Class 8 Rational Numbers
Rational Numbers
The numbers which can be expressed in the form p/q, where p and q are integers and q ≠ 0, are called rational numbers.
1. Find the additive inverse of :
i) 9/13 = -9/13
ii) 16/7 = -16/7
iii) -3/23 = 3/23
iv) 8/-11 = 8/11
v) -22/15 = 22/15
vi) -11/-9 = -11/9
2. Find the sum
i) -7/17 + 6/17 = -7+6/17 = -1/17
ii) -5/12 + 7/-12 = -5/12 + (-7/12) = -5-7/12 = -12/12 = -1
iii) 8/15 +5/12
here denominator are different, so for addition we have to make it same.
first find out LCM of 15 and 12
LCM is 60
so, 8*4/15*4 + 5*5/12*5
= 32/60 + 25/60
= 57/60
iv) -11/18 + 5/-12
= -11/18 + (-5/12)
here denominator are different, so for addition we have to make it same.
first find out LCM of 18 and 12
LCM is 36
so, -11*2/18*2 - 5*3/12*3
= -22/36 - 15/36
= -22-15/36
= -37/36
= -11/36
v) -11/6 + -3/4 + 5/8 + -7/3
here denominator are different, so for addition we have to make it same.
first find out LCM of 3, 4, 8, and 3
LCM is 24
so, -11*4/6*4 - 3*6/4*6 + 5*3/8*3 - 7*8/3*8
= -44/24 -18/24 + 15/24 -56/24
= -44-18+15-56/24
= -103/24
= -47/24
vi) 4/7 + 2/-3 +5/21 + -8/9
= 4/7 - 2/3 + 5/21 - 8/9
here denominator are different, so for addition we have to make it same.
first find out LCM of 7 , 3, 21 , 9
LCM is 63
so, 4*9/7*9 - 2*21/3*21 + 5*3/21*3 - 8*7/9*7
= 36/63 -42/63 + 15/63 - 56/63
= 36-42+15-56/63
= - 47/63
3. Subtract
i) 2/3 from 5/6
means 5/6 - 2/3
here denominator are different, so we have to make it same.
first find out LCM of 6 , 3
LCM is 6
so, 5/6 - 2*2/3*2
= 5/6 - 4/6
= 1/6
ii) -2/5 from -5/7
means -5/7 - (-2/5)
= -5/7 +2/5
here denominator are different, so we have to make it same.
first find out LCM of 7 and 5
LCM is 35
so, -5*5/7*5 + 2*7/5*7
= -25/35 + 14/35
= -25+14/35
= -11/35
iii) 4/9 from -7/8
means -7/8 -4/9
first find out LCM of 8 and 9
LCM is 72
so, -7*9/8*9 - 4*8/9*8
= - 63/72 - 32/72
= -63-32/72
= -95/72
= -123/72
iv) -11/6 from 8/3
means 8/3 - (-11/6)
8/3 +11/6
first find out LCM of 3 and 6
LCM is 6
so, 8*2/3*2 + 11/6
= 16/6 + 11/6
= 27/6
= 43/6
= 41/2
4. The sum of two rational number is -4/9. If one of them is 13/6 , then find the other.
let other number be x
so
x + 13/6 = -4/9
x = -4/9 - 13/6
first find out LCM of 9 and 6
LCM is 18
x = -4*2/9*2 - 13*3/6*3
x = -8/18 -39/18
x = -8-39/18
x = -47/18
x = -211/18
other number is -211/18
5. What number should be added to -2/3 to get -1/7
Let the number be x
so,
-2/3 + x = -1/7
x = -1/7 +2/3
first find out LCM of 7 and 3
LCM is 21
x = -1*3/7*3 + 2*7/3*7
x = -3/21 + 14/21
x = 11/21
11/21 is added to -2/3 to get -1/7
6. What number should be subtracted from -2 to get 7/11
let the number be x
so,
-2 - x =7/11
-x = 7/11 +2
-x = 7/11 + 2*11/1*11
-x = 7/11 + 22/11
-x = 7+22/11
-x = 29/11
x = -29/11
x = -27/11
-27/11 should be subtracted from -2 to get 7/11
7.What number should be added to -1 so as to get 5/7
let the number be x
-1 +x = 5/7
x = 5/7 +1
x = 5/7 + 1*7/1*7
x = 5+7/7
x = 12/7
8 . What number should be subtracted from -2/3 to get -1/6
Let the number be x
so,
-2/3 -x = -1/6
-x = -1/6 +2/3
-x = -1/6 + 2*2/3*2
-x = -1/6 +4/6
-x = 3/6
-x = 1/2
x = -1/2
Multiplication of Rational Numbers Class 8
Properties of Multiplication of Rational numbers
Property 1 : Closure Property : The product of two rational numbers is always a rational number
Example : -2/9 and 15/4 are rational numbers.
-2/9 * 15/4 = -30/36 = -5/6
which is also rational number.
Property 2 : Commutative Law : If a/b and c/d are any two rational numbers,
then (a/b)*(c/d) = (c/d)*(a/b)
Example : -36/25 * -15/28 = -15/28 * -36/25
Property 3 : Associative Law : If a/b, c/d, and e/f are three rational numbers,
then (a/b * c/d)*e/f = a/b * (c/d * e/f)
Example : 2/5, -5/12 and -7/15 are three rational numbers
(2/5 * -5/12) * -7/15 = 2/5 * (-5/12 * -7/15)
Property 4 : Existence of Multiplicative Identity : The rational number 1 is the multiplication identity for rational. So, a/b * 1 = 1*a/b = a/b
Example : (-13/19*1) = (1*-13/19) = -13/19
Property 5 : Existence of Multiplicative Inverse : Every non-zero rational number a/b has a multiplicative inverse b/a (reciprocal ) so, a/b*b/a = b/a*a/b =1
Example : 16/15 is reciprocal of 15/16 ,
16/15*15/16 = 15/16* 16/15 = 1
Property 6 : Distributive law of Multiplication over Addition : For any three rational numbers
a/b, c/d and e/f , a/b * (c/d + e/f) = (a/b * c/d) + (a/b*e/f)
Example : (-3/7) * (8/11+4/9) = (-3/7 * 8/11) + (-3/7*4/9)
Property 7 : Multiplicative Property of Zero : For any rational number a/b,
(a/b*0) = (0*a/b) = 0
Example : 3/16 *0 = 0*3/16 = 0
1. Find the Products
i) 4/9 * 7/12
ii) -9*7/18
iii) -3/16 * 8/-15
iv) 6/7 * -21/12
v) 5/-18 *-9/20
vi) -13/15 * -25/28
vii) 7/24* (-48)
viii) -13/5 * (-10)
4. Find the area of a rectangular part which is 363/5 m long and 162/5 m broad
Properties Of Division of Rational Numbers
Property 1 : Closure Property : If a/b and c/d are any two rational numbers such that c/d ≠0, then(a/b ÷ c/d) is also a rational number.
Example : Consider the rational numbers -3/40 and 11/24.
-3/40 ÷ 11/24 = -3/40 * 24/11 = -72/440 = -18/110, which is also rational.
Property 2 : For any rational number a/b, we have :
(a/b ÷ 1) = a/b.
Example : 5/7 ÷ 1 = 5/7 ÷ 1/1 = 5/7 *1/1 = 5/7
Property 3 : For any non-zero rational number a/b, we have :
(a/b ÷ a/b ) = 1
Examples : 5/6 ÷ 5/6 = 5/6 *6/5 = 30/30 = 1
1. Find the quotient :
i) 17/8 ÷ 51/4
ii) -16/35 ÷ 15/14
iii) -12/7 ÷ (-16)
iv) -9 ÷ (-5/18)
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