How to Simplify Fractions

Introduction: A fraction is of the form $\frac{a}{b}$ where b $\neq$ 0.

For example, $\frac{2}{3}$ , $\frac{5}{4}$

Types of fractions.
1. Like fractions
2. unlike fractions.
3. Proper fractions
4. Improper fractions.
5. Mixed Fractions.
1. Like fractions :Fractions which have the same denominator but different numerators are called like fractions..
Example : $\frac{3}{7}$, $\frac{4}{7}$, $\frac{5}{7}$, $\frac{6}{7}$
2. Unlike Fractions: Fractions which have  different denominators are called unlike fractions.
$\frac{8}{11}$, $\frac{9}{21}$
3. Proper Fractions:Fractions in which the numerator is smaller than the denominator are called proper fractions.
Example : $\frac{4}{9}$, $\frac{7}{9}$
4. Improper Fractions: Fractions in which the numerator is greater than the denominator are called improper fractions.
Example : $\frac{7}{2}$, $\frac{14}{3}$
5. Mixed Fractions: Fractions  which are a combination of a whole number and a proper fraction.
Example : 3 $\frac{1}{2}$, 5$\frac{2}{3}$

Simplification of Fractions

Simplification of the fraction means reducing the fraction to its simplest form. To simplify a fraction, divide the numerator and denominator by the highest number that can divide into both numbers exactly, without changing the value of the fraction.

Solved Examples

Question 1: Simplify $\frac{20}{25}$
Solution:
Here 5 is the common factor for the numerator and the denominator.

Hence $\frac{20\div 5}{25\div5}$ = $\frac{4}{5}$

We see that 4 and 5 are co-prime numbers which do not have any other factors other than 1 and itself.
 

Question 2: Simplify $\frac{126}{90}$

Solution:
$\frac{126}{90}$  = $\frac{126\div2}{90\div2}$

=
$\frac{63}{45}$

= $\frac{63\div3}{45\div3}$

=
$\frac{21}{15}$

= $\frac{21\div3}{15\div3 }$

= $\frac{7}{5}$


$\frac{7}{5}$ is the simplest form of the fraction $\frac{126}{90}$
 

How to Solve Fraction

Solve the fraction means to find the simplest possible form of fraction whose value is equal to that of the given fraction. We divide the numerator and the denominator by the HCF also called Highest Common Divisor of the numerator and the denominator to get the simplest fraction.  

Steps for simplifying fraction:


Step 1: Find the highest common divisor of the numerator and the denominator.

Step 2: Divide the numerator and the denominator by the highest common divisor.

Step 3:  Resultant fraction is the simplest form of the given fraction.

Solved Examples

Question 1: Simplify  $\frac{45}{54}$




Solution:
Step 1:

Let us find the HCF of 45 and 54.

Factors of 45 = 3 x 3 x 5

Factors of 54 = 2 x 3 x 3 x 3

The common factors are 3 x 3 = 9

⇒ Highest Common Factor = 9

Step 2:

Dividing the fraction by Highest Common Factor 9

$\frac{45}{54}$ = $\frac{45\div9}{54\div9}$

= $\frac{5}{6}$

Hence the simplest form of $\frac{45}{54}$ is $\frac{5}{6}$

 

Question 2: Simplify $\frac{32}{36}$
Solution:

Step 1:

Let us first find the HCF of  32 , 36

Factors of 32 = 2 x 2 x 2 x 2 x 2

Factors of 36 = 2 x 2 x 3 x 3

The common factors are 2 x 2 = 4

Step 2:


$\frac{32}{36}$ = $\frac{32\div4}{36\div4}$

= $\frac{8}{9}$

 

Prime Factorization Method

Steps to solve the fractions by using prime factorization method:

Step 1: First all the prime factors of the numerator and the denominator.

Step 2: Cancel the common factors in the numerator and the denominator.

Step 3: Multiply the remaining factors, if any.

Solved Example

Question: Simplify $\frac{180}{210}$ by using prime factorization method.

Solution:
Step 1:

Prime factors of 180 and 210

180 = 2 x 2 x 3 x 3 x 5

210 = 2 x 3 x 5 x  7

Step 2:


$\frac{180}{210}$ = $\frac{2\times2\times3\times3\times5}{2\times3\times5\times7}$

=
$\frac{2\times3}{7}$

=$\frac{6}{7}$
 

Practice Questions


Practice Problems

Question 1: Write the simplest form of  $\frac{18}{36}$
Question 2: Simplify $\frac{35}{55}$
Question 3:

Simplify $\frac{125}{225}$ by using prime factorization method.