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#### Page No 4:

(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:

(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:

(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:

(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:

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

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

#### Page No 5:

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:

(i) −41   (∵ Associative property)

(ii) −83   (∵ Associative property)

(iii)  53  (∵ Commutative property)

(iv)  −76  (∵ Commutative property)

(vii)  (−60) − (−59) = −1

(viii)  (−40) − (−31) = −9

#### Page No 5:

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

#### Page No 5:

36 − (−64) = 36 + 64 = 100

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

Here, 100 $\ne$ −100

Thus, they are not equal.

#### Page No 5:

(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:

Here, (a − b) = −9 − (−6) = −3

Similarly, (b − a) = −6 − (−9) = 3

∴ (a−b) ≠ (b−a)

#### Page No 5:

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

53 + a = −16
a = −16 − 53 = −69

∴ The other integer is −69.

#### Page No 5:

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

∴ The other integer is 96.

#### Page No 5:

We have:

a − (−6) = 4
a = 4 + (−6) = −2

a = −2

#### Page No 5:

(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:

(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:

(i) 16 $×$ 9 = 144
(ii) 18 $×$ (−6) = −108
(iii) 36 $×$ (−11) = −396
(iv)  (−28) $×$14 = −392
(v) (−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) = −275

#### Page No 9:

(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:

(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.
$\left(-3{\right)}^{6}$ = 729
(vi) Since the number of negative integers in the product is odd, the product will be negative.
$\left(-5{\right)}^{5}$ = −3125
(vii) Since the number of negative integers in the product is even, the product will be positive.
$\left(-1{\right)}^{200}$= 1
(viii) Since the number of negative integers in the product is odd, the product will be negative.
$\left(-1{\right)}^{171}$ = −1

#### Page No 9:

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:

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:

(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:

(i) (−6) × (x) = 6

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:

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:

(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 $\frac{1}{a}$, which is not an integer.

#### Page No 12:

(i) 65 $÷$ (−13) = −5

(ii) (−84) $÷$ 12 =  −7

(iii) (−76) $÷$ 19 = −4

(iv) (−132) $÷$ 12 = −11

(v) (−150) $÷$ 25 = −6

(vi) (−72) $÷$ (−18) =

(vii)  (−105) $÷$ (−21) = 5

(viii) (−36) $÷$ (−1) = 36

(ix) 0 $÷$ (−31) =  0

(x)  (−63) $÷$ 63 = −1

(xi)  (−23) $÷$ (−23) = 1

(xii) (−8) $÷$ 1 =  −8

(i)
72 ÷ (x) = −4

(ii)
−36 ÷ (x) = −4

(iii)
(x) ÷ (−4) = 24

(iv)
(x) ÷ 25 = 0

(v)
(x) ÷ (−1) = 36

(vi)
(x) ÷ 1 = −37

(vii)
39 ÷ (x) = −1

(viii)
1 ÷ (x) = −1

(ix)
−1 ÷ (x) = −1

#### Page No 12:

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

(iii) False (F).

(iv)  True (T).

(v)  False (F).

(vi) True (T).

(c) 14
Given:
6 − (−8)
= 6 + 8
= 14

(b) −3
Given:
−9 − (−6)
= −9 + 6
= −3

#### Page No 13:

(d) 5
We can see that

−3 + 5 = 2

Hence, 2 exceeds −3 by 5.

#### Page No 13:

(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:

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

Hence, −6 is four (4) less than −2.

#### Page No 13:

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

#### Page No 13:

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

#### Page No 13:

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

#### Page No 13:

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

#### Page No 13:

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

#### Page No 13:

(a) 4
(−36) ÷ (−9) = 4

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

#### Page No 13:

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

#### Page No 13:

(c) not defined

Dividing any integer by zero is not defined.

#### Page No 13:

(b) −11 < −8

Negative integers decrease with increasing magnitudes.

#### Page No 13:

(b) 9

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

#### Page No 13:

(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:

(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:

(c) 6

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

#### Page No 14:

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

#### Page No 14:

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

#### Page No 14:

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

#### Page No 14:

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

#### Page No 14:

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

(b) 90

#### Page No 15:

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:

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

Hence, the value of p is −5.

#### Page No 15:

Product of (−16) and (−9) = = 144
Now, gives the quotient −22.

∴ 144 + (−22) = 122

#### Page No 15:

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:

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

$a÷\left(-7\right)=12$
a = $-\frac{7}{12}$

Hence, $-\frac{7}{12}$ should be divided by −7 to obtain 12.

(i) −450
(ii)  360
(iii) −1080
(iv)  −600
(v)

(vi)

#### Page No 15:

(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:

(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:

(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:

(b)108

(108) ÷ (−18) = −6

#### Page No 15:

(a) 370
We have:

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

#### Page No 15:

(c) −250

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

(b) −3

(−9) − (−6)
= (−9) + 6
= −3

#### Page No 15:

(b) −6

−8 − (−6) = 2

Hence, −8 is −6 less than −2.

#### Page No 15:

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