## Fractions

## Fraction :-

Fraction is a part of a whole. A whole can be a region or a collection.

For example- | , 1 is a part of 4. | |

4 |

For example- | is a fraction, | |

7 |

## Simple Fraction :-

If the fraction has denominator other than power of 10, is called simple fraction.

For example- | , | , | etc. | |||

5 | 7 | 15 |

## Types of simple Fractions :-

### Like Fractions :-

Fractions having same denominator is called like fractions.

For example- | , | , | , | etc. | ||||

5 | 5 | 5 | 5 |

### Unlike Fractions :-

Fractions having different denominator is called unlike fractions.

For example- | , | , | , | etc. | ||||

3 | 5 | 7 | 4 |

### Unit Fraction :-

Fraction having numerator 1 is called unit fraction.

For example- | , | , | , | etc. | ||||

4 | 5 | 7 | 3 |

### Proper Fraction :-

Fraction in which numerator is less than its denominator is called proper fraction.

For example- | , | , | etc. | |||

3 | 11 | 18 |

### Improper Fraction :-

Fraction in which numerator is greater than its denominator is called improper fraction.

For example- | , | , | etc. | |||

3 | 11 | 18 |

### Mixed Fraction :-

Fraction in which whole part and fraction part is called mixed fraction.

For example- 5 | , 4 | , 1 | etc. | |||

3 | 5 | 15 |

### Compound Fraction :-

Fraction in which numerator or denominator or both are in fraction, then it is called compound fraction.

For example- | , | , | etc. | |||

2 / 3 | 5 / 7 | 3 / 8 |

### Inverse Fraction :-

When we inverse the numerator and the denominator of a fraction, then the new fraction is called inverse fraction.

For example- Inverse fraction of | is | . | ||

7 | 2 |

### Decimal Fraction :-

Fraction in which denominator in the powers of 10 is called decimal fraction.

For example- | = 0.2, | = 0.7 . | ||

10 | 10 |

## Types of Decimal Fractions :-

### Recurring Decimal Fraction :-

Decimal fraction in which one or more decimal digits are repeated is called recurring decimal fraction.

For example- | = 0.333…. = 0.3 , | = 8.3333…. = 8.3 | ||

3 | 3 |

### Pure Recurring Decimal Fraction :-

Decimal fraction in which all digits are repeated after the decimal point is called pure recurring decimal fraction.

For example- 0.3, 0.489 etc.

### Mixed Recurring Decimal Fraction :-

Decimal fraction in which some digits are repeated and some are not repeated after decimal point is called mixed recurring decimal fraction.

For example- 0.36, 3.223 etc.

### Conversion of Mixed fraction into Improper fraction

To convert mixed fraction into improper fraction, first multiply the whole part by denominator, after multiplication add the numerator then write the answer.

For example- 3 | = | = | |||

7 | 7 | 7 |

And 4 | = | = | |||

12 | 12 | 12 |

### Conversion of Improper fraction into Mixed fraction :-

To convert improper fraction into mixed fraction, first divide the numerator by the denominator and find quotient and remainder, then quotient will be whole part, remainder will be numerator and denominator will be same.

For example- | = 5 | ||

3 | 7 |

And | = 20 | ||

7 | 7 |

## Lowest term :-

A fraction is in the lowest term when the only common factor between the numerator and denominator is 1.

or, A fraction is in the lowest term when the HCF of numerator and denominator is 1.

For example- | , | , | etc. | |||

5 | 7 | 7 |

To reduced any fraction into lowest term, first find the HCF of numerator and denominator, then divide both numerator and denominator by HCF.

For example- | = 20 | |

30 |

= | = | ||

30 ÷ 6 | 5 |

### Reciprocals

Two numbers whose product is 1 are called reciprocals.

For example - | and | = | × | = 1 | ||||

5 | 3 | 5 | 3 |

To find the reciprocals of any numbers interchange the numerator and denominator.

For example - Reciprocal of | is | . | ||

11 | 8 |

### Equivalent Fractions :-

Fractions which are equal to the other fractions when they are reduced to lowest term is called equivalent fractions.

For example - Find the equivalent fraction of | which has a denominator of 72 ? | |

9 |

Solution : - |
= | = | |||

9 | 9 × 8 | 72 |

## Operations on Simple Fractions :-

### Addition of Simple Fractions :-

#### Addition of like fractions :-

When denominators of fractions are same, then numerators of fraction are added and addition is divided by denominator.

For example- | + | = | = | ||||

7 | 7 | 7 | 7 |

#### Addition of unlike fraction :-

When denominators of fractions are different, then make their denominators equal by taking their LCM and then add their numerators.

For example- | + | ||

3 | 5 |

= | = | ||

15 | 15 |

#### Addition of mixed fractions :-

To add mixed fraction first convert the mixed fracton into improper fraction, then add them. After addition convert the result into mixed fraction.

For example :- 2 | + 7 | ||

5 | 5 |

= | + | ||

5 | 5 |

= | + | ||

5 | 5 |

= | = | ||

5 | 5 |

### Subtraction of Simple Fractions :-

#### Subtraction of like fractions :-

When denominators of fractions are same, then numerators of fraction are subtracted and their subtraction is divided by denominator.

For example :- | - | ||

11 | 11 |

= | = | ||

11 | 11 |

#### Subtraction of unlike fractions :-

When denominators of fractions are different, then make their denominators equal taking their LCM and then subtract their numerator.

For example :- | - | ||

4 | 3 |

= | = | ||

12 | 12 |

= | |

12 |

#### Subtraction of mixed fractions

To subtract mixed fraction first convert mixed fraction into improper fraction then subtract them. And after subtraction convert the result into mixed fraction

For example- 9^{2}/_{3} - 4^{1}/_{3}

= {( 9 × 3 ) + 2} /3 - {( 4 × 3 ) + 1} /3

= 29/3 - 13/3

= ( 29 - 13 ) /3

= 16/3

= 5^{1}/_{3}

### Multiplication of Simple Fractions

To multiply fractions, first simply the fraction in lowest term, then multiply their numerator by numerator and denominator by denominator.

For example- 3/4 × 7/5 = (3 × 7 )/ (4 × 5 ) = 21/20

#### Multiplication of mixed fraction

To multiply mixed fraction, first convert mixed fraction into improper fraction, then multiply them and after multiplication convert the result into mixed fraction.

For example- Multiply 2^{3}/_{4} × 5^{4}/_{3}

= {( 2 × 4 ) + 3} / 4 × {( 5 × 3 ) + 4} /3 = 11/4 × 19/3

= (11 × 19) /(4 × 3 )

= 209/12

= 17^{5}/_{12}

### Division of Simple Fractions

To divide fractions, first find the reciprocal of the second fraction then multiply the first fraction by the reciprocal of the second fraction.

For example- 3/7 ÷ 4/5

= 3/7 × 5/4

= ( 3 × 5 ) / ( 7 × 4 )

= 15/28

### Division of mixed fractions

To divide mixed fractions, first convert the mixed fractions into improper fractions, then find the reciprocal of second fraction and multiply the first fraction by the reciprocal of second fraction.

For example- 8^{1}/_{3} ÷ 10^{5}/_{6}

= ( 8 × 3 ) + 1 /3 ÷ ( 10 × 6 ) + 5 /6

= 24 + 1 /3 ÷ 60 + 5 /6 = 25/3 ÷ 65/6

= 25/3 × 6/65

= 10/13

### Operations on Decimal Fractions

#### Addition of Decimal Fractions

To add two or more decimal fractions, write the given numbers in column such that the decimal points lie in one column then add them.

For example- 3424.5 + 254.42 = 3678.92

#### Subtraction of Decimal Fractions

To subtract two or more decimal fractions, write the given numbers in column such that the decimal points lie in one column then subtract them.

For example- 2345.67 - 364.09

= 2709.58

#### Multiplication of Decimal Fractions

To multiply decimal fraction, first multiply the given numbers without considering the decimal points and after multiplication put decimal from the right side in the result as many places of decimal as the sum of the numbers of decimal places in the multiplier and the multiplicand together.

For example- 5.2 × 0.23

= 52 × 23

= 1196

Sum of the decimal places = ( 1 + 2 ) = 3

= 1.196

#### Division of Decimal Fractions by an Integers

To divide decimal fraction, first divide the given numbers without considering the decimal point and place the decimal point as many places of decimal as in the dividend

For example- 0.729 ÷ 9

= 729/9

= 81 = 0.081

### Comparison of Simple Fractions

In this, method first find the LCM of the denominators of all fractions, so that the denominators of all fractions are same then compare the numerators.

For example - Arrange the following fractions in ascending order.

7/12, 2/3, and 3/8 Solution:- LCM of 12, 3 and 8 is 24.

7/12 = 7 × 2 / 12 × 2 = 14/24

2/3 = 2 × 8 / 3 × 8 = 16/24

3/8 = 3 × 3 / 8 × 3 = 9/24

9/24 < 14/24 < 16/24

3/3 < 7/12 < 2/3