#### 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

### Solved Examples

**Question 1:**Simplify $\frac{20}{25}$

**Solution:**

Here

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.

**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 $\frac{126}{90}$

= $\frac{63}{45}$

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

= $\frac{21}{15}$

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

= $\frac{7}{5}$

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

## How to Solve Fraction

Steps for simplifying 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 36 = 2 x 2 x 3 x 3

The common factors are 2 x 2 =

**4**

Step 2:

Step 2:

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

= $\frac{8}{9}$

## 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:

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.