Fractions - An Introduction

What is a Fraction?

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We define fraction as a part of a whole number or fractions are for counting part of something. For example, we have a pizza which is to be shared among 4 friends so what is the share for each friend?





Below figures represent pizza and the share for one person respectively.

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Types of fractions

•    Proper fractions

•    Improper fractions

•    Mixed fractions
 
Proper Fraction – A fraction, whose numerator is less than the denominator is called Proper Fraction.
 For example – 7/9, 3/11
Improper Fraction – A fraction whose numerator is greater than the denominator is called Improper Fraction.
For example – 17/5, 47/31
Mixed Fraction - A combination of a whole number and a proper fraction is called a Mixed Fraction.
For example – 2 3/5, 7 4/15

Conversion between Improper and Mixed fractions

In order to convert a mixed fraction into an improper fraction, we may use the following formula:

Improper Fraction = (Whole Number X Denominator) + Numerator
                                                       Denominator


For example – 3 2/5 = 3 X 5 +2 = 15 +2 = 17                                                  
                                      5         5       5
Now, to express an improper fraction as a mixed fraction, we first divide the numerator by denominator and calculate the quotient and remainder and then we write the mixed fraction as
                Remainder
Quotient    
                Denominator

For example – 19      = 4 ¾                     [Quotient = 4, Remainder = 3]
                      4


Fractions – Standard Form

Types of fractions used in reducing fractions to their lowest or standard form

Equivalent Fractions – A given fraction and various fractions obtained by multiplying (or dividing) its numerator and denominator by the same non – zero number, are called Equivalent fractions.
For example – 3 X 2 = 6,   3 X 3 = 9,    3 X 4 = 12
                      4 X 2    8   4 X 3   12     4 X 4      16

Like Fractions – Fractions having the same denominators are called like fractions.
For example – 2, 7
                     15 15

Unlike Fractions – Fractions with different denominators are called unlike fractions

For example – 2, 7
                     13 24

How to write fraction in its standard form

Fraction In Lowest Terms – A fraction is in its lowest terms if its numerator and denominator have no common factor other than 1.

For example - Reduce 144
                              180
First we find the HCF of 144 and 180 by factorization method.
The factors of 144 are: 1,2,3,4,6,8,9,12,16,18,24,36,48,72and 144
The factors of 180 are: 1,2,3,4,5,6,10,12,15,18,30,36,45,60,90 and 180
The common factors of 144 and 180 are: 1, 2, 3,4,6,12,18 and 36
So, HCF of 144 and 180 is 36.
Dividing numerator and denominator by the HCF of 144 and 180 i.e., 36
Now, 144    = 144 ÷ 36 = 4      
        180       180 ÷ 36   5

Comparing Fractions

Comparing Fractions – In order to compare fractions, we may use the following steps:
•    Find the LCM of the denominators of the given fractions.
•    Convert each fraction to its equivalent fraction with denominator equal to the LCM obtained in step 1.
•    Arrange the fractions in ascending or descending order by arranging numerators in ascending or descending order.
For example, which is larger ¾ or 5/12?
Let us first find the LCM of 4 and 12.
We have,
2  4   12
2  2     6
3  1     3
    1     1

LCM of 4 and 12 is 2 X 2 X 3= 12


Now we convert the given fractions to equivalent fractions with denominator 12.
3 = 3 X 3 = 9
4    4 X 3   12
We know that 9 > 5
9 > 5         3 > 5
12  12       4     12

Conversion of unlike fractions to like fractions

To convert unlike fractions to like fractions we follow the following steps:
•    Find the LCM of the denominators of the given fractions.
•   Convert each of the given fractions into an equivalent fraction having denominator equal to the LCM obtained in previous step.

For example – Convert the unlike fractions 7/6, 5/9 and 5/12 into like fractions.
We have, LCM (6, 9, 12) = (3 X 2 X 3 X 2) = 36
Now, 7 = 7 X 6 = 42 ; 5 = 5 X 4 = 20 ; 5 = 5 X 3 = 15
        6     6 X 6    36   9     9 x 4    36  12   12 X 3   36
Clearly, 42/36, 20/36 and 15/36 are like fractions.


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