**Fractions:**

The number of the form (a/b), where *a* and *b* are natural numbers are known as fractions. In a fraction (a/b), a is known as numerator and *b* is known as denominator. Note:

- Fraction is a part of whole.
- The numerator and denominator of a fraction are called its terms.

**Types of Fractions:**

*Decimal Fractions:* A fraction whose denominator is any of the numbers 10, 100, 1000 etc., is called a decimal fraction.

*Vulgar Fraction: *A fraction whose denominator is a natural number other than the numbers 10, 100, 1000, etc., is called a Vulgar Fraction.

*Proper Fraction:* A fraction whose numerator is less than its denominator is called proper fractions.

*Improper Fraction:* A fraction whose numerator is greater than or equal to its denominator, is called an improper fraction.

*Mixed Number (or Mixed Fraction):* When an improper fraction is written as a combination of whole number and a proper fraction, it is called a mixed number (or mixed fraction).

How to Convert a Mixed Fraction into an Improper Fraction?

The numerator is obtained by multiplying the whole number part with the denominator of the fractional part and then adding the numerator of the fractional part to the product. The denominator of the fractional part forms the denominator of the improper fraction.

How to convert improper fraction into a Mixed Fraction?

*Divide the numerator of the given improper fraction by the denominator. The quotient obtained forms the whole number part, the remainder obtained forms the numerator of the fractional part and the divisor forms the denominator of the fractional part.*

*Simple Fraction:* A fraction each of whose term is a whole numbers is called a simple fraction.

*Complex Fraction:* A fraction at least one of whose terms is itself a fraction is called a complex fraction.

*Equivalent Fractions:* Fractions having the same value are called equivalent fractions.

*Like Fractions:* Fractions having the same denominators are called like fractions.

*Unlike Fractions:* Fractions having different denominators are called unlike fractions.

How to convert Unlike Fractions into Like Fractions?

- Find the L.C.M of the denominators of all the given unlike Fractions.
- Convert each of the fractions into an equivalent fraction having denominator equal to the L.C.M of the denominator.
- All the fractions would have the same denominator now and would be like fractions.

Fractions in Lowest Terms or in Simplest Form

A fraction is said to be in lowest or simplest form if the H.C.F of the numerator and denominator is 1.

The fraction can be reduced to its lowest form if you divide both the numerator and the denominator by their H.C.F.

H.C.F of 126 and 162 = 18

Comparison of Fractions

*Comparison of Fractions with Like Denominators and Unlike Numerators:* If the fraction has the same denominator, then the fraction with larger Numerator is the larger fraction.

*Comparison of Fractions with Like Numerators by unlike Denominator:* If the two fractions have the same numerator but different Denominator, then the fraction with the smaller Denominator is the larger fraction.

*Comparison of Fractions with Unlike Numerators and Unlike Denominators*

- Method 1
- Step 1: First take the LCM of the denominators of the fractions
- Step 2: Then convert the given unlike fractions into equivalent fractions with the LCM as the common denominator
- You can compare the like fractions so obtained

- Method 2
- Short Cut Method (Cross Multiplication Method)
- If are two fractions to be compared then

To Insert Fractions between Fractions

Example

**Operations on Fractions**

Addition of Fractions

*Addition of Like Fractions**Addition of unlike Fractions*- Step 1: Find the LCM of Denominators of unlike fractions
- Step 2: Convert unlike fractions into equivalent Fractions with their LCM as the common Denominators
- Step 3: Add the like fractions so obtained

Subtraction of Fractions

The same rules are followed as above. Instead of + operator, – operator comes into play.

Multiplication of Fractions

The answer obtained is reduced to the lowest form by dividing the numerator and denominator by their HCF.

Easier way of multiplying fractions is that if there are common factors in one of the numerators and one of the denominators, then we can cancel that before multiplying.

Division of Fractions

In order to divide two fractions, we just multiply the dividend by the reciprocal of the Divisor.

Reciprocal of a Non-Zero Fraction

Let (a/b) be a non-zero fraction. Then *a* and *b* are natural numbers.

The Fraction (b/a) is called the reciprocal of (a/b)

**Simplification of Expressions Involving Fractions**

Use of BODMAS Rule (remember the word BODMAS)

- We simplify the expressions by applying the operations strictly in the order
- Brackets
- Of
- Division
- Multiplication
- Addition
- Subtraction

- Removal of Brackets: Follow this order
- Bar or Vinculum (—)
- Parenthesis ( )
- Curly Brackets { }
- Square Brackets [ ]