Addition and Subtraction of Unlike Fractions.

ADDING AND SUBTRACTION OF UNLIKE FRACTIONS:

Steps to find Addition or Subtraction of Unlike fractions:

a. We need to convert Unlike fractions into Equivalent fractions with common denominators. i.e; Like fractions.

 (i) Need to find L.C.M of denominators.

 (ii) Converting Unlike fractions into Equivalent fractions.

 (iii) Thus, formed Like fractions are Required fractions.

b. Addition or Subtraction of Fractions = We need to add or subtract the Numerators, which gives the required solution.

 

ADDITION OF UNLIKE FRACTIONS:

1)  3/8 + 1/6

Solution:

Given fractions: 3/8 , 1/6 ; Unlike Fractions. Denominators = 8, 6.

a. Converting Unlike fractions into Like fractions:

   (i) L.C.M of denominators = L.C.M ( 8,6 ) = 24.

   (ii) Equivalent fractions: 3/8 = 9/24 ;  1/6 = 4/24.

   (iii) Thus, Required fractions = 9/24 , 4/24 ; Like fractions. 

b. Addition of given fractions = Addition of Equivalent fractions.

   = 3/8 + 1/6  

   =  9/24 + 4/24  

   =  13/24.

 

2) 3/7 + 2/3 + 1/3

Solution:

Given fractions: 3/7, 2/3, 1/3; Unlike fractions. Denominators = 7 , 3, 3.

a. Converting Unlike fractions into Like fractions:

    (i) L.C.M of denominators = L.C.M ( 7 , 3, 3 ) = 21.

    (ii) Equivalent fractions: 3/7 = 9/21 ;  2/3 = 14/21 ;  1/3 = 7/21.

    (iii) Thus, Required fractions = 9/21, 14/21, 7/21 ; Like fractions.

b. Addition of given fractions =  3/7 + 2/3 + 1/3

    = 9/21 + 14/21 + 7/21

    = 30/21.

3)  3/5 – 2/9

Solution:

Given fractions = 3/5 , 2/9 ; Unlike fractions. Denominators = 5, 9.

a. Converting Unlike fractions into Like fractions.

   (i) L.C.M of denominators = L.C.M ( 5,9 ) = 45.

   (ii) Equivalent fractions: 3/5 = 21/45 ; 2/9 = 10/45.

   (iii) Thus, Required fractions = 21/45 , 10/45 ; Like fractions.

b. Subtracting  given fractions = 3/5 – 2/9

    = 21/45 – 10/45

    = 11/45.

     

4) 5 + 6/7

Solution:

Given fractions: 5/1 , 6/7; Unlike fractions. Denominators = 1, 7.

a. Converting Unlike fractions into Like fractions.

    (i) L.C.M of denominators = L.C.M ( 1 , 7 ) = 7.

    (ii) Equivalent fractions:  5/1 = 35/1 ;  6/7 = 6/7.

    (iii) Thus, required fractions = 35/1 , 6/7 ; Like fractions.

b. Addition of given fractions = 5 + 6/7

    = 35/1 + 6/7

    = 41/7.  

 

5) 3 – 1/5

Solution:

Given fractions: 3/1 , 1/5 ; Unlike fractions. Denominators = 1 , 5.

a. Converting Unlike fractions into Like fractions.

    (i) L.C.M of denominators = L.C.M ( 1 , 5 ) = 5.

    (ii) Equivalent fractions:  3/1 = 15/1 ;  1/5 = 1/5.

    (iii) Thus, required fractions = 15/1 , 1/5 ; Like fractions.

b. Subtracting given fractions = 3 – 1/5

    = 15/1 – 1/5

    = 14/5.  

Converting Unlike Fractions into Like Fractions.

CONVERTING UNLIKE FRACTIONS INTO LIKE FRACTIONS:

An Unlike fraction is converted into Equivalent fractions by using the below steps

1. We need to find the L.C.M of the denominators.

2. The given fraction is multiplied with the L.C.M, which gives the new numerator of the given fraction.

3. The required fraction is New numerator upon L.C.M.   

Required Fraction = New Numerator / L.C.M.

 

Convert Unlike fractions into Equivalent fractions:

1) 1/2 , 1/3

Solution:

Given fractions: 1/2 , 1/3 ; Denominators = 2 , 3.  

a. We need to find the L.C.M of the denominators: 2, 3

L.C.M ( 2, 3 ) = 6.

b. New numerator = (Given fraction) x (L.C.M )

  Therefore, Numerator of the fraction 1/2  becomes :  (1/2 ) x 6 = 3  

Numerator of the fraction 1/3 becomes = (1/3 ) x 6 = 2

c. Required fractions = new numerator / L.C.M = 1/2 = 3/6  ;  1/3 = 2/6; Equivalent fractions.

Thus, New Fractions = 3/6 , 2/6 ; Like fractions .  

 

2)  2/5, 3/4, 7/3

Solution:

Given Denominators = 5, 4, 3.

a. L.C.M ( 5, 4, 3 ) = 60.

b. New Numerators = (2/5) x 60 = 24;

      (3/4 ) x 60 = 45 ;  ( 7/3 ) x 60 = 140.

c. Required fractions:

 2/5 = 24 /60  ;  3/4 = 45/60 ;   7/3 = 140/60 ; Equivalent fractions.      

Thus, New fractions : 24/60 , 45/60 , 140/60 ; Like fractions.

 

3) 7/8 , 6/4 , 9/2

Solution:

Given Denominators = 8, 4, 2.

a. L.C.M ( 8, 4, 2 ) = 8.

b. New Numerators =

    (7/8) x 8 = 7 ;   (6/4) x 8 = 12 ;   (9/2) x 8 = 36.

c. Required fractions:

 7/8 = 7/8  ;  6/4 = 12/8 ;  9/2 = 36/8 ; Equivalent fractions.

Thus, New Fractions = 7/8 , 12/8 , 36/8 ; Like fractions.   

  

Comparision of Like fractions.

COMPARISION OF LIKE FRACTIONS:

When two or more like fractions are compared, the Largest fraction is the one with the greater numerator and the Smallest fraction is the one with the smallest numerator.

 

Compare the fractions and write < , > , = ;between them:

1)  9/12, 1/12, 5/12

Solution:

In the given fractions the numerators are = 9, 1, 5.

Common denominator = 12.

(i)Arranging the numerators in Ascending order = 1 < 5 < 9.

Ascending order of the fractions = 1/12 <  5/12 < 9/12.

(ii) Descending order of the numerators = 9 > 5 > 1.

Descending order of the fractions = 9/12 > 5/12 > 1/12.

 

2) 3/8, 1/8, 7/8

Solution:

The numerator of the fractions are = 3, 1, 7.

Common denominator = 8.

(i) Ascending order of the numerators = 1 < 3 < 7.

Ascending order of the fractions = 1/8 < 3/8 < 7/8.

(ii) Descending order of the numerators = 7 > 3 > 1.

Descending order of the fractions = 7/8 > 3/8 > 1/8.

 

3) 12/5 , 3/5 , 9/5, 16/5

Solution:

Numerators of the given fractions = 12, 3, 9, 16.

Common denominator = 5.

(i) Ascending order of the numerator = 3 < 9 < 12 < 16.

Ascending order of the fractions = 3/5 < 9/5 < 12/5 < 16/5.

(ii) Descending order of the numerators = 16 > 12 > 9 > 3.

Descending order of the fractions = 16/5 > 12/5 > 9/5 > 3/5.

 

4) 4/11, 2/11, 6/11, 9/11, 5/11

Solution:

Numerators of the given fraction = 4, 2, 6, 9, 5.

Common denominator = 11.

(i) Ascending order of the numerators = 2 < 4 < 5 < 6 < 9.

Ascending order of the fractions = 2/11 < 4/11 < 5/11 < 6/11 < 9/11.

(ii) Descending order of the numerators = 9 > 6 > 5 > 4 > 2.

Descending order of the fractions = 9/11 > 6/11 > 5/11 > 4/11 > 2/11.

 

5) 5/7, 3/7, 2/7

Solution:

Numerators of the given fractions = 5, 3, 2.

Common denominator = 7.

(i) Ascending order of the numerators = 2 < 3 < 5.

Ascending order of the fractions = 2/7 < 3/7 < 5/7.

(ii) Descending order of the numerators = 5 > 3 > 2.

Descending order of the fractions = 5/7 > 3/7 > 2/7.  

Addition and Subtraction of Like Fractions.

ADDITION AND SUBTRACTION OF LIKE FRACTIONS:

a) Addition of Like Fractions:

Sum of two or more Like fractions = Sum of Numerators/ Common denominator.

 

1) 11/14 + 2/14

Solution:

Given: Numerators of fractions = 11 , 2 ; Common Denominator = 14.

Sum of numerators = 11 + 2 = 13.

Sum of the Like fractions = Sum of numerators / Common denominator.  

                                             = 13/14.

 

2) 3/10 + 1/10

Solution:

Given: Numerators = 3 ,1 ; Common denominator = 10.

Sum of numerators = 3 + 1 = 4.

Sum of Like Fractions = Sum of numerators/ common denominator.

                                        =  4/10.

 

3) 8/17 + 4/17 + 3/17

Solution:

Given: Numerators = 8, 4, 3; Common denominators = 17.

Sum of numerators = 8 + 4 + 3 = 15.

Sum of Like fractions = Sum of numerators / common denominator.

                                      = 15/17.  

 

 

b) Subtraction of Like fractions:

Difference of two Like fractions = Difference of the numerators / Common denominator. 

 

1) 15/21 from 19/21

Solution:

Given: Numerators = 15, 19 ; Common denominator = 21.

Difference of the numerators = 19 – 15 = 4.

Difference of two fractions = Difference of numerators/ common denominator

 = 4/21.

 

2) 7/11 from 9/11

Solution:

Given: Numerators = 7, 9 ; Common denominator = 11.

Difference of the numerators = 9 – 7 = 2.

Difference of two fractions = Difference of numerators / common denominator

 = 2/11.

 

3) 9/13 from 11/13

Solution:

Given: Numerators = 9, 11; Common denominator = 13.

Difference of the numerators = 11 – 9 = 2.

Difference of two fractions = Difference of numerators/common denominator.

 = 2/13.  

Converting Mixed Fraction into Improper Fraction.

CONVERTING MIXED FRACTION INTO IMPROPER FRACTION:

A mixed fraction can be converted into an improper fraction by using the below steps:

1. Find the product of the whole number and the denominator.

2. Add the numerator of the mixed fraction to the product; gives the numerator of the improper fraction.

3. Thus, Required Improper fraction = Numerator of Improper fraction /                      denominator.  

 

Covert the Mixed fractions into Improper fractions:

1)  6  3/4

Solution:

Given: whole number = 6 ; numerator = 3 ; denominator = 4.

a. Product of Whole number = 6 and the denominator = 4.

  Whole number x denominator = 6 x 4 = 24.

b. Adding numerator of the mixed fraction = 3 and the product obtain in before step.

Numerator + Product = 3 + 24 = 27: Numerator of the improper fraction.

c. Thus, the Improper fraction =  27/4.

 

2) 5  8/9

Solution:

Given: whole number = 5 ; numerator = 8 ; denominator = 9.

a. Product of whole number = 5 and the denominator = 9

Product = 5 x 9 = 45.

b. Numerator of the improper fraction = Adding numerator of the mixed fraction and product.

   Numerator of the improper fraction = 8 + 45 = 53.

c. Thus, Required Improper function =  53/9.

 

3) 8  4/9

Solution:

Given: Whole number = 8 ; numerator = 4 ; denominator = 9.

a. Product = whole number x denominator = 8 x 9 = 72.

b. Numerator of the improper fraction = Numerator + product

     =  4 + 72 = 76.

c. Thus, required improper fraction = 76/9.

 

4) 2  6/7

Solution:

Given: Whole number = 2 ; numerator = 6 ; denominator = 7.

a. Product = whole number x denominator = 2 x 7 = 14.

b. Numerator of the Improper fraction = Numerator + product

    = 6 + 14 = 20.

c. Thus, required Improper fraction = 20/7.

 

5) 8  4/5

Solution:

Whole number = 8 ; numerator = 4 ; denominator = 5.

a. Product = whole number x denominator = 8 x 5 = 40.

b. Numerator of the Improper fraction = Numerator + product

    = 4 + 40 = 44.

c. Thus, required Improper fraction = 44/5.   

 

Converting Improper Fractions into Mixed Fractions.

Converting Improper Fractions into Mixed Fraction:

An Improper fraction can be converted into a mixed fraction by using the below steps:

a. Divide the numerator(dividend)  with the denominator(divisor).

b. Write the quotient as the whole number part of the mixed fraction.

c. Write the remainder as the numerator of the proper fraction and the number of parts of the whole (divisor) as the denominator.

 

Convert the Improper fractions into Mixed fractions:

1)  39/8

Solution:

a. Divide: numerator: N = 39 and denominator: D = 8.

dividend = 39 ; divisor = 8;

After dividing : Quotient Q = 4 ; Remainder R = 7.

b. The whole number part of the mixed fraction is: Q = 4.

c. The proper fraction part of the mixed fraction is :

 Remainder / divisor = 7/8.

Thus, the required Mixed fraction = Q  R / divisor = 4  7/8 .

 

2) 73/4

Solution:

a. Divide the numerator( dividend): N = 73 ; denominator(divisor): D = 4.

After dividing: Quotient: Q = 18 ;  Remainder R = 1.

b. The whole number part of the mixed fraction is: Q = 18.

c. The proper fraction part of the mixed fraction is :

Remainder / divisor = 1/4.

Thus, the required Mixed fraction = Q  R / divisor = 18  1/4.

 

3) 23/5

Solution:

a. Divide : dividend: 23 with divisor: 5.   

We get Quotient ( whole part) Q = 4 ;

Remainder ( Numerator of proper fraction) R = 3.

The required Mixed fraction is = Q  R / divisor =  4  3/5.

 

4) 82/9

Solution:

Divide: numerator(dividend) = 82 with denominator(divisor) = 9.

We get: Quotient Q = 8 : Whole part of the mixed fraction.

Remainder R = 10 : Numerator of the Proper fraction.

Thus, the required Mixed fraction: Q  R/ divisor = 8  10/9.

   

Types of Fractions.

 TYPES OF FRACTIONS:

1. Like and Unlike Fractions.

2. Unit Fractions.

3. Proper and Improper Fractions.

4. Mixed Fractions.

 

1. Like Fractions:

Fractions with the same Denominators are called Like fractions.

In Like fractions, the whole is divided into the same number of parts.

The fractions : 1/4 ;  2/4 ;  3/4

The fractions have same denominators. i.e; 4. The whole is divided into equal number of parts. Therefore, the fractions are called Like fractions.

 

2. Unlike Fractions:

Fractions with different denominators are called Unlike Fractions.

In Unlike fractions, the whole is divided into a different number of parts.

The fractions: 1/2 ; 1/3 ; 5/8 , have different denominators.

The whole is divided into different parts, thus forming Unlike Fractions.   

 

3. Unit Fractions:

A fraction with numerator as 1 is called a Unit Fraction.

The  fractions: 1/4 , 1/8 ,  1/5. The numerator is same in all the fractions, thus forming Unit fractions.

 

4. Proper Fractions:

A fraction in which the numerator is smaller than the denominator is called a Proper Fraction.

Proper fractions represents a part of the whole object.

     Numerator < Denominator.

The Proper fractions are: 2/3,  5/8, 4/7.

 

5. Improper Fractions:

A Fraction in which the numerator is either equal to or greater than the denominator is called Improper fraction.

Improper fractions represent one or more than one object.

    Numerator > Denominator.

The Improper fractions are: 3/3 , 6/5 ,7/ 4.

 

6. Mixed Fractions:

A fraction which is made up of a whole number and a proper fraction is called a Mixed fraction.

Mixed fractions are: 1  2/3  ,  2  3/4 ,  1 5/6.

The whole numbers in the given mixed fractions are : 1 , 2 , 1.

The Proper fractions are: 2/3 , 3 / 4 , 5/6.

Reducing the Fractions to their Lowest Terms.

REDUCING A FRACTION:

Reducing a fraction means making the fraction as simple as possible.

Reducing a fraction is called as simplifying the fraction. We say the fraction in its Lowest terms.       

We need to divide the numerator and denominator by a common factor.

We can keep reducing a fraction till the numerator and the denominator in the reduced fraction have no common factor other than 1.

 

Simplify (or) Reduce the Fractions (or) the fractions to their Lowest Terms:

1) 48/108 =

Solution:

(i) We need to find out the common factor of both Numerator and Denominator. The Common factor is 2.

(ii) Divide the numerator and denominator with that common factor:2

Numerator: N = 48 ÷ 2 = 24.

Denominator:D = 108 ÷ 2 = 54.

(iii) We need to simplify the fraction with their common factor, till we get the common factor as 1.

Numerator: N    = 48 ÷ 2  =  24 ÷ 2  = 12 ÷ 3  = 4.

Denominator: D = 108 ÷ 2 =  54 ÷ 2  = 27 ÷ 3  = 9.

The Simplified or Reduced fraction is: 4/9, for which the common factor is 1.   

     

2) 168/144 =

Solution:

(i) The common factor of the given fraction numerator, N: 168 and denominator, D: 144 is 4.

(ii) Divide the numerator and denominator with the common factor in each step.

(iii) Reduced the fraction till the common factor of the fraction is 1.

Numerator: N    = 168 ÷ 4 = 42 ÷ 3 = 14 ÷ 2 = 7.

Denominator: D = 144 ÷ 4 = 36 ÷ 3 = 12 ÷ 2 = 6.

The Common factor of the Reduced fraction: 7/6  is 1.   

Thus, the simplified or Reduced fraction of the given fraction is:

168/144 =  42/36  =  14/12 =  7/6.

 

3) 20/24 =

Solution:

(i) The common factor of the given fraction is: 4.

(ii) Dividing the given fraction with their common factor at each step till the common factor of the reduced fraction is 1.

Numerator: N    = 20 ÷ 4 = 5.

Denominator: D = 24 ÷ 4 = 6. 

Thus, the Reduced fraction = 20/24 = 5/6.

                             (OR)

If we choose the common factor for the given fraction as 2, then

Numerator: N    = 20 ÷ 2 = 10 ÷ 2 = 5.

Denominator: D = 24 ÷ 2 = 12 ÷ 2 = 6.

The Reduced fraction = 20/24 = 10/12 = 5/6.

 

Note: We can choose different common factor in reducing the fraction, the Reduced fraction is same.

 

4) 32/60 =

Solution:

Let the common factor of the given fraction =  4.

Numerator: N    = 32 ÷ 4 = 8.

Denominator: D = 60 ÷ 4 = 15.   

Thus, the Reduced Fraction is : 32/60 = 8/15.

                             (OR)

Let the common factor of the given fraction = 2.

Numerator: N     = 32 ÷ 2  =  16 ÷ 2  =  8.

Denominator: D = 60 ÷ 2  =  30 ÷ 2  = 15.

Thus, the Reduced fraction in its lowest terms:

32/60 = 16/30 = 8/15.

 

 5) 27/81=

Solution:

If the common factor of the given fraction is: 9.

Numerator: N    =  27 ÷ 9  =  3 ÷ 3  =  1.

Denominator: D =  81 ÷ 9  =  9 ÷ 3  =  3.

Thus, the given fraction in its Lowest terms = 27/81 = 3/9 = 1/3.

                          (OR)

If the common factor of the given fraction is: 3.

Numerator: N    =  27 ÷ 3  =  9 ÷ 3    =  3 ÷ 3  =  1.

Denominator: D =  81 ÷ 3  =  27 ÷ 3  =  9 ÷ 3  =  3.

Thus, the given fraction in its Lowest terms =

27/81  =   9/27  =  3/9  =  1/3.