCF of (18 and 48) = 6
      \( \dfrac{18}{48} = \dfrac{18 \div 6}{48 \div 6} = \dfrac{3}{8} \)

    2. \( \dfrac{14}{63} \)
      GCF of (14 and 63) = 7
      \( \dfrac{14}{63} = \dfrac{14 \div 7}{63 \div 7} = \dfrac{2}{9} \)

    3. \( \dfrac{35}{165} \)
      5 is a common factor to both numerator and denominator:
      \( \dfrac{35}{165} = \dfrac{35 \div 5}{165 \div 5} = \dfrac{7}{33} \)

    4. \( \dfrac{1230}{2340} \)
      Both numerator and denominator are divisible by 10:
      \( \dfrac{1230}{2340} = \dfrac{1230 \div 10}{2340 \div 10} = \dfrac{123}{234} \)
      Both are divisible by 3:
      \( \dfrac{123}{234} = \dfrac{123 \div 3}{234 \div 3} = \dfrac{41}{78} \)

    5. \( \dfrac{33x^2}{44x} \)
      Write as the product of two fractions:
      \( \dfrac{33x^2}{44x} = \dfrac{33}{44} \times \dfrac{x^2}{x} \)
      Simplify using common factors 11 and \( x \):
      \( = \dfrac{3}{4} \times \dfrac{x}{1} = \dfrac{3x}{4} \)

    6. \( \dfrac{15xy}{45xy^2} \)
      Write as the product of two fractions:
      \( \dfrac{15xy}{45xy^2} = \dfrac{15}{45} \times \dfrac{xy}{xy^2} \)
      Simplify using common factors 15 and \( xy \):
      \( = \dfrac{1}{3} \times \dfrac{1}{y} = \dfrac{1}{3y} \)

    7. \( \dfrac{12(x-1)}{18(x-1)^2} \)
      Write as the product of two fractions:
      \( \dfrac{12(x-1)}{18(x-1)^2} = \dfrac{12}{18} \times \dfrac{x-1}{(x-1)^2} \)
      Simplify using common factors 6 and \( x - 1 \):
      \( = \dfrac{2}{3} \times \dfrac{1}{x-1} = \dfrac{2}{3(x-1)} \)

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