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Page No 4:

Answer:

(i) 15 + (−8) = 7

(ii) (−16) + 9 = −7

(iii) (−7) + (−23) = −30

(iv) (−32) + 47 = 15

(v) 53 + (−26) = 27

(vi) (−48) + (−36) = −84

Page No 4:

Answer:

(i) 153 + (−302) = −149

(ii)  1005 + (−277) = 728

(iii) (−2035) + 297 = −1738

(iv)  (−489) + (−324) = −813

(v)  (−1000) + 438 = −562

(vi) (−238) + 500 = 262

Page No 4:

Answer:

(i) Additive inverse of −83 = −(−83) = 83

(ii) Additive inverse of 256 = −(256) = −256

(iii) Additive inverse of 0 = −(0) = 0

(iv) Additive inverse of 2001 = −(−2001) = 2001



Page No 5:

Answer:

(i) 42 28 = (42) + (28) = 70

(ii) 42 (36) = 42 + 36 = 78

(iii) -53 - (-37) = (-53) - (-37) = -16

(iv)  -34 - (-66) = -34 + 66 = 32

(v) 0 - 318 = -318

(vi)  (-240) - (-153) = -87

(vii)  0 - (-64) = 0 + 64 = 64

(viii) 144 - (-56) = 144 + 56 = 200

Page No 5:

Answer:

Sum of −1032 and 878 = −1032 + 878
                                    = -154

Subtracting the sum from −34, we get
−34 − (−154)
= (−34)+ 154
= 120

Page No 5:

Answer:

First, we will calculate the sum of 38 and −87.
38 + (−87) = −49

Now, subtracting −134 from the sum, we get:
−49 − (−134)
=(−49) + 134
= 85

Page No 5:

Answer:

(i) −41   (∵ Associative property)

(ii) −83   (∵ Associative property)

(iii)  53  (∵ Commutative property)

(iv)  −76  (∵ Commutative property)

(v) 0  (∵ Additive identity)

(vi)  83  (∵ Additive inverse)

(vii)  (−60) − (−59) = −1

(viii)  (−40) − (−31) = −9

Page No 5:

Answer:

{−13 − (−27)} + {−25 − (−40)}
= {−13 + 27} + {−25 + 40}
=14 + 15
= 29

Page No 5:

Answer:

36 − (−64) = 36 + 64 = 100

Now, (−64) − 36 = (−64) + (−36) = −100

Here, 100 −100

Thus, they are not equal.

Page No 5:

Answer:

(a + b) + c = (−8 + (−7)) + 6 = −15 + 6 = −9

a + (b + c) = −8 + (−7 + 6) = −8 + (−1) = −9

Hence, (a + b) + c = a + (b + c)   [i.e., Property of Associativity]

Page No 5:

Answer:

Here, (a − b) = −9 − (−6) = −3

Similarly, (b − a) = −6 − (−9) = 3

∴ (a−b) ≠ (b−a)

Page No 5:

Answer:

Let the other integer be a. Then, we have:

53 + a = −16
a = −16 − 53 = −69

∴ The other integer is −69.

Page No 5:

Answer:

Let the other integer be a.
Then, −31 + a = 65
⇒ a = 65 − (−31) = 96

∴ The other integer is 96.

Page No 5:

Answer:

We have:

a − (−6) = 4
a = 4 + (−6) = −2

a = −2

Page No 5:

Answer:

(i)  Consider the integers 8 and −8. Then, we have:
8 + (−8) = 0

(ii) Consider the integers 2 and (−9). Then, we have:
 2 + (−9)= −7, which is a negative integer.

(iii)  Consider the integers −4 and −5. Then, we have:
(−4) + (−5) = −9, which is smaller than −4 and −5.

(iv) Consider the integers 2 and 6. Then, we have:
 2 + 6 = 8, which is greater than both 2 and 6.

(v)  Consider the integers 7 and −4. Then, we have:
7 + (−4) = 3, which is smaller than 7 only.

Page No 5:

Answer:

(i)  F (false). −3, −90 and −100 are also integers. We cannot determine the smallest integer, since they are infinite.

(ii)  F (false). −10 is less than −7.

(iii)  T (true). All negative integers are less than zero.

(iv)  T (true).

(v)  F (false). Example: −9 + 2 = −7

 



Page No 9:

Answer:

(i) 16 × 9 = 144
(ii) 18 × (−6) = -(18×6) = −108
(iii) 36 × (−11) = - (36×11) = −396
(iv)  (−28) ×14 = -(28×14) = −392
(v) (−53) × 18 = -(53×18) = −954
(vi) (−35) × 0 = 0  
(vii) 0 × (−23) = 0
(viii) (−16) × (−12) = 192
(ix) (−105) × (−8) = 840
(x) (−36) × (−50) = 1800
(xi) (−28) × (−1) = 28
(xii)  25 × (−11) = - (25×11) = −275

Page No 9:

Answer:

(i) 3 × 4 × (−5) = (12) × (−5) = −60
(ii) 2 × (−5) × (−6) = (−10) × (−6) = 60
(iii) (−5) × (−8) × (−3) = (−5) × (24) = −120
(iv)  (−6) × 6 × (−10) = 6 × (60) = 360
(v)  7 × (−8) × 3 = 21 × (−8) = −168
(vi)  (−7) × (−3) × 4 = 21 × 4 = 84

Page No 9:

Answer:

(i)  Since the number of negative integers in the product is even, the product will be positive.
    (4) × (5) × (8) × (10) = 1600
(ii) Since the number of negative integers in the product is odd, the product will be negative.
  −(6) × (5) × (7) × (2) × (3) = −1260
(iii) Since the number of negative integers in the product is even, the product will be positive.
   (60) × (10) × (5) × (1) = 3000
(iv) Since the number of negative integers in the product is odd, the product will be negative.
   −(30) × (20) × (5) = −3000
(v) Since the number of negative integers in the product is even, the product will be positive.
    (-3)6 = 729
(vi) Since the number of negative integers in the product is odd, the product will be negative.
   (-5)5 = −3125
(vii) Since the number of negative integers in the product is even, the product will be positive.
    (-1)200= 1
(viii) Since the number of negative integers in the product is odd, the product will be negative.
     (-1)171 = −1

Page No 9:

Answer:

Multiplying 90 negative integers will yield a positive sign as the number of integers is even.
Multiplying any two or more positive integers always gives a positive integer.
The product of both(the above two cases) the positive and negative integers is also positive.
Therefore, the final product will have a positive sign.

Page No 9:

Answer:

Multiplying 103 negative integers will yield a negative integer, whereas 65 positive integers will give a positive integer.
The product of a negative integer and a positive integer is a negative integer.

Page No 9:

Answer:

(i) (−8) × (9 + 7)   [using the distributive law]
= (−8) × 16 = −128

(ii)  9 × (−13 + (−7))  [using the distributive law]
= 9 × (−20) = −180

(iii)  20 × (−16 + 14)    [using the distributive law]
= 20 × (−2) = −40

(iv) (−16) × (−15 + (−5))  [using the distributive law]
= (−16) × (−20) = 320

(v) (−11) × (−15 +(−25))  [using the distributive law]
= (−11) × (−40)
= 440

(vi) (−12) × (10 + 5)   [using the distributive law]
= (−12) × 15 = −180

(vii) (−16 + (−4)) × (−8)  [using the distributive law]
= (−20) × (−8) = 160

(viii) (−26) × (72 + 28)    [using the distributive law]
= (−26) ×100 = −2600

Page No 9:

Answer:

(i) (−6) × (x) = 6
x = 6-6 = -66= -1

Thus, x = (−1)

(ii) 1      [∵ Multiplicative identity]
(iii) (−8)      [∵ Commutative law]
(iv) 7         [∵ Commutative law]
(v) (−5)   [∵ Associative law]
(vi) 0    [∵ Property of zero]

Page No 9:

Answer:

We have 5 marks for correct answer and (−2) marks for an incorrect answer.

Now, we have the following:

(i) Ravi's score = 4 × 5 + 6 × (−2)
= 20 + (−12) =8

(ii) Reenu's score = 5 × 5 + 5 × (−2)
= 25 − 10 = 15

(iii) Heena's score = 2 × 5 + 5 × (−2)
= 10 − 10 = 0

Page No 9:

Answer:

(i) True.
(ii) False. Since the number of negative signs is even, the product will be a positive integer.
(iii) True. The number of negative signs is odd.
(iv) False. a × (−1) = −a, which is not the multiplicative inverse of a.
(v) True. a × b = b × a
(vi) True. (a × b) × c = a × (b × c)
(vii) False. Every non-zero integer a has a multiplicative inverse 1a, which is not an integer.



Page No 12:

Answer:

(i) 65 ÷ (−13) = 65-13 = −5

(ii) (−84) ÷ 12 = -8412 =  −7

(iii) (−76) ÷ 19 = -7619 = −4

(iv) (−132) ÷ 12 = -13212 = −11

(v) (−150) ÷ 25 = -15025 = −6

(vi) (−72) ÷ (−18) = -72-18 = 4

(vii)  (−105) ÷ (−21) = -105-21  = 5

(viii) (−36) ÷ (−1) = -36-1 = 36

(ix) 0 ÷ (−31) =  0-31  = 0

(x)  (−63) ÷ 63 = -6363 = −1

(xi)  (−23) ÷ (−23) = -23-23 = 1

(xii) (−8) ÷ 1 =  -81 = −8

Page No 12:

Answer:

(i)
72 ÷ (x) = −4
 72x = -4x = 72-4 = -18 

(ii)
−36 ÷ (x) = −4
-36x = -4x = -36-4  = 9

(iii)
(x) ÷ (−4) = 24
x-4 = 24x = 24×(-4) = -96

(iv) 
(x) ÷ 25 = 0
x25 = 0x = 25×0 = 0

(v)
(x) ÷ (−1) = 36
x-1 = 36x = 36×(-1) = -36

(vi)
(x) ÷ 1 = −37
x1= -37x = -37×1 = -37

(vii)
39 ÷ (x) = −1
39x = -1x = -1×39 = -39

(viii) 
1 ÷ (x) = −1
1x= -1x = -1×1 = -1

(ix)
−1 ÷ (x) = −1
-1x = -1x = -1-1= 1

Page No 12:

Answer:

(i) True (T). Dividing zero by any integer gives zero.
(ii) False (F). Division by zero gives an indefinite number.

(iii) False (F). -5-1 = 5 

(iv)  True (T). -81= -8

(v)  False (F). -1-1 = 1

(vi) True (T). -9-1 = 9

Page No 12:

Answer:

(c) 14
Given:
6 − (−8)
= 6 + 8
= 14

Page No 12:

Answer:

(b) −3 
Given:
−9 − (−6)
= −9 + 6
= −3



Page No 13:

Answer:

(d) 5
We can see that

−3 + 5 = 2

Hence, 2 exceeds −3 by 5.

Page No 13:

Answer:

(a)  5
Let the number to be subtracted be x.
To find the number, we have:
−1 − x = −6
x = −1 + 6 = 5

Page No 13:

Answer:

(c) 4 
 We can see that
(−2) − (−6) = (−2) + 6 = 4

Hence, −6 is four (4) less than −2.

Page No 13:

Answer:

(b) −8
Subtracting 4 from −4, we get:
(−4) − 4 = −8

Page No 13:

Answer:

(b) 2
Required number = (−3) − (−5) = 5 − 3 = 2

Page No 13:

Answer:

(c) 6
(−3) − x = −9
∴ x = (−3) + 9 = 6
Hence, 6 must be subtracted from −3 to get −9.

Page No 13:

Answer:

(c) −11
Subtracting 6 from −5, we get:
(−5) − 6 = −11

Page No 13:

Answer:

(c) 5
Subtracting −13 from −8, we get:
(−8) − (−13)
= −8 + 13
= 5

Page No 13:

Answer:

(a) 4
(−36) ÷ (−9) = 4

Here, the negative signs in both the numerator and denominator got cancelled with each other.

Page No 13:

Answer:

(b) 0
Dividing zero by any integer gives zero as the result.

Page No 13:

Answer:

(c) not defined

Dividing any integer by zero is not defined.

Page No 13:

Answer:

(b) −11 < −8

Negative integers decrease with increasing magnitudes.

Page No 13:

Answer:

(b) 9

Let the other integer be a. Then, we have:
−3 + a = 6
∴ a = 6 − (−3) = 9

Page No 13:

Answer:

(a) −10
Let the other integer be a. Then, we have:
6 + a = −4
∴ a = −4 − 6 = −10

Hence, the other integer is −10.

Page No 13:

Answer:

(a) 22
Let the other integer be a. Then, we have:
−8 + a = 14
a = 14 + 8 = 22

Hence, the other integer is 22.

Page No 13:

Answer:

(c) 6

The additive inverse of any integer a is −a.
Thus, the additive inverse of −6 is 6.



Page No 14:

Answer:

(b) −150
We have (−15) × 8 + (−15) × 2
= (−15) × (8 + 2)    [Associative property]
= −150

Page No 14:

Answer:

(b) −24
We have (−12) × 6 − (−12) × 4
= (−12) × (6 − 4)       [Associative property]
= −24

Page No 14:

Answer:

(b) 810
(−27) × (−16) + (−27) × (−14)
= (−27) × (−16 + (−14))    [Associative property]
=(−27) × (−30)
= 810

Page No 14:

Answer:

(a)  −270
30 × (−23) + 30 × 14
= 30 × (−23 + 14)     [Associative property]
=  30 × (−9)
= −270

Page No 14:

Answer:

(c) 152
Let the other integer be a. Then, we have:
−59 + a = 93
∴ a = 93 + 59 = 152

Page No 14:

Answer:

(b) 90

x ÷ (-18) = -5x-18 = -5 x = -18 ×-5 = 90



Page No 15:

Answer:

Let the other integer be a. Then, we have:
a + (−12) = 43
a = 43 − (−12) = 55

Hence, the other integer is 55.

Page No 15:

Answer:

Given:
p − (−8)= 3
p = 3 + (−8)
p = −5

Hence, the value of p is −5.

Page No 15:

Answer:

Product of (−16) and (−9) = (-16) ×(-9) = 144
Now, (-132) ÷ 6 gives the quotient −22.

∴ 144 + (−22) = 122

Page No 15:

Answer:

Suppose that a divides −240 to obtain 16. Then, we have:

(−240) ÷ a = 16
a = (−240) ÷ 16 = −15

Hence, −15 should divide −240 to obtain 16.

Page No 15:

Answer:

Let a be divided by (−7) to obtain 12. Then, we have:

a÷(-7)=12
a = -712

Hence, -712 should be divided by −7 to obtain 12.

Page No 15:

Answer:

(i) −450
(ii)  360
(iii) −1080
(iv)  −600
(v) (-5)5 =-3125

(vi)  (-1)25 = -1

Page No 15:

Answer:

(i) (−16) × 12 + (−16) × 8
= (−16) × (12 + 8)   [Associative property]
=  (−16) × 20
= −320

(ii) 25 × (−33) + 25 × (−17)
= 25 × ((−33) + (−17))  [Associative property]
= 25 × (−50) = −1250

(iii)  (−19) × (−25) + (−19) × (−15)
=  (−19) × ((−25) + (−15))  [Associative property]
=  (−19) × (−40) = 760

(iv) (−47) × 68 − (−47) × 38
= (−47) × (68 − 38)  [Associative property]
= (−47) × 30 = −1410

(v)  (−105) ÷ 21 = −5

(vi)  12

(vii)  0 (zero). Dividing 0 by any integer gives 0.

(vii)  Not defined. Dividing any integer by zero is not defined.

Page No 15:

Answer:

(d) −8
Let the other integer be a. Then, we have:
2 + a = −6
a = −6 − 2 = −8

∴ The other integer is −8.

Page No 15:

Answer:

(b) 8
Suppose that a is subtracted from −7. Then, we have:

−7 − a = −15
a = −7 + 15 = 8

∴ 8 must be subtracted from −7 to obtain −15.

Page No 15:

Answer:

(b)108

(108) ÷ (−18) = −6

Page No 15:

Answer:

(a) 370
We have:

(−37) × (−7) + (−37) × (−3)
= (−37) × {(−7) + (−3)}  [Associative property]
= (−37) × (−10)
= 370

Page No 15:

Answer:

(c) −250

(−25) × 8 + (−25) × 2
= (−25) × (8 + 2)  [Associative property]
= −250

Page No 15:

Answer:

(b) −3

(−9) − (−6)
= (−9) + 6
= −3

Page No 15:

Answer:

(b) −6

−8 − (−6) = 2

Hence, −8 is −6 less than −2.

Page No 15:

Answer:

(i)  −1
(ii)  1
(iii) (−16)   [Commutative property]
(iv) 0   [Property of zero]
(v)  −7
(vi)  −19
(vii)  0
(viii) 152

Page No 15:

Answer:

(i) True (T).
(ii) False (F). Dividing any integer by zero is not defined.
(iii) False (F). (−1) ÷ (−1) = 1
(iv) True (T).
(v) True (T).
(vi) False (T). 68 ÷ (−17) = −4



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