Simplifying Fractions

Simplifying fractions mean reduce the fraction to lowest terms or write the fraction in the lowest terms. A fraction is simply the means by which a whole of something is divided into smaller components. Various operations are applied on fractions like addition, subtraction, multiplication, division and simplification of fractions. Here we discussed about the simplification of the fractions. Simplifying fractions means to make the fraction as simple as possible. Divide the top and bottom by the highest number that can divide into both numbers exactly.

Simplify the fractions mean that the numerator and denominator have no common factors other than 1.

Easiest Way to Simplify Fractions

There are different methods to simplify a fraction:

Method 1:
Both the numerator and denominator can evenly divided by the same number without changing the value of the fraction. We may have to divide more than once to find the simplest form.

Example: Simplify $\frac{18}{24}$

Step 1: Divide by 2

=> $\frac{18}{24}$ = $\frac{9}{12}$

Step 2: Divide by 3

=> $\frac{9}{12}$ = $\frac{3}{4}$

=> $\frac{18}{24}$ is reduced to $\frac{3}{4}$

Method 2:
Factor the numerator and denominator to find the greatest common factor, then divide numerator and denominator by that GCD.

Example: Simplify $\frac{10}{5}$

Step 1:
Greatest common factor of 10 and 5 is 5

Step 2:
Multiply numerator and denominator by 5

=> $\frac{10}{5}$ = $\frac{10}{5}$ * $\frac{5}{5}$

= $\frac{2}{1}$

= 2

Method 3:
Write the numerator and denominator as a product of prime numbers. Split each fraction into two fractions, the first with the common prime numbers. This puts the fraction in the form of 1 times another fraction.

Solved Example
Question: Simplify, $\frac{6}{9}$
Solution:
Step 1: Prime factors of 6 and 9

Factors of 6 = 2 * 3

Factors of 9 = 3 * 3

Step 2: Now write the common factors as a separate fraction:

$\frac{6}{9}$ = $\frac{2 * 3}{3 * 3}$

=
$\frac{3}{3}$  * $\frac{2}{3}$

=
$\frac{2}{3}$

Simplifying Fractions

Simplify the fractions mean that the numerator and denominator have no common factors other than one. To simplify a fraction, divide the numerator and denominator by the greatest common factor or find the prime factors of the numerator and denominator and reduced the fraction by cancelling common factors. When a fraction is simplified, then no common factor between numerator and denominator other than one.

Solved Examples

Question 1: Simplify, $\frac{75}{30}$
Solution:
Given, $\frac{75}{30}$

Step 1: Prime factors of 75 and 30

Factors of 75 = 3 * 5 * 5

Factors of 30 = 2 * 3 * 5

Step 2: Now write the common factors as a separate fraction:

$\frac{75}{30}$ = $\frac{3 * 5 * 5}{2 * 3 * 5}$

$\frac{3 * 5}{3 * 5}$ * $\frac{ 5}{2}$

= 1 * $\frac{5}{2}$

= $\frac{5}{2}$

=> $\frac{75}{30}$  is simplifies to  $\frac{5}{2}$

Question 2: Simplify, $\frac{150}{250}$
Solution:
Given, $\frac{150}{250}$

Step 1: Prime factors of 150 and 250

Factors of 150 = 2 * 3 * 5 * 5

Factors of 250 = 2 * 5 * 5 * 5

Step 2: Now write the common factors as a separate fraction:

$\frac{150}{250}$ = $\frac{2 * 3 * 5 * 5}{2 * 5 * 5 * 5}$

= $\frac{2 * 5 * 5}{2 * 5 * 5}$ * $\frac{3}{5}$

= 1 * $\frac{3}{5}$

= $\frac{3}{5}$

=> $\frac{150}{250}$ = $\frac{3}{5}$

Simplify Algebraic Fractions

Algebraic fractions have properties which are the same as those for numerical fractions, the only difference being that the the numerator and denominator are both algebraic expressions. Simplification of a algebraic fraction is same as we simplify fractions. Simplifying fraction is a way to avoid common fraction. Fractions written in simplest form have no factor other than one.

Solved Examples

Question 1: Simplify $\frac{x^2 - 1}{x + 1}$ to simplest form
Solution:
Given, $\frac{x^2 - 1}{x + 1}$

$\frac{x^2 - 1}{x + 1}$  = $\frac{(x - 1)(x + 1)}{x + 1}$

[Using identity a2 - b2 = (a - b)(a + b)]

= x - 1

=> $\frac{x^2 - 1}{x + 1}$  is reduced to (x - 1).

Question 2: Simplify $\frac{15a^2b^3}{25a^3b^2}$
Solution:
Given $\frac{15a^2b^3}{25a^3b^2}$

Step 1: Find the factors of numerator and denominator

15a2 b3 = 3 * 5 * a * a * b * b * b

25a3 b2 = 5 * 5 * a * a * a * b * b

Step 2:

$\frac{15a^2b^3}{25a^3b^2}$  =
$\frac{3 * 5 * a * a * b * b * b}{5 * 5 * a * a * a * b * b}$

=
$\frac{3b}{5a}$

=> $\frac{15a^2b^3}{25a^3b^2}$  = $\frac{3b}{5a}$

Question 3: Simplify $\frac{(2x + 3)(x - 3)}{x^2 - 9}$
Solution:
Given  $\frac{(2x + 3)(x - 3)}{x^2 - 9}$

Step 1: Find the factors of x2 - 9

x2 - 9 = x2 - 32 = (x - 3)(x + 3)

Step 2:

$\frac{(2x + 3)(x - 3)}{x^2 - 9}$ = $\frac{(2x + 3)(x - 3)}{(x - 3)(x + 3)}$

=
$\frac{2x + 3}{x + 3}$

=>
$\frac{(2x + 3)(x - 3)}{x^2 - 9}$ = $\frac{2x + 3}{x + 3}$.