Properties of Multiplication of Fractions.

PROPERTIES OF MULTIPLICATION OF FRACTIONAL NUMBERS:

The properties of multiplication of Whole numbers apply to the multiplication of Fractions as well.

 

1) The product of a Fraction and Zero is Zero.

    2/3  x  0  =   0;   3 5/4  x  0  =  0.

 

2) The product of a Fraction and One(1) is the Fraction itself.

     3/8  x 1 = 3/8;   4 7/3 x 1 = 4 7/3.

 

3) Two fraction can be multiplied in any order, the product remains the same.

    1/6 x  2/7 =  (1x2) / (6x7)  = 2/42  ;  2/7 x 1/6 = (2x1)/(7x6) = 2/42.

 

4) While multiplying more than two fractions, they can be grouped in any order. The product remains the same.

Let the fractions be:  2/5 x 1/7 x 2/3

2/5 x (1/7 x 2/3) = (2x1x2) / (5x7x3) = 4/105.

                    (or)                                                           

(2/5 x 1/7) x 2/3 = (2x1x2) / (5x7x3) = 4/105.

                   (or)

(2/5 x 2/3) x 1/7 = (2x2x1) / (5x3x7) = 4/105.

Therefore, 2/5 x (1/7 x 2/3) = (2/5 x 1/7) x 2/3 = (2/5 x 2/3) x 1/7 = 4/105.

 

Fill in the blanks using Multiplication Properties:

1) 3/8 x 6/7 =  6/7 x 3/8.

2) (1/5 x 4/6) x 2/9 =  1/5 x  (4/6 x 2/9).

3) 5/9 x 0 = 0.

4) 5/7 x 1 = 5/7.  

5) 3/7 x 2/7 = 2/7 x 3/7.

6)  6/5 x (3/7 x 4/5) = (6/5 x 3/7) x 4/5.    

Properties of Division of Fractions.

PROPERTIES OF DIVISION OF FRACTIONS:

 1) When a fraction is divided by 1, the quotient is the fraction itself.

   2/5 ÷  1 = 2/5 ;    3/7 ÷ 1 = 3/7 ;   2  3/5  ÷ 1 =  2 3/5.

 

2) When Zero is divided by a fraction, the quotient is always Zero.

  0 ÷ 2/5  =  0 ;   0 ÷ 3 4/5 =  0 ;   0 ÷ 3/8  =  0.

Note: We cannot divide a fraction by Zero.

 

3) When a fraction is divided by itself, the quotient is One (1).

  2/5 ÷ 2/5 = 1 ;   3 4/5 ÷ 3 4/5 = 1  ;  3/7 ÷ 3/7 = 1.

 

Fill in the blanks using properties of division:

1)  4/7 ÷ 4/7 = 1 

2)  3/8 ÷ 1 = 3/8.

3) 4/9 ÷  4/9 = 1.

4) 0 ÷ 5/9 = 0.

5) 0 ÷ 2/9 = 0.

Division of a Fractional number by a Whole number.

DIVISION OF A FRACTION BY A WHOLE NUMBER:

Steps to find Division of fraction by a whole number:

i) Find out the reciprocal of the Whole number.

ii) Multiply the Fraction with reciprocal of the Whole number.

iii) Simplify into its lowest terms.

Required Answer = Fraction x Reciprocal of the Whole number.

 

Solve the Following:

1) 5/9 ÷  4

Solution :

Given: Fraction = 5/9 ; Whole number = 4.

i) Reciprocal of the Whole number = 1/4

Required Answer =  Fraction x Reciprocal of the Whole number

                             =  (5/9)  x  (1/4) =  5/36.           

                                                                                   

2) 5/6 ÷ 4

Solution:

Given: Fraction = 5/6 ; Whole number = 4.

Reciprocal of the Whole number = 1/4

Required Answer =  Fraction x Reciprocal of the Whole number

                             =  (5/6) x (1/4) =  5/24.

 

3) 3/7 ÷ 3

Solution:

Given: Fraction = 3/7 ; Whole number = 3.

Reciprocal of the Whole number = 1/3.

Required Answer =  Fraction x Reciprocal of the Whole number

                             =  (3/7) x (1/3) = 7.

Division of a Whole number by a Fraction.

DIVISION OF A WHOLE NUMBER BY A FRACTION:

Steps to find Division of whole number by a fraction:

i) Find out the reciprocal of the fraction.

ii) Multiply the whole number with reciprocal of the fraction.

iii) Simplify into its lowest terms.

Required Answer = Whole number x Reciprocal of the Fraction.

 

Solve the following:

1) 8 ÷  1/5

Solution:

Given: Whole number = 8; Fraction = 1/5.  

i) Reciprocal of the fraction = 5.

ii) Multiplying whole number with reciprocal of the fraction.

  8 x 5 = 40 , required answer.        

 

2) 5 ÷   1/6

Solution:

Given: Whole number = 5 ; Fraction = 1/6.

Reciprocal of the fraction = 6.

Required Answer = Whole number x Reciprocal of the fraction.

                                =  5 x 6 = 30.

 

3) 9 ÷  2/3

Solution:

Given: Whole number = 9 ; Fraction = 2/3.

Reciprocal of the fraction = 3/2.

Required answer = Whole number x Reciprocal of the fraction.

                             = 9 x  (3/2) = 27/2.

 

4) 4 ÷ 2/3

Solution:

Given: Whole number = 4 ; Fraction = 2/3.

Reciprocal of the fraction = 3/2.

Required answer = Whole number x Reciprocal of the fraction.

                             = 4 x (3/2) = 6.

Division of Fraction by a Fraction.

DIVISION OF FRACTION BY A FRACTION:

Steps to find the result when a fraction is divided by another fraction:

i) Write the divisor in terms of its reciprocal.

ii) Multiply the dividend with reciprocal of the divisor.

iii) Simplify into its lowest terms.

Required Answer = Dividend x Reciprocal of the Divisor.

 

Solve the following:

1)  15/6  ÷  3/4

Solution:

Given: Dividend = 15/6 ; Divisor = 3/4

i) Reciprocal of the Divisor = 4/3.

ii)   Required Answer =  Dividend x  Reciprocal of the divisor

                                       =  (15/6)   x  (4/3)

iii) Simplifying into its lowest terms:

=  (15x4) / (6x3) = (5x2) / 3 = 10/3.

 

2)  21/28  ÷  3/7

Solution:

Given: Dividend = 21/28 ; Divisor = 3/7

Reciprocal of divisor = 7/3.

Required Answer = Dividend x Reciprocal of the Divisor 

=  (21/28) x (7/3) = (21x7) / (28x3)

 = 7/4.   ( simplify into Lowest terms :  3x7  /  4x3 =  7/4 )

 

3)  2 4/5  ÷  7/2

Solution:

Given: Dividend =  2 4/5 ; Divisor = 7/2

Converting Dividend: mixed fraction into fraction

2 4/5 = 14/5.

Reciprocal of the Divisor = 2/7.

Required Answer  =  Dividend x  Reciprocal of the divisor 

                                = 14/5  x  2/7  =  (14x2) / (5x7) =  4/5.

 

4) 8/9 ÷  5 1/3

Solution:

Given: Dividend = 8/9 ; Divisor = 5 1/3

Converting Divisor(mixed fraction)  into Fraction:

 5 1/3 = 16/3.

Reciprocal of the Divisor = 3/16.

Required Answer = Dividend x  Reciprocal of the divisor 

                               =  (8/9)  x  (3/16)  =  (8x3) / (9x16)  

                               =  1/6.

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.