 # Reduce Fractions Step by Step

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 at the bottom of the page. Reducing fractions including variables are also included.

    

## 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 forms

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

b)
The following fractions are in NOT in reduced forms
$\quad \dfrac{2}{6}$ : because the numerator $2 = 2 \times 1$ and the numerator $6 = 3 \times 2$ and they therefore have a common factor equal to 2.

$\quad \dfrac{12}{16}$ : because the numerator $12 = 3 \times 4$ and the numerator $16 = 4 \times 4$ and they therefore have a common factor equal to 4.

$\quad \dfrac{10}{50}$ : because the numerator $10 = 10 \times 1$ and the numerator $50 = 5 \times 10$ 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 the lowest terms.

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.

$\quad \dfrac{12}{28} = \dfrac{12 \color{red}{\div 4}}{28 \color{red}{\div 4}} = \dfrac{3}{7}$

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 the lowest terms.

Solution to Example 3
The numerator and denominator are multiple of ten, we can therefore start by dividing the numerator and denominator by $10$ to obtain an equivalent fraction.

$\quad \dfrac{240}{360} = \dfrac{240 \color{red}{\div 10}}{360 \color{red}{\div 10}} = \dfrac{24}{36}$

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.

$\quad \dfrac{240}{360} = \dfrac{24}{36} = \dfrac{24 \color{red}{\div 12}}{36 \color{red}{\div 12}} = \dfrac{2}{3}$

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 the prime numbers: 2, 3, 5, 7, 11 .... until the numerator and denominator have no common factor except 1.

Example 4
Reduce the fraction $\dfrac{36}{168}$ to the lowest terms.

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

$\quad \dfrac{36}{168} = \dfrac{36 \div 2}{ 168 \div 2} = \dfrac{18}{84}$

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

$\quad \dfrac{18}{84} = \dfrac{18 \div 2}{ 84 \div 2} = \dfrac{9}{42}$

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

$\quad \dfrac{9}{42} = \dfrac{9 \div 3}{ 42 \div 3} = \dfrac{3}{14}$

$\quad \dfrac{3}{14}$ 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 $\dfrac{21 x}{36 x^2}$ to the lowest terms.

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.

$\quad \dfrac{21 x}{36 x^2} = \dfrac{21}{36} \times \dfrac{x}{x^2}$

Reduce fraction in numerical form using any method. The GCF of 21 and 36 is equal to 3. Hence

$\quad \dfrac{21}{36} = \dfrac{21 \div 3}{36 \div 3 } = \dfrac{7}{12}$

Reduce fraction in algebraic form. The common factor to $x$ and $x^2$ is $x$. We therefore divide the numerator and the denominator by the common factor $x$.

$\quad \dfrac{x}{x^2} = \dfrac{x \div x}{x^2 \div x} = \dfrac{1}{x}$

We now rewrite the given fraction in reduced form

$\quad \dfrac{21 x}{36 x^2} = \dfrac{21}{36} \times \dfrac{x}{x^2}$

$\quad = \dfrac{7}{12} \times \dfrac{1}{x} = \dfrac{7}{12 x}$

$\quad \dfrac{7}{12 x}$ is the reduced form of the given fraction

## Questions

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. $\dfrac{2 x }{3 x }$

7. $\dfrac{7 x y}{11 y }$

2. Write in reduced forms the following fractions
1. $\dfrac{18}{48}$

2. $\dfrac {14}{63}$

3. $\dfrac {35}{165}$

4. $\dfrac{1230}{2340}$

5. $\dfrac{33 x^2 }{44 x}$

6. $\dfrac{15 x y }{45 x y^2}$

7. $\dfrac{12 (x-1) }{18 (x-1)^2}$

## Answers to the Above Questions

1. The following fractions are NOT in reduced form.
a) $\dfrac{20}{155}$ , numerator and denominator are divisible by 5 and therefore have common factor 5.

c) $\dfrac{246}{1246}$ , numerator and denominator are even numbers and therefore divisible by 2; hence have common factor 2.

e) $\dfrac{9}{27}$ , numerator and denominator are divisible by 3 and therefore have common factor 3.

f) $\dfrac{2 x }{3 x }$ , numerator and denominator have common factor $x$.

g) $\dfrac{7 x y}{11 y }$ , numerator and denominator have common factor $y$.

2. Write in reduced form.

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

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

3. $\dfrac {35}{165}$
5 is a common factor to both numerator and denominator. Hence

$\dfrac {35}{165} = \dfrac {35 \div 5 }{165 \div 5} = \dfrac {7}{33}$

4. $\dfrac{1230}{2340}$
Both numerator and denominator are divisible by 10. Hence

$\dfrac{1230}{2340} = \dfrac {1230 \div 10 }{2340 \div 10} = \dfrac {123}{234}$

Both numerator and denominator are divisible by 3. Hence

$\dfrac {123}{234} = \dfrac {123 \div 3 }{234 \div 3} = \dfrac {41}{78}$

5. $\dfrac{33 x^2 }{44 x}$
Write as the product of two fractions.
$\dfrac{33 x^2 }{44 x} = \dfrac{33 }{44 } \times \dfrac {x^2}{x}$

Both $33$ and $44$ have common denominator $11$ and $x^2$ and $x$ have common factor $x$. Hence

$\dfrac{33 x^2 }{44 x} = \dfrac{33 \div 11 }{44 \div 11} \times \dfrac {x^2 \div x }{x \div x}$

Simplify

$= \dfrac {3}{4} \times \dfrac{x}{1} = \dfrac{3 x}{4}$

6. $\dfrac{15 x^2 y }{45 x^2 y^2}$
Write as the product of two fractions.
$\dfrac{15 x^2 y }{45 x^2 y^2} = \dfrac{15 }{45 } \times \dfrac {x^2 y}{x^2 y^2}$

Both $15$ and $45$ have common factor $15$ and $x^2 y$ and $x^2 y^2$ have common factor $x^2 y$. Hence

$\dfrac{15 x^2 y }{45 x^2 y^2} = \dfrac{15 \div 15 }{45 \div 15} \times \dfrac { x^2 y \div x^2 y }{ x^2 y^2 \div x^2 y}$

Simplify

$= \dfrac {1}{3} \times \dfrac{1}{y} = \dfrac{1}{3 y}$

7. $\dfrac{12 (x-1) }{18 (x-1)^2}$ Write as the product of two fractions.

$\dfrac{12 (x-1) }{18 (x-1)^2} = \dfrac{12 }{18 } \times \dfrac {(x-1)}{(x-1)^2}$

Both $12$ and $18$ have common factor $6$ and $x - 1$ and $(x-1)^2$ have common factor $x - 1$. Hence

$\dfrac{12 (x-1) }{18 (x-1)^2} = \dfrac{12 \div 6 }{18 \div 6} \times \dfrac { (x - 1) \div (x - 1) }{ (x-1)^2 \div (x - 1)}$

Simplify

$= \dfrac {2}{3} \times \dfrac{1}{x - 1} = \dfrac{2}{3 (x-1)}$