Multiplication of a Whole number by a Fractional number.

MULTIPLICATION OF A WHOLE NUMBER BY A FRACTIONAL NUMBER:

Steps to multiply a whole number by a fractional number:

i) Write the whole number as a fractional number.

ii) Multiply the numerators of the fractions.

iii) Multiply the denominators of the fractions.

iv) Simplify into lowest terms.

 

Find the Product:

1) 10 x  3/5

Solution:

Given: whole number = 10 ; fraction = 3/5.

Required product = 10/1 x 3/5 = (10x3) / 5 = 30/5 = 6.

2) 16 x  5/4

Solution:

Given: Whole number = 16 ;  fraction = 5/4.

Required Product = 16/1  x  5/4 =  ( 16x5 ) / 4 = 20.

 

3) 6 2/7  of 7.

Solution:

Given: Whole number = 7; mixed fraction = 6 2/7.

Converting mixed fraction into fraction:

((Whole number x denominator) + numerator ) / denominator.

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

 6 2/7 = 44/7.

Required Product = 7 of  44/7 =  7 x  44/7  

                              = ( 7/1  x  44/7 ) = 44.

 

4) 1 1/15  of  15

Solution:

Given: Whole number = 15 ; Mixed fraction = 1 1/15

Converting mixed fraction into fraction:

1 1/15 = 16/15.

Required Product = 15  of  16/15  =  15 x  16/15

                              = ( 15/1  x  16/15 ) =  16.

 

5)  5 1/5 of 10.

Solution:

Given: Whole number = 10; Mixed fraction = 5 1/5.

Converting mixed fraction into fraction:

5 1/5 = 26/5.

Required Product = 10 of  26/5 = 10  x  26/5

                              = ( 10/1  x  26/5 ) = 2 x 26 = 52. 

   

Multiplication of Fractions.

MULTIPLICATION OF FRACTIONAL NUMBERS:

Steps to Multiply Fractions:

i) Multiply the numerators of the given fractions, which is the new numerator.

ii) Multiply the denominators of the given fractions, which is the new denominators.

iii) Simplify the obtained new numerators and denominators into its Lowest terms, if required (cancel out the common factors wherever possible) .

 

Solve the Following:

1)  5/8  x  8/15

Solution:

Given: Numerators = 5, 8 ; Denominators = 8, 15.

New numerator =  5 x 8 = 40.

New Denominator = 8 x 15 = 90.

Thus, required Product is = 40/90 = 4/9.

 

2)  26/33 x 22/39

Solution:

Given: Numerators = 26 , 22 ; Denominators = 33, 39.

Required Product =  (26x22) / (33x39) ; reducing into lowest terms.

                           = 4/9;  (26/39 = 2/3 ; 22/33 = 2/3; Thus: (2/3) x (2/3) = 4/9).

 

3)  (2/11) x  (3 /4).

Solution:

Given : Numerators = 2, 3 ; Denominators = 11, 4

 Required Product = (2x3) / (11x4) ; reducing into Lowest terms

                               =  (1x3) / (11x2)  ( Since : 2/4 = 1 /2)

                              = 3 / 22.

 

4)  1/5  x  3/ 4

Solution:

Given: Numerator = 1, 3 ; Denominator = 5, 4.

Required Product =  (1x3) / (5x4)

                                = 3 / 20.

 

5)  7/7 x  2/6

Solution:

Given: Numerator = 7, 2 ; Denominator = 7, 6.

Required Product = (7x2) / (7x6); reducing into lowest terms.

                               = 1/3.  (7/7 = 1; 2/6 = 1/3; Thus: 1 x 1/3 =1/3).

Multiplication of Mixed fractions.

MULTIPLICATION OF MIXED FRACTIONS:

Steps to find product of mixed fractions:

i) Convert Mixed fractions into Improper fractions.

ii) Multiply the formed Improper fractions.  (use the steps to find the product of Fractions).

iii) Simplify into its Lowest terms.

 

Solve the following:

1)  3 1/5  of  2 3/4  

 Solution:

Given: mixed fractions = 3 1/5; 2 3/ 4

Converting into Improper fractions:

3 1/5 = 16/5 ;  2 3/ 4 = 11/4.

Required product = 16/5 x 11/4 = (16x11) / (5x4)  =  44/5.

 

2)  3 2/6  of  6 2/5

Solution:

Given: Mixed fractions = 3 2/6 ; 6 2/5

Converting into Improper fractions:

3 2/6 = 20/6 ; 6 2/5 = 32/5.

Required Product = 20/6 x 32/5 = (20x32) / (6x5) = 64/3.

 

3)  2 1/15 of  2 1/5

Solution:

Given: Mixed fractions= 2 1/15 ; 2 1/5

Converting into Improper fractions:

2 1/15 = 31/15 ; 2 1/5 = 11/5.

Required Product = (31x11) / (15x5) =  341/75.

Addition and Subtraction of Mixed Fractions.

ADDITION AND SUBTRACTION OF MIXED FRACTIONS:

Steps to solve Mixed fractions:

We  can solve the Mixed fractions in two methods:

Steps for Method-1:

a) Solve (Add or Subtract) the whole number part.

b) Solve the Fractional part.

c) The required Solution is writing the obtained Whole part and Fractional part together.  

 

Steps for Method-2:

a) Converting the mixed number into Improper fractions.

b) Solve the Improper fractions.

c) Convert again the obtained New Improper fractions into Mixed fractions gives the required solution.

 

Add the following:

1)  3 3/4  and  2 1/8

Solution:

Given: Whole number = 3, 2 ; Fractional part = 3/4 , 1/8.

Method-1:

a) Add the whole numbers: 3 + 2 = 5.

b) Add the fractional numbers:

 3/4 + 1/8 ; Unlike fractions.

Convert Unlike fractions into Like fractions:

L.C.M of 4,8 = 8

Equivalent fractions: 3 /4 = 6/8 ; 1/8 = 1/8.

Therefore, 3 /4 + 1/8 = 6/8 + 1/8 = 7/8.

c) Thus required solution of 3 3 /4 + 2 1/8  = 5 7/8.

                   ( OR )

Method-2:

a) Converting Mixed fractions into Improper fractions:

3 3/ 4 = 15/4  ;  2 1/8 = 17/8.

b) Solve : 15/4 + 17/8 ; Unlike fractions.

Convert Unlike into Like fractions.

L.C.M of 4, 8 = 8.

Therefore, Equivalent fractions: 15/4 = 30/8  ; 17/8 = 17/8.

Add:  30/8 + 17/8 =  47/8.

c) Convert obtained Improper fraction into Mixed fraction:

Divide 47/8 ; we get Quotient = 5, Remainder = 7, denominator = 8.

Thus, required solution is :  Q R/denominator = 5  7/8.

 

Subtract the Mixed fractions:

2) 5 2/3 – 2 1/2

Solution:

Given: Whole numbers = 5, 2 ; Fractional part = 2/3 , 1 /2.

Method-1:

a) Subtract the whole numbers: 5 – 2 = 3.

b) Subtract Fractional part: 2/3 – 1 /2; Unlike fractions.

L.C.M of  3, 2 = 6.

Equivalent fractions: 2/3 = 4/6 ; 1/ 2 = 3/6.

Therefore, 4/6 – 3/6 = 1/6.

c) Therefore, required solution: 5 2/3 – 2 1 /2 = 3 1/6.   

 

                             (OR)

Method- 2 :

a) Convert Mixed fractions into Improper fractions:

5 2/3 = 17/3 ; 2 1/ 2 = 5/2.

b) Solve the unlike fractions: L.C.M of 3, 2 = 6.

Equivalent fractions: 17/3 = 34/6 ;  5/2 = 15/6.

Subtract the fractions: 34/6 – 15/6 = 19/6.

c) Convert Improper fraction into Mixed fraction:

Divide: 19/6; Quotient Q = 3; Remainder R = 1; Denominator = 6.

Thus, required solution = 3 1/6.

 

3) Solve: 2 2/3 – 1 ¼  + 3 1/6

Solution:

Given: Whole numbers = 2,  1 , 3 ; Fractional part = 2/3, 1 /4 , 1/6.

a) Solve the whole numbers:

  2 – 1 + 3 = 4.

b) Solve the Fractional part:

 2/3 – 1/ 4 + 1/6 =

L.C.M( 3, 4, 6) = 12.

Equivalent fractions:  2/3 = 8/12 ; 1/ 4 = 3/12 ; 1/6 = 2/12.  

Therefore, 8/12 – 3/12 + 2/12 = 7/12.

c) Thus, required solution is:  4 7/12.

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.