Download Rational Numbers 4

MATHEMATICS
GRADE VIII
RATIONAL NUMBERS
MULTIPLICATION AND DIVISION OF RATIONAL NUMBERS
Strategies discussed/Scope of the session
The multiplication of rational numbers possesses the following properties:
Closure property: The product or multiplication of any two rational numbers is always a rational number.
Commutativity: The multiplication of rational numbers is commutative. That is, if
a
c
and
are any two
b
d
rational number, then.
Associativity: The multiplication of rational numbers is associative. That is, if
a c
e
, and are three
b d
f
a c  e a c e 
     
b d  f b d f 
rational numbers, then 
Existence of multiplicative Identity: If
a
a
a
a
is any rational number, then 1  1 1 is called the
b
b
b
b
multiplicative identity for rational numbers.
Multiplication By 0: Every rational number when multiplied with 0 gives 0 . That is, if
number, then
a
is any rational
b
a
a
 0  0
b
b
Distributivity of Multiplication over Addition: The multiplication of rational numbers is distributive over
their addition. That is, if
a c e a c a e
      
b d f  b d b f
Existence of Multiplicative Inverse or Reciprocal of a non-zero Rational: For every non-zero rational
number
a
a c
c a
c
c
there exists a rational number
such that   1   . The rational number
is
b
b d
d b
d
d
1
a
a
For any non-zero rational number ,
b
b
called the multiplicative inverse or reciprocal of a denoted be  
we have
a c
c a
c
 1   . The rational number is called the multiplicative inverse or reciprocal of a
b d
d b
d
a
denoted be  
b
1
For any non-zero rational number
a b a  b ab
b a b  a ba
a
, we have  

 1 and,  

1
b a b  a ab
a b a  b ab
b
a b
b a
b
a
is .
  1   If follows from this result that the multiplicative inverse or reciprocal of
b a
a b
a
b
1
b
a
That is,   =
a
b
DIVISION of rational numbers: If x and y are two rational numbers such that y  0, then the result of
dividing x by y is the rational number obtained on multiplying x by the reciprocal of y.
When x is divided by y, we write x  y = x 
1
y
1
If
a
c
c
a c a c
a d
and
are two rational numbers such that  0, then
     
b
d
d
b d b d 
b c
Dividend: The number to be divided is called the dividend.
Divisor: The number which divides the dividend is called the divisor.
Quotient: When dividend is dividend by the divisor, the result of the division is called the quotient.
1
a
a
c
c
a c a c
a d
If
is divided by
, then
is the dividend,
is the divisor and        is the
b
b
d
d
b d b d 
b c
quotient.
NOTE : It should be noted that division by 0 is not defined.
Properties of division of Rational Numbers.
Property I : If
a
c
a c
c
and
are two rational numbers such that  0, then  is always a rational
b
d
b d
d
number. That is, the set of all non-zero rational numbers is closed under division.
Property II : For any rational number
a
a
a
a
a a
, we have
 1  and  (1)   
b
b
b
b
b b
Property III : For every non-zero rational number
(a)
a a
 1
b b
(b)
a  a
     1
b  b
a
, we have
b
 a  a
(c) 
   1
 b  b
Remark: The division of rational numbers is neither commutative nor associative.
WORKSHEET
1 . Verify the property : x  y = y  x by taking :
1
3
I)x=- , y
2
7
II ) x =
3
11
,y
5
13
III ) x = 2, y =
7
8
iv) x = 0, y =
15
8
2. Verify the property : x  (y  z) = (x  y)  z by taking:
I)x=
7
12
4
,y  ,z 
3
5
9
II ) x = 0, y =
III ) x =
1
5
7
,y
,z 
2
4
5
IV ) x =
3
9
,Z=
5
4
5
12
7
,y
,z 
7
13
18
3. Verify the property : x  ( y + z ) = x  y + x  z by taking :
I)x=
3
12
5
,y  ,z 
7
13
6
II ) x =
12
15
8
,y
,z 
5
4
3
III ) x =
8
5
13
,y  ,z 
3
6
12
IV ) x =
3
5
7
,y
,z 
4
2
6
4. Use the distributivity of multiplication of rational numbers over their addition to simplify :
i)
3  35 10 
  
5  24 1 
ii )
5  8 16 
  
4 5 5 
iii)
2  7 21 
  
7  16 4 
iv)
3 8

   40 
4 9

5. Find the multiplicative inverse (reciprocal ) of each of the following rational numbers :
i) 9
ii) -7
viii) -2 x
3
ix) -1
5
12
5
0
x)
3
iii)
iv)
7
3
v)
9
5
vi)
2 9

3 4
vii)
5 16

8 15
6. Name the property of multiplication of rational numbers illustrated by the following statements :
i)
5 8
8 5
  
16 15 15 16
ii)
17
17
9  9
5
5
iii)
7  8 13  7 8 7 13
 
   
4  3
12  4 3 4 12
iv)
5  4 9   5 4  9
13
13
13
v)
      
1 
 1
9  15 8   9 15  8
17
17
17
vii)
2
2
 0  0  0
13
13
viii)
3 5 3 7 3  5 7 
  

 

2 4 2 6
2 4 6 
vi)
11 16

1
16 11
7. Fill in the blanks :
i) The product of two positive rational numbers is always……
ii) The product of two positive rational number and a negative rational number is always ……
iii) The product of two negative rational numbers is always. …..
iv) The reciprocal of a positive rational number is ……
v) The reciprocal of a negative rational number is …..
vi) Zero has ……. reciprocal
vii) The Product of a rational number and its reciprocal is ……
viii) The numbers …….. and …. are their own reciprocals.
ix) If a is reciprocal of b, then the reciprocal of b is …..
x) The number 0 is …… the reciprocal of any number.
xi) Reciprocal of
1
,   0 is ……

xii) 17 12   17 1  ........
1
8. Fill in the blanks :
(i) 4 
iii)
7 7
  ......
9 9
1  3 5  1
5
      ........  ....... 
2  4 12  2
12
ii)
5 3 3


 ........
11 8
8
iv)
4  5 8   4
 8
       .......  
5 7 9   5
 9
9. Divide :
5
7
3
2 7
vii)
by  6 viii) by
4
3 12
i) 1 by ½
ii) 5 by
3
9
by
4
16
3
ix) -4 by
5
iii)
7 21
by
8
16
3 4
x)
by
13 65
iv)
v)
7
63
by
4 64
vi) 0 by
7
5
10. Find the value and express as a rational number in standard form :
i.
2 26

5 15
ii .
10 35

3 12
 8 

 17 
iii. 6  
iv.
40
22 110
 (20) v.

99
27
18
vi.
36 3

125 75
11. The product of two rational numbers is 15. If one of the numbers is -10. Find the other.
12. The product of two rational numbers is
13. By what number should be multiply
8
4
. If one of the numbers is
, find the other.
9
15
23
1
so that the product may be
?
9
6
14. By what number should we multiply
15
5
so that the product may be
?
28
7
15. By what number should we multiply
8
so that the product may be 24 ?
13
16. By what number should
3
2
be multiplied in order to produce ?
4
3
17. Find (x+y) ( x  y) , if
i) x 
2
3
2
1
5
1
ii) x = , y 
iii) x = , y 
,y
3
2
5
2
4
3
iv) x =
2
4
,y
7
3
v) x =
18. The cost of 7
2
3
metres of rope is Rs 12 . Find its cost per metre.
3
4
19. The cost of 2
1
1
metres of cloth is Rs 75 . Find the cost of cloth per metre.
3
4
20. By what number should
1
3
,y
4
2
11
33
be divided to get
?
4
16
21.Divide the sum of
13
12
31
1
by the product of
and
and
5
7
7
2
22. Divide the sum of
12
65
and
by their difference.
7
12
23. If 24 trousers of equal size can be prepared in 54 metres of cloth, what length of cloth is required
for each trouser?
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