Rd Sharma 2018 Solutions for Class 6 Math Chapter 5 Negative Numbers And Integers are provided here with simple step-by-step explanations. These solutions for Negative Numbers And Integers are extremely popular among Class 6 students for Math Negative Numbers And Integers Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rd Sharma 2018 Book of Class 6 Math Chapter 5 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rd Sharma 2018 Solutions. All Rd Sharma 2018 Solutions for class Class 6 Math are prepared by experts and are 100% accurate.

#### Question 2:

Find the sum of:

(i) −557 and 488

(ii) −522 and −160

(iii) 2567 and −325

(iv) −10025 and 139

(v) 2547 and −2548

(vi) 2884 and −2884

#### Question 1:

Find the additive inverse of each of the following integers:

(i) 52
(ii) −176
(iii) 0
(iv) 1

#### The additive inverse of the number a  is the number added to a,  yields zero.  So, we find the following:

(i) 52 + (−52) = 0
Here, −52 is the additive inverse of 52.
(ii) (−176) + 176 = 0
Here, 176 is the additive inverse of −176.
(iii) 0 + 0 = 0
Here, 0 itself is its inverse.
(iv) 1 + (−1) = 0
Here, −1 is the additive inverse of 1.

#### Question 2:

Find the successor of each of the following integers:

(i) −42
(ii) −1
(iii) 0
(iv) −200
(v) −99

For the integer a, + 1 will be its successor.
(i) −41 is the successor of −42.
(ii) 0 is the successor of −1.
(iii) 1 is the successor of 0.
(iv) −199 is the successor of 200.
(v) −98 is the successor of −99.

#### Question 3:

Find the predecessor of each of the following integers:

(i) 0
(ii) 1
(iii) −1
(iv) −125
(v) 1000

For the integer a, the predecessor is (− 1).
(i)  −1 is the predecessor of 0.
(ii) 0 is the predecessor of  1.
(iii) −2 is the predecessor of −1.
(iv) −126 is the predecessor of −125.
(v) 999 is the predecessor of 1000.

#### Question 4:

Which of the following statements are true?

(i) The sum of a number and its opposite zero.
(ii) The sum of two negative integer is positive integer.
(iii) The sum of a negative integer and a positive integer is always a negative integer.
(iv) The successor of −1 is 1.
(v) The sum of three different integers can never be zero.

(i) True − It is the definition of additive inverse; for example, 5 + (−5) = 0.
(ii) False − For example, −2 − 3 = −5; it is a negative integer.
(iii) False − −3 +  5 = 2; it is a positive integer.
(iv) False − 0 is the successor of −1.
(v) False − It can be zero like (−2) + (−1) + (3).

#### Question 5:

Write all integers whose absolute values are less than 5.

Let be an integer such that |x| < 5.
∴ -5 < x < 5
=> x = ] -4, 4 [
These are the nine integers whose absolute values are less than 5, namely, -4, -3, -2, -1, 0, 1, 2, 3 and 4.

#### Question 6:

Which of the following is false:

(i) $\left|4+2\right|=\left|4\right|+\left|2\right|$
(ii) $\left|2-4\right|=\left|2\right|+\left|4\right|$
(iii) $\left|4-2\right|=\left|4\right|-\left|2\right|$
(iv) $\left|\left(-2\right)+\left(-4\right)\right|=\left|-2\right|+\left|-4\right|$

(i) |4|+ |2| = 6 = |4| + |2|; True
(ii) |2 − 4| = |−2| = 2 ≠ 2 + 4 = 6; False
(iii) |4 − 2| = 2 = |4| − |2|; True
(iv) |(−2) + (−4)| = |−6| = 6 = |−2| + |(−4)|; True

#### Question 7:

Complete the following table:

 + −6 −4 −2 0 2 4 6 6 10 4 2 8 0 −6 −2 −4 0 −6 −6

From the above table

(i) Write all the pairs of integers whose sum is 0.
(ii) is (−4) + (−2) = (−2) + (−4)?
(iii) is 0 + (−6) = −6?

(i) (6, −6), (4, −4), (2, −2), (0, 0), (−2, 2), (−4, 4) and (−6,6).
(ii) (−4) + (−2) = −6 = (−2) + (−4); Yes
(iii) 0 + (−6) = −6; Yes

 + −6 −4 −2 0 2 4 6 6 0 2 4 6 8 10 12 4 −2 0 2 4 6 8 10 2 −4 −2 0 2 4 6 8 0 −6 −4 −2 0 2 4 6 −2 −8 −6 −4 −2 0 2 4 −4 −10 −8 −6 −4 −2 0 2 −6 −12 −10 −8 −6 −4 −2 0

#### Question 8:

Find an integer x such that

(i) x + 1 = 0
(ii) x + 5 = 0
(iii) −3 + x = 0
(iv) x + (−8) = 0
(v) 7 + x = 0
(vi) x + 0 = 0

(i) x +  1 = 0 ⇒ = −1
(ii) x + 5 = 0 ⇒ x = −5
(iii) −3 + x = 0 ⇒ x = 3
(iv) x + (−8) = 0 ⇒ x = 8
(v) 7 + x = 0 ⇒ x = −7
(vi) x + 0 = 0  ⇒ x = 0

#### Question 1:

Subtract the first integer from the second in each of the following:

(i) 12, −5
(ii) −12, 8
(iii) −225, −135
(iv) 1001, 101
(v) −812, 3126
(vi) 7560, −8
(vii) −3978, −4109
(viii) 0, −1005

(i) (−5) − 12 = −17
(ii) 8 − (−12) = 8 + 12 = 20
(iii) −135 − (−225) =  −135 + 225 = 90
(iv) 101 − 1001 = −900
(v) 3126 − (−812) = 3126 + 812 = 3938
(vi) −8 − 7560 = −7568
(vii) −4109 − (−3978) =  −4109 + 3978 = −131
(viii) −1005 − 0 = −1005

#### Question 2:

Find the value of:

(i) −27 − (−23)
(ii) −17 − 18 − (−35)
(iii) −12 − (−5) − (−125) + 270
(iv) 373 + (−245) + (−373) + 145 + 3000
(v) 1 + (−475) + (−475) + (−475) + (−475) + 1900
(vi) (−1) + (−304) + 304 + 304 + (−304) + 1

(i) −27 − (−23)
= −27 + 23
= −4

(ii) −17 −18 − (−35)
= −17 − 18 + 35
= −35 + 35
= 0

(iii) −12 − (−5) − (−125) + 270
= −12 + (5 + 125 + 270)
= −12 + 400
= 388

(iv) 373 + (−245) + (−373) + 145 + 3000
= 373 − 245 − 373 + (145 + 3000)
= 128 − 373 + 3145
= −245 + 3145
= 2900

(v) 1 + (−475) + (−475) + (−475) + (−475) + 1900
= 1 (−475 − 475 − 475 − 475 ) + 1900
= 1 − 1900 + 1900
= 1

(vi) (−1) + (−304) + 304 + 304 + (−304) + 1
= −1 + (−304 + 304) + ( 304 − 304) + 1
= −1 + 0 + 0 + 1
= 0

#### Question 3:

Subtract the sum of −5020 and 2320 from −709.

We have to subtract the sum of −5020 and 2320 from −709.
Sum:
−5020 + 2320 = −2700
Now,
−709 − (−2700) = −709 + 2700 = 1991

#### Question 4:

Subtract the sum of −1250 and 1138 from the sum of 1136 and −1272.

Sum of −1250 and 1138 = (−1250) + 1138 = −112
Sum of 1136 and −1272 = 1136 + (−1272) = −136
Now,
−136 − (−112) = −136 + 112 = −24

#### Question 5:

From the sum of 233 and −147, subtract −284.

We have to subtract −284 from the sum of 233 and −147.
Sum of 233 and (−147) = 233 + (−147) = 233 − 147 = 86
Now, we will subtract −284 from 86.
86 − (−284) = 86 + 284 = 370

#### Question 6:

The sum of two integers is 238. If one of the integers is −122, determine the other.

Let x and y be two integers such that x + y =  238.
Given: x = −122
Now,
x + y = 238
⇒ −122 + y = 238
⇒  y = 238 + 122 = 360
So, the other integer is 360.

#### Question 7:

The sum of two integers is −233. If one of the integers is 172, find the other.

Let and be two integers such that x + y = −223.
Given: x = 172
Now,
x + y = −223
⇒ 172 + y = −223
⇒ y = −223 − 172
⇒ y = −395

#### Question 8:

Evaluate the following:

(i) −8 − 24 + 31 − 26 − 28 + 7 + 19 − 18 − 8 + 33
(ii) −26 −20 + 33 − (−33) + 21 + 24 − (−25) −26 − 14 − 34

(i) −8 − 24 + 31 − 26 − 28 + 7 + 19 − 18 − 8 + 33
= (−8 − 24) + (31 − 26) + (−28 + 7) + (19 − 18) + (−8 + 33)
= (−32 + 5 − 21) + (1 + 25)
= −48 + 26
= −22

(ii) −26 − 20 + 33 − (−33) + 21 + 24 − (−25) − 26 − 14 − 34
= (−26 − 20) + (33 + 33) + (21 + 24) + (25 − 26) + (−14 − 34)
= (−46 + 66) + (45 − 1 − 48)
= 20 − 4
= 16

#### Question 9:

Calculate:

1 − 2 + 3 − 4 + 5 − 6 +......+ 15 − 16

1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + 9 − 10 + 11 − 12 + 13  − 14 + 15 − 16
= (1 + 3 + 5 + 7 + 9 + 11 + 13 + 15) − (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16)
= 64 − 72
= −8

#### Question 10:

Calculate the sum:

5 + (−5) + 5 (−5) + .....

(i) if the number of terms is 10.
(ii) if the number of terms is 11.

(i) If the number of terms is 10, then 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) = 0.
(ii) If the number of terms is 11, then 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 + (−5) + 5 = 5.

#### Question 11:

Replace * by '<' or '>' in each of the following to make the statement true:

(i) (−6) + (−9) * (−6) − (−9)
(ii) (−12) − (−12) * (−12) + (−12)
(iii) (−20) − (−20) * 20 − (65)
(iv) 28 − (−10) * (−16) − (−76)

(i) (−6) + (−9) = −15 < (−6) − (−9) = −6 + 9 = 3
(ii) (−12) − (−12) = −12 + 12 = 0 > (−12) + (−12) = −12 − 12 = −24
(iii) (−20) − (−20) = −20 + 20 = 0 > 20 − 65 = −45
(iv) 28 − (−10) = 28 + 10 = 38 < −16 − (−76) = −16 + 76 = 60

#### Question 12:

If ∆ is an operation on integers such that a ∆ b = −a + b − (−2) for all integers a, b. Find the value of

(i) 4 ∆ 3
(ii) (−2) ∆ (−3)
(iii) 6 ∆ (−5)
(iv) (−5) ∆ 6

(i) −4 + 3 − (−2)
= −4 + (3 + 2)
= −4 + 5
= 1

(ii) −(−2) + (−3) − (−2)
= (2 − 3) + 2
= −1 + 2
= 1

(iii) −6 + (−5) − (−2)
= −6 + (−5 + 2)
= −6 − 3
= −9

(iv) −(−5) + 6 − (−2)
= 5 + (6 + 2)
= 5 + 8
= 13

#### Question 13:

If a and b are two integers such that a is predecessor of b. Find the value of a − b.

a and are integers such that is the predecessor of b, that is, = b − 1.
∴ (ab)
= (b − 1) − b
b − 1 − b
= −1

#### Question 14:

If a and b are two integers such that a  is the successor of b. Find the value of a − b.

and are two integers such that is the successor of b, that is, a = b + 1.
∴​
= b + 1 − b
1

#### Question 15:

Which of the following statements are true:

(i) −13 > −8 − (−2)
(ii) −4 + (−2) < 2.
(iii) The negative of a negative integer is positive.
(iv) If a and b are two integers such that  a > b, then ab is always positive integer,
(v) The difference of two integers is an integer.
(vi) Additive inverse of a negative integer is negative.
(vii) Additive inverse of a positive integer is negative.
(viii)  Additive inverse of a negative is positive.

(i) False; It should be −13 < −8 + 2 = −6.
(ii) True; −4 − 2 = −6 < 2
(iii) True; For example: −(−2) = 2
(iv) True; a > b
(v) True; For example: 3 − 2 = 1, which is a integer.
(vi) False; For example: −2 + 2 = 0. Here, 2 is the additive inverse of −2; it is positive.
(vii) True
(viii) True

#### Question 16:

Fill in the blanks:

(i) −7 + ...... = 0
(ii) 29 + ...... = 0
(iii) 132 + (−132) = ......
(iv) −14 + ....= 22
(v) −1256 + ..... = −742
(vi) ..... −1234 = −4539.

(i) −7 + 7 = 0  (−a and a are the negative and additive inverses of each other.)
(ii) 29 + (−29) = 0  (−a and a are the negative and additive inverses of each other.)
(iii) 132 + (−132) = 0  (−a and a are the negative and additive inverses of each other.)
(iv) −14 + 36 = 22
(v) −1256 + 514 = −742
(vi) −3305 − 1234 = −4539

#### Question 1:

Which of the following statement is true?
(a) − 7 > − 5              (b) − 7 < − 5                 (c) (− 7) + (− 5) > 0                       (d) (− 7) − (− 5) > 0

(a)
On the number line, − 7 is to the left of − 5, so − 7 < − 5.
(c)
Here, (− 7) + (− 5) = − ( 7 + 5) = − 12.
On the number line, − 12 is to the left of 0, so (− 7) + (− 5) < 0.
(d)
Here, (− 7) − (− 5) = (− 7) + (additive inverse of − 5) = (− 7) + (5) = − (7 − 5) = − 2
On the number line, − 2 is to the left of 0, so (− 7) − (− 5) < 0.
Hence, the correct option is (b).

#### Question 2:

5 less than − 2 is

(a) 3                            (b) − 3                        (c) − 7                                 (d) 7

Here, 5 less than − 2 = (− 2) − (5) = − 2 − 5 = − 7.
Hence, the correct option is (c).

#### Question 3:

6 more than − 7 is

(a) 1                            (b) − 1                        (c) 13                                 (d) − 13

6 more than − 7 = (− 7) + 6
= − ( 7 − 6)
= − 1
Hence, the correct option is (b).

#### Question 4:

If x is a positive integer, then

(a) x + |x| = 0              (b) x − |x| = 0                    (c) x + |x| = −2x                (d) x = − |x|

If x is positive integer, then |x| = x. So, x + |x| = x + x = 2x and x − |x| = xx = 0.
Hence, the correct option is (b).

#### Question 5:

If x is a negative integer, then

(a) x + |x| = 0              (b) x − |x| = 0                    (c) x + |x| = 2x                (d) x − |x| = − 2x

If x is negative integer, then |x| = −x. So
x + |x| = xx = 0
x − |x| = x − (− x) = x + x = 2x
Hence, the correct option is (a).

#### Question 6:

If x is greater than 2, then |2 − x| =

(a) 2 − x              (b) x − 2                    (c) 2 + x                (d) − x − 2

If a is negative integer, then |a| = − a.
Here, x is greater than 2. So, 2 − x is negative.
Therefore, |2 − x| = − (2 − x) = − 2 + x = x − 2.
Hence, the correct option is (b).

#### Question 7:

9 + |− 4| is equal to

(a) 5                         (b) − 5                       (c) 13                            (d) −13

Here, |− 4| = 4. Therefore
9 + |− 4| = 9 + 4 = 13
Hence, the correct option is (c).

#### Question 8:

(− 35) + (− 32) is equal to

(a) 67                         (b) − 67                       (c) − 3                            (d) 3

(− 35) + (− 32) = − (35 + 32) = − 67
Hence, the correct option is (b).

#### Question 9:

(− 29) + 5 is equal to

(a) 24                         (b) 34                       (c) − 34                            (d) − 24

(− 29) + 5 = − (29 − 5) = − 24
Thus, the correct option is (d).

#### Question 10:

|− |− 7| − 3| is equal to

(a) − 7                       (b) 7                              (c) 10                                    (d) − 10

|− |− 7| − 3| = |− 7 − 3|                                 (∵ |− 7| = 7)
= |− 10|
= 10                                          (∵ |− 10| = 10)
Hence, the correct option is (c).

#### Question 11:

The successor of − 22 is

(a) − 23                       (b) − 21                              (c) 23                                    (d) 21

If a is an integer, then its successor is a + 1. So
Successor of − 22 = − 22 + 1 = − (22 − 1) = − 21
Hence, the correct option is (b).

Q

ans

#### Question 13:

If the sum of two integers is − 26 and one of them is 14, then the other integer is

(a) − 12                       (b) 12                              (c) − 40                               (d) 40

Sum of two integers = − 26
One of the two numbers = 14
∴ second number = − 26 − 14
= − (26 + 14)
= − 40
Hence, the correct option is (c).

#### Question 14:

Which of the following pairs of integers have 5 as a difference?

(a) 10, 5                       (b) − 10, − 5                          (c) 15, − 20                               (d) both (a) and (b)

(a) 10 − 5 = 5
(b) (− 5) − (− 10) = − 5 + 10 = 5
(c) 15 − (− 20) = 15 + 20 = 35
Hence, the correct option is (d).

#### Question 15:

If the product of two integers is 72 and one of them is − 9, then the other integers is

(a) − 8                       (b) 8                          (c) 81                               (d) 63

Product of integers = 72
One of the two integers = − 9
So, the other integer is 72 $\text{÷}$ (− 9) = − 8.
Hence, the option is (a).

#### Question 16:

On subtracting − 7 from − 14, we get

(a) − 12                       (b) − 7                          (c) −14                               (d) 21

Required number = − 14 − (− 7)
= − 14 + 7
= − (14 − 7)
= − 7
Hence, the correct option is (b).

#### Question 17:

The largest number that divides 64 and 72 and leave the remainders 12 and 7 respectively, is

(a) 17                             (b) 13                                       (c) 14                                      (d) 18

Subtract 12 and 7 from 64 and 72 respectively.
64 − 12 = 52
72 − 7 = 65
The required number is the HCF of 52 and 65.
Now
52 = 4 $\text{×}$ 13
65 = 5 $\text{×}$ 13
∴ HCF 52 and 65 = 13
Therefore, the largest number that divides 64 and 72 and leave the remainder 12 and 7 respectively, is 13.
Hence, the correct option is (b).

#### Question 18:

The sum of two integers is − 23. If one of them is 18, then the other is

(a) −14                             (b) 14                                    (c) 41                                    (d) −41

Sum of integers = − 23
One of the numbers = 18
Other number = (− 23) − (18)
= − 23 − 18
= − (23 + 18)
= − 41
Therefore, the other number is − 41.
Hence, the correct option is (d).

#### Question 19:

The sum of two integers is − 35. If one of them is 40, then the other is

(a) 5                             (b) − 75                                    (c) 75                                    (d) − 5

Sum of integers = − 35
One of the numbers = 40
Other number = (− 35) − (40)
= − 35 − 40
= − (35 + 40)
= − 75
Therefore, the other number is − 75.
Hence, the correct option is (b).

#### Question 20:

On subtracting − 5 from 0, we get

(a) − 5                             (b) 5                                    (c) 50                                    (d) 0

Here, 0 − (− 5) = 0 + 5 = 5.
Therefore, on subtracting − 5 from 0, we get 5.
Hence, the correct option is (b).

#### Question 21:

(− 16) + 14 − (− 13) is equal to

(a) − 11                             (b) 12                                    (c) 11                                    (d) − 15

(− 16) + 14 − (− 13) = (− 16) + 14 + 13           (Additive inverse of − 13 is 13)
= (− 16) + 27
= 27 − 16
= 11
Hence, the correct option is (c).

#### Question 22:

(− 2) $\text{×}$ (− 3) $\text{×}$$\text{×}$ (− 1) is equal to

(a) 36                             (b) − 36                                (c) 6                                    (d) − 6

(− 2) $\text{×}$ (− 3) $\text{×}$$\text{×}$ (− 1) = (2 $\text{×}$ 3) $\text{×}$$\text{×}$ (− 1)
= 6 $\text{×}$$\text{×}$ (− 1)
= 36 $\text{×}$ (− 1)
= − (36 $\text{×}$ 1)
= − 36
Hence, the correct option is (b).

#### Question 23:

86 + (−28) + 12 + (−34) is equal to

(a) − 36                             (b) 40                                (c) 36                                    (d) − 40

86 + (−28) + 12 + (−34) = 86 + (−28) − (34 − 12)
= 86 + (−28) − 22
= (86 − 28) − (34 − 12)
= 58 − 22
= 36
Hence, the correct option is (c).

#### Question 24:

(−12) $\text{×}$ (−9) − 6 $\text{×}$ (−8) is equal to

(a) 156                             (b) 60                                (c) −156                                    (d) − 60

(−12) $\text{×}$ (−9) − 6 $\text{×}$ (−8) = (12 $\text{×}$ 9) − 6 $\text{×}$ (−8)
= 108 − 6 $\text{×}$ (−8)
= 108 + 6 $\text{×}$ 8
= 108 + 48
= 156
Hence, the correct option is (a).

#### Question 1:

The successor of − 79 is

(a) − 80                            (b) − 78                                 (c) 80                                   (d) 78

If a is an integer, then its successor is a + 1. So
Successor of − 79 = − 79 + 1
= − (79 − 1)
= − 78
Hence, the correct option is (b).

#### Question 2:

The predecessor of − 99 is

(a) − 98                            (b) − 100                                 (c) 98                                   (d) 100

If a is an integer, then its predecessor is a − 1. So
Predecessor of − 99 = − 99 − 1
= − (99 + 1)
= − 100
Hence, the correct option is (b).

#### Question 3:

The integer 8 more than − 12 is

(a) 4                            (b) − 4                                 (c) − 20                                   (d) 20

The integer 8 more than − 12 is (− 12) + 8.
Now
(− 12) + 8 = − (12 − 8) = − 4
Hence, the correct option is (b).

#### Question 4:

What should be added to 18 to get − 34?

(a) 52                            (b) − 52                                 (c) − 16                                   (d) 16

The required number is (− 34) − 18.
Now
(− 34) − 18 = − (34 + 18) = − 52
Hence, the correct option is (b).

#### Question 5:

The additive inverse of 17 is

(a) − 17                            (b) 17                                 (c) $\frac{1}{17}$                                   (d) $-\frac{1}{17}$

If a is an integer, then its additive inverse is − a.
So
Additive inverse of 17 = − 17
Hence, the correct option is (a).

#### Question 6:

If an integer a is greater than 7, then |7 − a| =
(a) 7 − a                            (b) a − 7                                 (c) 7 + a                                   (d) − 7 − a

If x is negative integer, then |x| = − x.
Here, a is greater than 7, so 7 − a is negative.
Therefore
|7 − a| = − (7 − a) = − 7 + a = a − 7
Hence, the correct option is (b).

#### Question 7:

The additive identity element in the set of integers is

(a) 1                            (b) − 1                                 (c) 0                                   (d) None of these

If a is an integer, then a + 0 = 0 + a = a. Here, 0 is the additive identity element in the set of integers.
Hence, the correct option is (c).

#### Question 8:

Which of the following pairs of integers have 9 as difference?

(a) 19, 10                        (b) − 19, − 10                                 (c) 19, − 10                                (d) (a) and (b) both

(a) 19 − 10 = 9
(b) − 10 − (− 19) = − 10 + 19 = 19 − 10 = 9
(c) 19 − (− 10) = 19 + 10 = 29
Thus, (a) and (b) both are correct.
Hence, the correct option is (d).

#### Question 9:

When 47 is subtracted from − 23, we get?

(a) 70                        (b) 24                             (c) − 24                                (d) − 70

(− 23) − (47) = − 23 − 47
= − (23 + 47)
= − 70
Thus, when 47 is subtracted from − 23, we get − 70.
Hence, the correct option is (d).

#### Question 10:

If ∆ is an operation on integers such that a b = ab − 2, for all integers a, b. Then, 7 ∆ (− 4) =

(a) 11                        (b) − 9                             (c) 9                                (d) 1

Here, the operation ∆ is defined as a b = ab − 2. So
7 ∆ (− 4) = 7 − (− 4) − 2
= 7 + 4 − 2
= 11 − 2
= 9
Hence, the correct option is (c).

#### Question 11:

Simplify: (− 145) + 97 + (− 365) + (− 71) + 8.

(− 145) + 97 + (− 365) + (− 71) + 8 = (− 145) + 97 + (− 365) − (71 − 8)
= (− 145) + 97 + (− 365) − 63
= (− 145) + 97 − (365 + 63)
= (− 145) + 97 − 428
= (− 145) − (428 − 97)
= (− 145) − 331
= − (145 + 331)
= − 476
Hence, (− 145) + 97 + (− 365) + (− 71) + 8 = − 476.

#### Question 12:

The sum of two integers is 84. If one of the integers is 44, determine the other.

Sum of two integers = 84
One of the integers = 44
Other integer = Sum of two integers − One of the integers
= 84 − 44
= 40
Hence, the other integer is 40.

#### Question 13:

Simplify: 9 $\text{×}$ (− 16) + (− 17) $\text{×}$ (− 16).

9 $\text{×}$ (− 16) + (− 17) $\text{×}$ (− 16) = − (9 $\text{×}$ 16) + (17 $\text{×}$ 16)
= − 144 + 272
= 272 − 144
= 128
Hence, 9 $\text{×}$ (− 16) + (− 17) $\text{×}$ (− 16) = 128.

#### Question 14:

If x = (− 23) + 22 + (− 23) + 22 + ... + (40 terms) and y = 11 + (− 10) + 11 + (− 10) + ... + (20 terms), then find yx.

x = (− 23) + 22 + (− 23) + 22 + ... + (40 terms)
= (− 23) + (− 23) + ... + (20 terms) + 22 + 22 + ... + (20 terms)
= 20 $\text{×}$ (− 23) + 20 $\text{×}$ 22
= 20 $\text{×}$ (− 23 + 22)
= 20 $\text{×}$ (− 1)
= − 20
Now
y = 11 + (− 10) + 11 + (− 10) + ... + (20 terms)
= 11 + 11 + ... + (10 terms) + (− 10) + (− 10) + ... + (10 terms)
= 11 $\text{×}$ 10 + (− 10) $\text{×}$ 10
= 110 − 100
= 10
Therefore, yx = 10 − (− 20) = 10 + 20 = 30.

#### Question 15:

Calculate 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + ... + 49 − 50.

1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + ... + 49 − 50
= (1 − 2) + (3 − 4) + (5 − 6) + (7 − 8) + ... + (49 − 50)
= (− 1) + (− 1) + (− 1) + ... + 25 terms
= (− 1) $\text{×}$ 25
= − 25
Hence, 1 − 2 + 3 − 4 + 5 − 6 + 7 − 8 + ... + 49 − 50 = − 25.

#### Question 16:

Evaluate 7 $\text{×}$ |− 15| − |− 9| $\text{×}$ 8.

∵ |− 15| = 15 and |− 9| = 9
∴ 7 $\text{×}$ |− 15| − |− 9| $\text{×}$ 8 = 7 $\text{×}$ 15 − 9 $\text{×}$ 8
= 105 − 72
= 33
Hence, 7 $\text{×}$ |− 15| − |− 9| $\text{×}$ 8 = 33.

#### Question 17:

Find the value of 38 − (− 25) − 58 + (− 15) + 23 − (− 8).

38 − (− 25) − 58 + (− 15) + 23 − (− 8)
= 38 − (− 25) − 58 + (− 15) + 23 +8
= 38 − (− 25) − 58 + (− 15) + 31
= 38 − (− 25) − 58 + (31 − 15)
= 38 − (− 25) − 58 + 16
= 38 − (− 25) − (58 − 16)
= 38 − (− 25) − 42
= 38 + 25 − 42
= 38 − (42 − 25)
= 38 − 17
= 21
Hence, 38 − (− 25) − 58 + (− 15) + 23 − (− 8) = 21.

#### Question 18:

Simplify: 5 + (− 5) + 5 + (− 5) + ...
(i) When the number of terms is 20              (ii) When the number of terms is 25

(i) When the number of terms is 20
5 + (− 5) + 5 + (− 5) + ... = (5 + 5 + 5 + ... + 10 terms) + (− 5) + 5 + (− 5) + ... + 10 terms
= 10 $\text{×}$ 5 + 10 $\text{×}$ (− 5)
= 50 − 50
= 0

(ii) When the number of terms is 25
5 + (− 5) + 5 + (− 5) + ... = (5 + 5 + 5 + ... + 12 terms) + (− 5) + 5 + (− 5) + ... + 13 terms
= 12 $\text{×}$ 5 + 13 $\text{×}$ (− 5)
= 60 − 65
= − (65 − 60)
= − 5

#### Question 19:

If ∆ is an operation on integers such that for integers a and b, a b = ab − (− 5). Find the values of

(i) (− 7) ∆ 3                   (ii) (− 9) ∆ (− 4)                      (iii) 2 ∆ 5                          (iv) 4 ∆ (− 5)

(i)
a b = ab − (− 5)
∴ (− 7) ∆ 3 = (− 7) − 3 − (− 5)
= − 7 − 3 + 5
= − (7 + 3) + 5
= − 10 + 5
= − (10 − 5)
= − 5
(ii)
(− 9) ∆ (− 4) = (− 9) − (− 4) − (− 5)
= − 9 + 4 + 5
= − 9  + 9
= 0

(iii)
2 ∆ 5 = 2 − 5 − (− 5)
= 2 − 5 + 5
= 2
(iv)
4 ∆ (− 5) = 4 − (− 5) − (− 5)
= 4 + 5 + 5
= 14

#### Question 20:

Evaluate: − 36 − 40 + 43 − (− 29) + 18 − (− 74).

− 36 − 40 + 43 − (− 29) + 18 − (− 74)
= − 36 − 40 + 43 − (− 29) + 18 + 74
= − 36 − 40 + 43 − (− 29) + 92
= − 36 − 40 + 43 + 29 + 92
= − 36 − 40 + 43 + 121
= − 36 − 40 + 164
= − (36 + 40) + 164
= − 76 + 164
= 164 − 76
= 88

#### Question 21:

The largest negative integer is ...............

The numbers ... − 5, − 4, − 3, − 2, − 1, 0, 1, 2, 3, 4, 5, ... form the set of integers.
Thus, the largest negative integer is − 1.

#### Question 22:

The smallest positive integer is .................

The collection ... − 4, − 3, − 2, − 1, 0, 1, 2, 3, 4, ... forms the set of integers.
Thus, the smallest positive integer is 1.

#### Question 23:

(− 22) + 21 + (− 22) + 21 + ... + 20 terms is equal to .............

∵ (− 22) + 21 = − (22 − 21) = − 1
∴  (− 22) + 21 + (− 22) + 21 + ... + 20 terms = (− 1) + (− 1) + (− 1) + ... + 10 terms
= 10 $\text{×}$ (− 1)
= − 10

#### Question 24:

(− 3) (− 4) (12) (− 1) = ............

(− 3) (− 4) (12) (− 1) = (− 3) (− 4) (− 12)                       [ ∵ (12) (− 1) = (− 12)]
= (− 3) (48)                                    [ ∵ (− 4) (− 12) = 48]
= − (3 $\text{×}$ 48)
= − 144
Hence, (− 3) (− 4) (12) (− 1) = − 144.

#### Question 25:

(− 1) (− 1) (− 1) (− 1) (− 1) = ...............

Here, − 1 is multiplied 5 times and since, 5 is an odd integer, therefore
(− 1) (− 1) (− 1) (− 1) (− 1) = − 1

#### Question 1:

Write the opposite of each of the following:

(i) Increase in population
(ii) Depositing money in a bank
(iii) Earning money
(iv) Going North
(v) Gaining a weight of 4 kg
(vi) A loss of Rs 1000
(vii) 25
(viii) −15

(i) Decrease in population
(ii)Withdrawing money from a bank
(iii) Spending money
(iv) Going South
(v) Losing weight of 4 kg
(vi) A gain of Rs 1,000
(vii) −25
(viii) 15

#### Question 2:

Indicate the following by using intergers:

(i) ${25}^{0}$ above zero
(ii) ${5}^{0}$ below zero
(iii) A profit of Rs 800

(iv) A deposit of Rs 2500

(v) 3 km above sea level

(vi) 2 km below sea level

(i) If temperature is above zero, then it should be positive, i.e., +25°.
(ii) If temperature is below zero, then it should be negative, i.e., 5°.
(iii) If there is a profit of Rs 800, then it should be +800.
(iv) Deposition of Rs 2,500 indicates that money in the account is increased; therefore, it should be +2500 .
(v) Here, the distance is above the sea level; therefore, it  should be +3.
(vi) Here, the distance is below the sea level; therefore, it should be −2.

#### Question 3:

Mark the following integers on a number line:

(i) 7
(ii) −4
(iii) 0

#### Question 4:

Which number in each of the following pairs is smaller?

(i) 0, −4
(ii) −3, 13
(iii) 8, 13
(iv) −15, −27

(i) 0 is greater than every negative integer, so −4 < 0.
(ii) Every positive integer is greater than every negative integer; therefore, −3 < 12.
(iii) Because 8 is to the left of 13 on a number line, 8 < 13.
(iv) Because −27 is to the left of −15 on a number line, −27 < −15.

#### Question 5:

Which number in each of the following pairs is larger?

(i) 3, −4
(ii) −12, −8
(iii) 0, 7
(iv) 12, −18

(i) Every positive integer is greater than every negative integer; therefore, 3 > −4.
(ii) Because −12 is to the left of −8 on a number line, −8 > −12.
(iii) Every positive integer is greater than zero; therefore, 7 > 0.
(iv) Every positive integer is greater than every negative integer; therefore, 12 > −18.

#### Question 6:

Write all integers between:

(i) −7 and 3
(ii) −2 and 2
(iii) −4 and 0
(iv) 0 and 3

(i) There are nine integers in between −7 and 3, namely, −6, −5, −4, −3, −2, −1, 0, 1 and 2.
(ii) There are three integers in between −2 and 2, namely, −1, 0 and 1.
(iii) There are three integers in between −4 and 0, namely, −3, −2 and −1.
(iv) There are two integers in between 0 and 3, namely, 1 and 2.

#### Question 7:

How many integers are between?

(i) −4 and 3
(ii) 5 and 12
(iii) −9 and −2
(iv) 0 and 5

(i) There are six integers in between −4 and 3, namely, −3, −2, −1, 0, 1 and 2.
(ii) There are six integers in between 5 and 12, namely, 6, 7, 8, 9, 10 and 11.
(iii) There are six integers in between −9 and −2, namely, −8, −7, −6, −5, −4 and −3.
(iv) There are four integers in between 0 and 5, namely, 1, 2, 3 and 4.

#### Question 8:

Replace * in each of the following by < or > so that the statement is true:

(i) 2 * 5
(ii) 0 * 3
(iii) 0 * −7
(iv) −18 * 15
(v) −235 * −532
(vi) −20 * 20

In the given pairs of numbers, the numbers that are to the left of the other numbers on a number line are smaller.

(i) 2 < 5
(ii) 0 < 3
(iii) 0 > −7
(iv) −18 < 15
(v) −235 > −532
(vi) −20 < 20

#### Question 9:

Write the following integers in increasing order:

(i) −8, 5, 0, −12, 1, −9, 15
(ii) −106, 107, −320, −7, 185

(i)  −12 < −9 < −8 < 0 < 1 < 5 < 15
(ii) −320 < −106 < −7 < 107 < 185

#### Question 10:

Write the following integers in decreasing order:

(i) −15, 0, −2, −9, 7, 6, −5, 8
(ii) −154, 123, −205, −89, −74

(i) 8 > 7 > 6 > 0 > −2 > −5 > −9 > −15
(ii) 123 > −74 > −89 > −154 > −205

#### Question 11:

Using the number line, write integer which is:

(i) 2 more than 3
(ii) 5 less than 3
(iii) 4more than −9

(i) We want an integer two more than 3. So, on the number line, we will start from 3 and move 2 units to the right to obtain 5, as shown on the number line.

(ii) We want an integer five less than 3. So, on the number line, we will start from 3 and move 5 units to the left to obtain −2, as shown on the number line.

(iii) We want an integer four more than −9. So, on the number line, we will start from −9 and move 4 units to the right to obtain −5, as shown on the number line.

(i)

(ii)

(iii)

#### Question 12:

Write the absolute value of each of the following:

(i) 14
(ii) −25
(iii) 0
(iv) −125
(v) −248
(vi) a −7, if a is greater than 7
(vii) a −7, if a −2 is less than 7
(viii) a + 4, if a is greater than −4
(ix) a + 4, if a is less than −4
(x) $\left|-3\right|$
(xi) $-\left|-5\right|$
(xii) $\left|12-5\right|$

(i) Absolute value of 14 is 14.
(ii) Absolute value of −25 is 25.
(iii) Absolute value of 0 is 0.
(iv) Absolute value of −125 is 125.
(v) Absolute value of −248 is 248.
(vi) Absolute value of (a − 7) is (a − 7) if a is greater than 7, that is, a − 7 > 0.
(vii) if a − 2 is less than 7, that is, a − 2 < 7 ⇒ a < 9 or a − 7 < 2
So absolute value of a−7 = a−7 if  7 < a < 9 that is a−2 is less than 7 but a−2 is greater than 5.
and absolute value will be −(a−7) if  a < 7 if a−7 is less than 5.
(viii) Absolute value of (a + 4) is (a + 4) if a is greater than −4, that is, a > −4 ⇒ a + 4 > 0.
(ix) Absolute value of (a + 4) is −(a + 4) if is less than −4, that is, a < −4 ⇒ a + 4 < 0.
(x) Absolute value of −3 is 3.
(xi) −|−5| is −5 and its absolute value is 5.
(xii) |12 − 5 | = | 7| and its absolute value is 7.

#### Question 13:

(i) Write 4 negative integers less than −10.
(ii) Write 6 negative integers just greater than −12.

(i) −9, −8, −7 and −6 are the four negative integers less than −10.
(ii) −11, −10, −9, −8, −7 and −6 are the six negative integers just greater than −12.

#### Question 14:

Which of the following statements are true?

(i) The smallest integer is zero.

(ii) The opposite of zero is zero.

(iii) Zero is not an integer.

(iv) 0 is larger than every negative integer.

(v) The absolute value of an integer is greater than the integer.

(vi) A positive integer is greater than its opposite.

(vii) Every negative integer is less than every natural number.

(viii) 0 is the smallest positive integer.

(i) False
Integers are negative also.
(ii) True
0 is neither positive nor negative.
(iii) False
0 is simply an integer that is neither positive nor negative.
(iv) True
Every negative integer is to the left of 0 on a number line.
(v) False
The absolute value of positive integer is integer itself. So both are equal.
(vi) True
Its opposite will be a negative integer and positive integer is always greater than negative integer.
(vii) True
Natural numbers start from 0, and 0 is greater than every negative integer.
(viii) False
0 is neither positive nor negative.

#### Question 1:

Draw a number line and represent each of the following on it:

(i) $5+\left(-2\right)$
(ii) $\left(-9\right)+4$
(iii) $\left(-3\right)+\left(-5\right)$
(iv) $6+\left(-6\right)$
(v) $\left(-1\right)+\left(-2\right)+2$
(vi) $\left(-2\right)+7+\left(-9\right)$

(i) If we start from 5 and move 2 units to the left of 5, we will obtain 3, as shown on the number line.

(ii) If we start from -9 and move 4 units to the right of -9, we will obtain -5, as shown on the number line.

(iii) If we start from -3 and move 5 units to the left of -3, we will obtain -8, as shown on the number line.

(iv) If we start from 6 and move 6 units to the left of 6, we will obtain 0, as shown on the number line.

(v) If we start from -1 and move 2 units to the left of -1, we will obtain -3; and then if we start from -3 and move 2 units to the left, we will obtain -1, as shown on the number line.

(vi) If we start from -2 and move 7 units to the right, we will obtain 5; and then if we start from 5 and move 9 units to the left, we will obtain -4, as shown on the number line.

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