Rational Numbers
Class 8th Mathematics RD Sharma Solution
- (i) -5/7 and 3/7 (ii) -15/4 and 7/4 (iii) -8/11 and -4/11 (iv) 6/13 and -9/13…
- (i) 3/4 and -5/8 (ii) 5/-9 and 7/3 (iii) -3 and 3/5 (iv) -7/27 and 11/18 (v)…
- Simplify: (i) 8/9 + -11/6 (ii) 3 + 5/-7 (iii) 1/-12 + 2/-15 (iv) -8/19 + -4/57…
- Add and express the sum as a mixed fraction: (i) therefore and 43/10 (ii) 24/7…
- Verify commutativity of addition of rational numbers for each of the following…
- Verify associativity of addition of rational numbers i.e., (x+y) + z = x + (y+z)…
- Write the additive invese of each of the following rational numbers: (i) -2/17…
- Write the negative (additive inverse) of each of the following: (i) -2/5 (ii)…
- Using commutativity and associativity of addition of rational numbers, express…
- Re-arrange sutably and find the sum in each of the following. (i) 11/12 + -17/3…
- Subtract the first rational number from the second in each of the following: (i)…
- Evalute each of the following: (i) + therefore (ii) - 4/7 - 2/-3 (iii) 4/7 -…
- The sum of the two numbers is 5/9 . If one of the numbers is 1/3 , find the…
- The sum of two numbers is -1/3 . If one of the numbers is -12/3 , find the other…
- The sum of two numbers is -4/3 . If one of the numbers is -5, find the other…
- The sum of two ratinal numbers is -8. If one of the numbers is -15/7 , find the…
- What should be added to -7/8 so as to get 5/9 ?
- What number should be added to -5/11 so as to get 26/33 ?
- What number should be addede to -5/7 to get -2/3 ?
- What number should be subtracted from -5/3 to get 5/6 ?
- What number should be subtracted from 3/7 to get 5/4 ?
- What should be added to (+) to get?
- What should be added to (+ +) to get 3?
- What should be subtracted from (-) to get ?
- Fill in the blanks: (i) -4/13 - -3/26 = (ii) -9/14 + l l = - 1 (iii) (iv) 15/23…
- Simplify each of the following and write as rational number of the form p/q :…
- Express each of the following as a rational number of the form p/q : (i) -8/3 +…
- Simplify: (i) -3/2 + 5/4 - 7/4 (ii) 5/3 - 7/6 + -2/3 (iii) a - x - % (iv) -2/5 -…
- (i) 7/11 by 5/4 (ii) 5/7 by -3/4 (iii) -2/9 by 5/11 (iv) -3/17 by -5/-4 (v) 9/-7…
- (i) -5/17 by 51/-60 (ii) -6/11 by -55/36 (iii) -8/25 by -5/16 (iv) 6/7 by -49/36…
- Simplify each of the following and express the result as a rational number in…
- Simplify: (i) (25/8 x 2/5) - (3/5 x -10/9) (ii) (1/2 x 1/4) + (1/2 x 6) (iii) (-…
- Simplify: (i) (x y i) + (v x) () (ii) (1/4 x 2/7) - (5/14 x -2/3) + (3/7 x 9/2)…
- Verify the property: x x y = y x x by taking: (i) x = - 1/3 , y = 2/7 (ii) x =…
- Verify the property: x x (y x z) = (x x y) x z by taking: (i) x = -7/3 , y =…
- Verify the property: x x (y+z) = x x y+x x z by taking: (i) x = -3/7 , y = 12/13…
- Use the distributivity of multiplication of rational numbers over their addition…
- Find the multiplicative inverse (reciprocal) of each of the following rational…
- Name the property of multiplication of rational numbers illustrated by the…
- (i) The product of two positive rational numbers is always.. (ii) The product of…
- (i) -4 x 7/9 = 7/9 x l l (ii) 5/11 x -3/8 = -3/8 x l (iii) 1/2 x (3/4 + -5/12) =…
- Divide: (i) 1by 1/2 (ii) 5 -5/7 (iii) -3/4 by 9/-16 (iv) -7/8 by -21/-16 (v)…
- Find the value and express as a rational number in standard from: (i) 2/5 /…
- The product of two rational numbers is 15. If one of the numbers is -10, find…
- The product of two rational numbers is -8/9 . If one of the numbers is -4/15 ,…
- By what number should we multiply -1/6 so that the product may be -23/9 ?…
- By what number should we multiply -15/28 so that the product may be -5/7 ?…
- By what number should we multiply -8/13 so theat the product may be 24?…
- By what number should -3/4 be multiplied in order to produce 2/3 ?…
- Find (x+y) / (x-y) if (i) x = 2/3 , y = 3/2 (ii) x = 2/5 , y = 1/2 (iii) x = 5/4…
- The cost of 7 2/3 metres of rope is Rs 12 3/4 .Find its cost per metre.…
- The cost of 2 1/3 metres of cloth is Rs 75 1/4 . Find the cost of cloth per…
- By what number should -33/16 be divided to get -11/4 ?
- Divide the sum of -13/5 and 12/7 by the product of -31/7 and -1/2…
- Divide the sum of 65/12 and 12/7 by their difference.
- If 24 trousers of equal size can be prepared in 54 metres of cloth. What length…
- Find a rational number between -3 and 1
- Find any five rational numbers less than 2.
- Find two rational numbers between -2/9 and 5/9
- Find two rational numbers between 1/5 and 1/2
- Find ten rational numbers between 1/4 and 1/2
- Find ten rational numbers between -2/5 and 1/2 .
- Find ten rational numbers between 3/5 and 3/4
Exercise 1.1
Question 1.Add the following rational numbers:
(i) 
and
(ii) 
and
(iii) 
and
(iv) 
and
Answer:
(i) Clearly,
Denominators of the given numbers are positive
The L.C.M of denominator 7 and 7 is 7
We have,
+ 
= 
= 
(ii) Clearly,
Denominators of the given numbers are positive
The L.C.M of denominator 4 and 4 is 4
We have,
+ 
= 
= 
= -2
(iii) Clearly,
Denominators of the given numbers are positive
The L.C.M of denominator 11 and 11 is 11
We have,
+
= 
= 
(iv) Clearly,
Denominators of the given numbers are positive
The L.C.M of denominator 13 and 13 is 13
We have,
+ 
= 
= 
Question 2.
Add the following rational numbers:
(i) 
and
(ii) 
and
(iii) 
and
(iv) 
and
(v) 
and
(vi) 
and
(vii) 
and
(viii) 
and
Answer:
(i) The denominator of given rational numbers are 4 and 8 respectively
The L.C.M of 4 and 8 is 8
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
and 
Therefore,
-
= 
= 
(ii) The denominator of given rational numbers are 9 and 3 respectively
The L.C.M of 9 and 3 is 9
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
and 
= 
Therefore,
+ = 
= 
(iii) The denominator of given rational numbers are 1 and 5 respectively
The L.C.M of 1 and 5 is 5
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
and 
Therefore,
+ = 
= 
(iv) The denominator of given rational numbers are 27 and 18 respectively
The L.C.M of 27 and 18 is 54
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
= 
And,
= 
= 
Therefore,
() +
= - 
= 
= 
(v) The denominator of given rational numbers are -4 and 8 respectively
The L.C.M of -4 and 8 is 8
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
= 
And,
Therefore,
(
) + ()
= -
= 
(vi) The denominator of given rational numbers are 36 and 12 respectively
The L.C.M of 36 and 12 is 36
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
And,
Therefore,
-
= 
= 
(vii) The denominator of given rational numbers are 16 and 24 respectively
The L.C.M of 16 and 24 is 48
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
= 
And,
= 
= 
Therefore,
+
= +
= 
(viii) The denominator of given rational numbers are -4 and 8 respectively
The L.C.M of 18 and 27 is 54
Now, we rewrite the given rational numbers into forms in which both of them have the same denominator
= 
= 
And,
= 
Therefore,
+
= 
= 
Question 3.
Simplify:
(i) 
(ii) 
(iii)
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
(x) 
Answer:
(i) The L.C.M of 9 and 6 is 18
= 
= 
And,
= 
= 
Therefore,
- 
= 
(ii) 
(iii) The L.C.M of -12 and -15 is 60
= 
= 
And,
= 
= 
Therefore,
- 
= 
(iv) The L.C.M of 19 and 57 is57
= 
= 
And,
= 
=
Therefore,
- 
= 
(v) The L.C.M of 9 and 4 is 36
= 
= 
And,
= 
= 
Therefore,
- 
= 
(vi) The L.C.M of 26 and -39 is 78
= 
= 
And,
= 
= 
Therefore,
- 
= 
(vii) The L.C.M of 9 and 12 is 108
= 
= 
And,
= 
= 
Therefore,
- 
= 
= 
(viii) The L.C.M of 8 and 36 is 72
= 
= 
And,
= 
= 
Therefore,
+ = 
(ix) The L.C.M of 0 and 5 is 0
Therefore,
0 - = 
(x) The L.C.M of 1 and 5 is 5
= 
= 
And,
= 
=
Therefore,
- 
= 
Question 4.
Add and express the sum as a mixed fraction:
(i) 
and
(ii) 
and
(iii) 
and
(iv) 
and 
Answer:
(i) The denominators of the given rational numbers 5 and 10 respectively.
The L.C.M of 5 and 10 is 10
Now,
We write the given rational numbers into forms in which both of them have the same denominator
= 
And,
= 
Therefore,
+ = 
= 
= 1
(ii) The denominators of the given rational numbers 7 and 4 respectively.
The L.C.M of 7 and 4 is 28
Now,
We write the given rational numbers into forms in which both of them have the same denominator
= 
And,
= 
Therefore,
-
= 
= 
= 1
(iii) The denominators of the given rational numbers 6 and 8 respectively.
The L.C.M of 6 and 8 is 24
Now,
We write the given rational numbers into forms in which both of them have the same denominator
= 
And,
= 
Therefore,
-
= 
= 
= -8 
(iv) The denominators of the given rational numbers 6 and 8 respectively.
The L.C.M of 6 and 8 is 24
Now,
We write the given rational numbers into forms in which both of them have the same denominator
= 
And,
= 
Therefore,
+ = 
= 
= 17 
Exercise 1.2
Question 1.Verify commutativity of addition of rational numbers for each of the following pairson of rational numbers:
(i) 
and
(ii) 
and
(iii) and
(iv) 
and
(v) 4 and
(vi) -4and
Answer:
(i) The addition of rational number is commutative
i.e, if 
and 
are any two rational numbers, then
+ = +
Verification: In order to verify this property,
Let us consider two expressions:
+ 
And,
+ 
We have:
+ = 
+ 
= 
= 
And,
+ = +
= 
=
Therefore,
+ = +
(ii) The addition of rational number is commutative
i.e, if and are any two rational numbers, then
+ = +
Verification: In order to verify this property,
Let us consider two expressions:
+ 
And,
+
We have:
+ = 
+
= 
= 
And,
+ = +
= 
=
Therefore,
+ = +
(iii) The addition of rational number is commutative
i.e, if and are any two rational numbers, then
+ = +
Verification: In order to verify this property,
Let us consider two expressions:
+ 
And,
+
We have:
+ = + 
= 
= 
And,
+ = +
= 
=
Therefore,
+ = +
(iv) The addition of rational number is commutative
i.e, if and are any two rational numbers, then
+ = +
Verification: In order to verify this property,
Let us consider two expressions:
+ 
And,
+
We have:
+ = 
+ 
= 
= 
And,
+ 
= +
= 
=
Therefore,
+ = +
(v) The addition of rational number is commutative
i.e, if and are any two rational numbers, then
+ = +
Verification: In order to verify this property,
Let us consider two expressions:
4 +
And,
+ 4
We have:
4 + = 
-
= 
= 
And,
+ 4 = +
= 
=
Therefore,
4 + = + 4
(vi) The addition of rational number is commutative
i.e, if and are any two rational numbers, then
+ = +
Verification: In order to verify this property,
Let us consider two expressions:
+ 
And,
- 4
We have:
-4 + = 
-
= 
= 
And,
- 4 = 
-
= 
=
Therefore,
-4 + = -4
Question 2.
Verify associativity of addition of rational numbers i.e., 
when:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) In order to verify this property, let us consider the following expressions:
Verification: 
+ [
+ (-)] = + [
- 
]
= + 
= 
= 
And,
( + ) + (
) = (
+ 
) -
= 
-
= 
=
Therefore,
The associative property of additional of rational numbers has been verified
(ii) In order to verify this property, let us consider the following expressions:
Verification: 
+ [
+ (-
)] = + [
- 
]
= 
+ 
= 
= 
And,
( + ) + (
) = (
+ 
) -
= 
-
= 
=
Therefore,
The associative property of additional of rational numbers has been verified
(iii) In order to verify this property, let us consider the following expressions:
Verification: 
+ [
+ (-
)] = + [
- 
]
= -
= 
= 
And,
(-
+ ) + (
) = (
-
) -
= 
-
= 
=
Therefore,
The associative property of additional of rational numbers has been verified
(iv) In order to verify this property, let us consider the following expressions:
Verification: -2 + [ + (-)] = -2 + [
- ]
= -2 -
= 
= 
And,
(-2 +) + (
) = (
+ ) -
= 
-
= 
=
Therefore,
The associative property of additional of rational numbers has been verified
Question 3.
Write the additive invese of each of the following rational numbers:
(i) 
(ii)
(iii) 
(iv) 
Answer:
(i) The additive inverse of 
is 
(ii) The additive inverse of is 
(iii) The additive inverse of 
is
(iv) The additive inverse of 
is 
Question 4.
Write the negative (additive inverse) of each of the following:
(i) 
(ii) 
(iii) 
(iv) 
(v) 0
(vi) 1
(vii) -1
Answer:
(i) The additive inverse of 
is 
(ii) The additive inverse of 
is 
(iii) The additive inverse of 
is 
(iv) The additive inverse of -5 is 5
(v) The additive inverse of 0 is 0
(vi) The additive inverse of 1 is -1
(vii) The additive inverse of -1 is 1
Question 5.
Using commutativity and associativity of addition of rational numbers, express each of the following as a rational number:
(i) 
(ii) 
(iii) 
(iv)
Answer:
(i) Reaaranging and grouping the numbers in pairs in such a way that each group contains a pair of rational numbers with equal denominator
We have,
+ 
+ + 
- + - 
= + 
= + 
= 
= 
(ii) Reaaranging and grouping the numbers in pairs in such a way that each group contains a pair of rational numbers with equal denominator
We have,
+ + 
+
- 
+ - = 
+ 
= +
= 
+ 
= 
(iii) Reaaranging and grouping the numbers in pairs in such a way that each group contains a pair of rational numbers with equal denominator
We have,
+ 
+ 
+ + 
+ + - - = 
+ 
-
= + +
= 
+ -
= 
= 
(iv) Reaaranging and grouping the numbers in pairs in such a way that each group contains a pair of rational numbers with equal denominator
We have,
+ 0 + 
+ 
+ 
- 
+ 
+ + = 
+ -
= 
+ -
= 
-
= 
– 
= 
- 
= 
Question 6.
Re-arrange sutably and find the sum in each of the following.
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Answer:
(i) 
- 
- 
= 
= 
(ii) 
- 
- 
-
= 
- -
= 
= 
(iii) + 
+ - - 
= 
-
= 
-
= 
= 
(iv) 
+ 
- 
- +
= 
- +
= 
- +
= 
= 
(v) + + -
= 
+ 
= 
-
= 
=
=
(vi) 
+ 
+ + 
+ - 
= 
+ + -
= 
= 
Exercise 1.3
Question 1.Subtract the first rational number from the second in each of the following:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Answer:
(i) - 
= 
= 
(Therefore, L.C.M of 8 and 8 is 8)
= 
(ii) - 
= 
= 
(Therefore, L.C.M of 9 and 9 is 9)
(iii) 
- 
= 
= (Therefore, L.C.M of 11 and 11 is 11)
(iv) 
- 
= 
= 
(Therefore, L.C.M of 13 and 13 is 13)
(v) 
- = 
= (Therefore, L.C.M of 8 and 4 is 8)
(vi) - = 
= 
(Therefore, L.C.M of 6 and 3 is 6)
= 
(vii) 
- = 
= 
(Therefore, L.C.M of 14 and 7 is 14)
(viii) 
- 
= 
= 
(Therefore, L.C.M of 22 and 33 is 66)
Question 2.
Evalute each of the following:
(i) +
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
(x) 
(xi) 
- 
Answer:
(i) -
= 
= 
= 
(ii) +
= 
= 
= 
(iii) - 
= 
= 
= 
(iv) -2 - 
= 
= 
= 
(v) +
= 
= 
= 
(vi) +
= 
= 
= 
(vii) 
+
= 
= 
=
(viii) - 
= 
= 
= 
(ix) 
+ 
= 
= 
= 
(x) - 
= 
= 
= 
(xi) +
= 
= 
= 
Question 3.
The sum of the two numbers is 
. If one of the numbers is 
, find the other
Answer:
It is given that,
The sum of the two numbers =
One of the number =
Since, the sum is
Therefore,
The other number = Sum of numbers - given numberOther number
Hence, the other number is 2/9.
Question 4.
The sum of two numbers is 
. If one of the numbers is 
, find the other
Answer:
It is given that,
The sum of the two numbers =
One of the number = 
Suppose the other number is x
Since, the sum is
Therefore,
x - 
=
=
3x = 12 - 1
3x = 11
x = 
Question 5.
The sum of two numbers is 
. If one of the numbers is -5, find the other
Answer:
It is given that,
The sum of the two numbers =
One of the number = -5
Suppose the other number is x
Since, the sum is
Therefore,
x - 5 =
3x – 15 = -4
3x = -4 + 15
3x = 11
x =
Question 6.
The sum of two ratinal numbers is -8. If one of the numbers is 
, find the other
Answer:
It is given that,
The sum of the two numbers = -8
One of the number =
Suppose the other number is x
Since, the sum is -8
Therefore,
x + = -8
= -8
7x - 15 = -56
7x = -56 + 15
7x = -41
x = 
Question 7.
What should be added to 
so as to get ?
Answer:
Suppose x be the rational number to be added to 
to get
Then,
+ x =
= + 
x = 
x = 
x = 
Therefore,
The required number x =
Question 8.
What number should be added to 
so as to get 
?
Answer:
Suppose x be the rational number to be added to 
to get 
Then,
+ x =
= + 
x = 
x = 
x = 
Therefore,
The required number x =
Question 9.
What number should be addede to
to get
?
Answer:
Suppose x be the rational number to be added to to get
Then,
+ x =
= -
x = 
x = 
x = 
Therefore,
The required number x =
Question 10.
What number should be subtracted from 
to get
?
Answer:
Suppose x be the rational number to be subtracted to 
to get 
Then,
- x =
– 
=
- 
=
=
-5 – 3x = 
-3x = + 5
-3x = 
-3x = 
-3x = 
-18x = 45
x = 
x = 
Therefore,
The required number x = 
Question 11.
What number should be subtracted from
to get
?
Answer:
Suppose x be the rational number to be subtracted to to get 
Then,
- x =
= -
-x = 
-x = 
x = 
Therefore,
The required number x =
Question 12.
What should be added to ( + ) to get
?
Answer:
Let the number be x
It is given that,
x + ( + ) =
=
15x = -2 -19
x = 
x =
Question 13.
What should be added to (
+ 
+ 
) to get 3?
Answer:
Let the three numbers be x
It is given that,
x + ( + + ) = 3
x + (
+ 
+ 
) = 3
Therefore, L.C.M of 2, 3 and 5 is 30
= 3
= 90
30x = 59
x = 
Question 14.
What should be subtracted from (
- ) to get 
?
Answer:
Let the number be x
It is given that,
( - ) – x =
-x + [
– 
] =
-x + [
- 
] =
=
- 12x + 1 = -2
-12x = -2 - 1
-12x = -3
x = 
x =
Question 15.
Fill in the blanks:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) + 
= – 
= 
= 
(ii) Let x be a rational number. Then,
+ x = -1
= -14
= 9 – 14
=
(iii) + x = 3
+ 
= 3
= 27
x = 
(iv) x + 
= 4
23x + 15 = 92
23x = 92 – 15
23x = 77
x = 
Exercise 1.4
Question 1.Simplify each of the following and write as rational number of the form
:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Answer:
(i) We have,
+ + ()
= + -
= 
+ 
+ 
Therefore,
L.C.M of 4, 6 and 8 is 24
= 
=
(ii) We have,
+ 
+ ()
= - -
= 
+ 
-
Therefore,
L.C.M of 3, 6 and 9 is 18
= 
=
(iii) We have,
+ + ()
= -
-
= 
-
-
Therefore,
L.C.M of 6, 2 and 8 is 24
= 
= 
(iv) We have,
+ + (
)
= - - 
= 
-
-
Therefore,
L.C.M of 5, 10 and 15 is 30
= 
= 
(v) We have,
+ 
+ (
)
= - - 
= 
-
-
Therefore,
L.C.M of 15, 10 and 20 is 60
= 
=
= 
(vi) We have,
+ 
+ (
) + 3
= - - + 3
= 
-
-
+ 
Therefore,
L.C.M of 3, 2, 3 and 1 is 6
= 
=
Question 2.
Express each of the following as a rational number of the form 
:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Answer:
(i) 
+ (
) + (
) + 
- 3
= - 
- 
+ 
- 3
= 
– 
– 
+ 
– 
= 
= 
= 
(ii) 
+ 1 + (
) + (
) + 
= 
+
–
+ 
– 
= 
+ – 
+ 
– 
= 
=
(iii) 
+ (
) + (
) + 6 + 
= + - 
+ 6 - 
= 
+
– 
+ 
– 
= 
= 
(iv) 
+ 0 + (
) + 
+ 
= +0 -
+ + 
= 
– 
+
+ 
= 
= 
(v) + (
) + (
) + 
+ 2
= + - - + 2
= 
+
– 
-
+
= 
= 
= 
Question 3.
Simplify:
(i) 
(ii) 
(iii) - -
(iv) 
(v) 
(vi) 
Answer:
(i) We have,
+ -
= 
(L.C.M of 2, 4 and 4 is 4)
= 
= 
=
= -2
(ii) We have,
- -
= 
(L.C.M of 3, 6 and 3 is 6)
= 
=
(iii) We have,
- + = 
(L.C.M of 4, 6 and 3 is 24)
= 
= 
= 
=
(iv) We have,
+ 
+ = 
(L.C.M of 5, 10 and 7 is 70)
= 
= 
= 
(v) We have,
- + = 
(L.C.M of 6, 5 and 15 is 30)
= 
= 
= 
(vi) We have,
+ 
- = 
(L.C.M of 8, 9 and 36 is 72)
= 
= 
= 
= 
Exercise 1.5
Question 1.Multiply:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Answer:
(i) *
= 
= 
(ii) * 
= 
= 
(iii) 
*
= 
= 
(iv) 
* 
= 
= 
(v) 
* 
= 
= 
(vi) 
*
= 
= 
= 
(vii) *
= 
= 
(viii) 
* 7
= 
= 
Question 2.
Multiply:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Answer:
(i) 
* 
= 
= 
=
(ii) 
* 
= 
=
(iii) 
*
= 
= 
(iv) 
* 
= 
= 
(v) 
* 
= 
= 
(vi) * 
= 
= 
Question 3.
Simplify each of the following and express the result as a rational number in standard from:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Answer:
(i) 
* 
= 
= 
(ii) * 
= 
= 
(iii) 
* 16
= 
= 
(iv) 
* 
= 
= 
=
(v) 
* 
= 
=
(vi) 
* 
= 
= 
(vii) 
* 
= 
= 
(viii) * 
= 
= 
=
Question 4.
Simplify:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi)
(vii) 
(viii)
Answer:
(i) 
– 
= 
+ 
= +
= 
= 
(ii) 
+ 
= + 
= + 3
= 
= 
(iii) 
+ 
= 
+ 
= +
= 
=
(iv) 
+ 
= 
+ 
= 
= 
=
(v) 
+ 
= 
+ 
= 
+
= 
(vi) 
+ 
= 
+ 
= 
= 
(vii) 
+ 
= 
+
= 
+ 
= 
= 
= 
(viii) 
– 
= - 
= 
= 
= 
Question 5.
Simplify:
(i) (
× 
) + ( × 
) – (
)
(ii) 
(iii) 
(iv) 
Answer:
(i) ( × ) + ( × ) – ( )
= + 
- 
= 
= 
= 
= 
(ii) 
+ 
+ 
= 
= 
= 
= 
(iii) 
+ 
+ 
= 
+ 
+
= 
= 
(iv) 
- 
+ 
= 15/66 - 1 + 1/13
Exercise 1.6
Question 1.Verify the property: 
by taking:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) We have,
x * y = * ()
= 
y * x = * ()
=
(ii) We have,
x * y = * ()
= 
y * x = * ()
=
(iii) We have,
x * y = 2 * (
)
= 
= 
y * x = * 2
= =
(iv) We have,
x * y = 0 * (
)
= 0
y * x = 
* 0
= 0
Question 2.
Verify the property: 
by taking:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) We have,
x = , y = and z =
x * (y * z) = * ( * )
= (
)
= 
(x * y) * z = ( * ) *
= ()
=
(ii) We have,
x =0, y = and z = 
x * (y * z) = 0 * ( * )
= 0
(x * y) * z = (0* ) *
= 0
(iii) We have,
x = , y = 
and z =
x * (y * z) = * ( * )
= ()
=
(x * y) * z = ( * ) *
= ()
=
(iv) We have,
x = , y = 
and z =
x * (y * z) = * ( * )
= 
(x * y) * z = ( * ) *
= 
()
=
Question 3.
Verify the property: 
by taking:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) (
) * (
+ )
= * + *
= 
+ 
= 
= 
= 
x * y + x * z = 
() * (
) = * 
=
() * ( + ) = () * () + () * ()
(ii) () * ( + )
= * ( 
)
= 
x * y + x * z
() * () + () * ()
=
(iii) 
* ( + 
) = () * () + () * ()
* (
) = 
+ 
= 
=
Therefore,
L.H.S = R.H.S
(iv) ( + ) = () * () + () * ()
* + * = (
+ 
)
+ 
= ( - 
)
= ( – 
)
= (
)
=
1 = 1
Therefore,
L.H.S = R.H.S
Question 4.
Use the distributivity of multiplication of rational numbers over their addition to simplify:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) 
* 
+ * 10
= 
+ 6
= 
+ 
= 
(ii) 
* (
)
= * 
= -6
(iii) * (
)
= 
* 
= 
(iv) 
* (
)
= * 
= 
Question 5.
Find the multiplicative inverse (reciprocal) of each of the following rational numbers:
(i) 9
(ii) -7
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) -1
(x) 
(xi)1
Answer:
(i) Reciprocal of 9 is 
(ii) Reciprocal of –7 is
(iii) Reciprocal of is
(iv) Reciprocal of is
(v) Reciprocal of 
is
(vi) * 
=
Reciprocal of is
(vii) * 
=
Reciprocal of is 
(viii) -2 * =
Reciprocal of is
(ix) Reciprocal of -1 is -1
(x) Reciprocal of 
is not defined as in the form of 
, q = 0, so it is not a rational number
(xi) Reciprocal of 1 is 1
Question 6.
Name the property of multiplication of rational numbers illustrated by the following statements:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Answer:
(i) In the above rational number the property of multiplication which is used is cummutativity because:
According to commutative law:
* = * , which is mentioned above
(ii) In the above rational number the property of multiplication which is used is cummutativity because:
According to commutative law:
* = * , which is mentioned above
(iii) In the above rational number the property of multiplication which is used is distributivity of multiplication over addition because:
According to this law:
( + 
) = ( *) + ( *), which is mentioned above
(iv) In the above rational number the property of multiplication which is used is associativity of multiplication because:
According to associative law:
* ( * ) = ( * ) * , which is mentioned above
(v) In the above rational number the property of multiplication which is used is existence of identity for multiplication which is mentioned above
(vi) In the above rational number the property of multiplication which is used is existence of multiplication inverse which is mentioned above
(vii) In the above rational number the property of multiplication which is used is multiplication by zerobecause:
According to this law:
* 0 = 0 * , which is mentioned above
(viii) In the above rational number the property of multiplication which is used is distributive law because:
According to distributive law:
* + * = * ( +), which is mentioned above
Question 7.
Fill in the blanks:
(i) The product of two positive rational numbers is always………..
(ii) The product of a 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 numbers is ………..
(v) The reciprocal of a negative rational number is………..
(vi) The product of a rational number and its reciprocal is……..
(vii) Zero has……. reciprocal
(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 
is
(xii) 
Answer:
(i) The product of two positive rational numbers is always positive
(ii) The product of a positive rational number and a negative rational number is always negative
(iii) The product of two negative rational numbers is always positive
(iv) The reciprocal of a positive rational numbers is positive
(v) The reciprocal of a negative rational number is negative
(vi) The product of a rational number and its reciprocal is 1
(vii) Zero has no reciprocal
(viii) The numbers 1 and -1 are their own reciprocals
(ix) If a is reciprocal of b, then the reciprocal of b is a
(x) The number 0 is not the reciprocal of any number
(xi) Reciprocal of is a
(xii) (17 * 12)-1 = 17-1 * 12-1
Question 8.
Fill in the blanks:
(i) 
(ii) 
(iii) 
(iv) 
Answer:
(i) -4 * = * -4
This is because of the use of cumutative law
According to commutative law:
* = *
(ii) * = *
This is because of the use of cumutative law
According to commutative law:
* = *
(iii) * ( + ) = * + *
This is because of the use of distributive law
According to distributive law:
* + * = * ( +)
(iv) * ( + ) = ( * ) + *
This is because of the use of distributive law
According to distributive law:
* + * = * ( +)
Exercise 1.7
Question 1.Divide:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
(ix) 
(x) 
Answer:
(i) 1 by
= 
= 1 
= 2
(ii) 5 by
= 5 ÷
= 
7
= -7
(iii) by 
= 
= 
=
(iv) by 
= 
= 
=
(v) 
by 
= 
= 
=
(vi) 0 by
= 
= 
= 0
(vii) by -6
= 
= 
=
(viii) by 
= 
= 
=
=
(ix) 
by 
= 
= 
= 
(x) by 
= 
= 
= 
Question 2.
Find the value and express as a rational number in standard from:
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
Answer:
(i) 
= 
= 
=
(ii) 
= 
= 
=
(iii) 
= 
= 
=
(iv) 
= 
= 
=
(v) 
= 
= 
=
(vi) 
= 
= 
=
Question 3.
The product of two rational numbers is 15. If one of the numbers is -10, find the other.
Answer:
It is that the product of two rational numbers is 15
If one of the number is -10
So, the other number is obtained by dividing the product by the given number.
Therefore,
Other number = 
=
Question 4.
The product of two rational numbers is 
. If one of the numbers is 
, find the other.
Answer:
It is given that the product of two rational numbers is .
If one of the number is -
we have to find the other number.
So, the other number is obtained by dividing the product by the given number.
Therefore,
Other number = 
= (8 × 15)/(9 × 4) = (2 × 5)/3 = 10/3
Question 5.
By what number should we multiply 
so that the product may be
?
Answer:
Let the required number be ‘x’
Now,
According to the question,
Question 6.
By what number should we multiply
so that the product may be
?
Answer:
Let the required number be ‘x’
Now,
According to the question,
Question 7.
By what number should we multiply
so theat the product may be 24?
Answer:
Let the required number be ‘x’
Now,
According to the question,
Question 8.
By what number should 
be multiplied in order to produce 
?
Answer:
Let the required number be ‘x’
Now,
According to the question,
Question 9.
Find
if
(i) 
(ii) 
(iii) 
(iv) 
(v) 
Answer:
(i) 
(ii)
9
(iii)
(iv)
(v)
Question 10.
The cost of 
metres of rope is Rs 
.Find its cost per metre.
Answer:
It is given that,
= 
metre of rope is Rs. 
Let cost of 1 metre be x
So,
x * =
x = 
= 
Therefore,
Cost of rope is Rs. per metre
Question 11.
The cost of
metres of cloth is Rs
. Find the cost of cloth per metre
Answer:
It is given that,
Cost of 
metres of cloth is Rs. 
Let the cost of the cloth per metre be x
So,
x × =
x = 
x = 32.55
Therefore,
Cost of cloth is Rs. 32.55 per metre
Question 12.
By what number should
be divided to get
?
Answer:
Let x be the required number
So,
= 
= x * 
= x
Therefore,
Question 13.
Divide the sum of 
and 
by the product of 
and
Answer:
According to question,
+ 
And,
×
Now,
We have to divide by 
Therefore
= × 
= 
Question 14.
Divide the sum of 
and by their difference.
Answer:
According to question,
= 
= 
= 
Question 15.
If 24 trousers of equal size can be prepared in 54 metres of cloth. What length of cloth is required for
each trouser?
Answer:
No.of trousers = 24
Total length of cloth = 54
Length of cloth required for each trousers = 
= 
= 
m
Exercise 1.8
Question 1.Find a rational number between -3 and 1
Answer:
We know that between two rational numbers x and y
Such that x< y there is a rational number 
So, rational number between -3 and 1 is 
= -1
Thus, we have -3 
1
Question 2.
Find any five rational numbers less than 2.
Answer:
Five rational numbers less than two are:
0, 
Question 3.
Find two rational numbers between 
and 
Answer:
A rational number lying between 
Now,
A rational number between
Therefore,
Question 4.
Find two rational numbers between 
and
Answer:
For finding rational numbers between two numbers:
Add the numbers are divide by 2, this will give a number between the numbers. Now take the new number and add with any of the numbers and repeat the process. You can keep on repeating the process and new numbers will be obtained.
A rational number lying between 
Now,
A rational number between
Hence, the two rational numbers lying between are 
Question 5.
Find ten rational numbers between 
and
Answer:
The LCM of denominators 4 and 2 is 4
Converting the given rational numbers into equivalent rational number having common denominator 4, we get:
Clearly,
21, 22, 23,…, 39 are integers between numerators 20 and 40
Hence, the rational numbers between 
are:
Question 6.
Find ten rational numbers between 
and
.
Answer:
The LCM of denominators -5 and 2 is -10
Converting the given rational numbers into equivalent rational number having common denominator 10, we get:
Clearly,
-7, -6, -5,…, 10 are integers between numerators -8 and 10
Hence, the rational numbers between 
are:
Question 7.
Find ten rational numbers between 
and
Answer:
The L.C.M of denominators 5 and 4 is 20
Converting the given rational numbers into equivalent rational number having common denominator 20, we get:
Clearly,
61, 62, 63,…, 74 are integers between numerators 60 and 75 of these equivalent rational numbers
Thus, we have
As rational number between and
We can take only 10 of these as required rational numbers
