RS Aggrawal 2020 2021 Solutions for Class 6 Maths Chapter 4 - Integers

Question 1:

Question 2:

(i) A decrease of 8
(ii) A gain of Rs 7
(iii) Losing a weight of 5 kg
(iv) 10 km below the sea level
(v) 5^oC above the freezing point
(vi) A withdrawal of Rs 100
(vii) Spending Rs 500
(viii) Going 6 m to the west
(ix) The opposite of 24 is -24.
(x) The opposite of -34 is 34.

Question 3:

(i) +Rs 600  
(ii) -Rs 800
(iii) -7^oC
(iv) -9
(v) +2 km
(vi) -3 km
(vii) + Rs 200
(viii) -Rs 300

Question 4:

(i) -5


(ii) -2

(iii) 0 


(iv) 7

Question 5:

(i)0, -2
  0 > -2
This is because zero is greater than every negative integer.

(ii) -3, -5
  -3 > -5
Since 3 is smaller than 5, -3 is greater than -5.

(iii) -5, 2
2 > -5
This is because every positive integer is greater than every negative integer.

(iv) -16, 8
8 > -16
This is because every positive integer is greater than every negative integer.
v) -365, -913
-365 > -913
Since 365 is smaller than 913,  -365 is greater than -913.
vi) -888, 8
8 > -888
This is because every positive integer is greater than every negative integer.

Question 6:

i) -7 < 6
This is because every positive integer is greater than every negative integer.
ii) -1 < 0
This is because zero is greater than every negative integer.
iii) -27 < -13
Since 27 is greater than 13, -27 is smaller than -13.
iv) -26 < 17
This is because every positive integer is greater than every negative integer.
v) -603 < -317
Since 603 is greater than 317, -603 is smaller than -317.
vi) -777 < 7
This is because every positive integer is greater than every negative integer.

Question 7:

i) 1, 2, 3, 4, 5

ii) -4, -3, -2, -1

iii) -2, -1, 0, 1, 2

iv) -6

Question 8:

i) 0 < 7
This is because 0 is less than any positive integer.
ii) 0 > -3
This is because 0 is greater than any negative integer.
iii) -5 < -2
Since 5 is greater than 2, -5 is smaller than -2.
iv) -15 < 13
This is because every positive integer is greater than every negative integer.
v) -231 < -132
Since 231 is greater than 132, -231 is smaller than -132.
vi) -6 < 6
This is because every positive integer is greater than every negative integer.

Question 9:

i) -7 < -2 < 0 < 5 < 8
ii) -100 < -23 < -6 < -1 < 0 < 12
iii) -501 < -363 < -17 < 15 < 165
iv) -106 < -81 < -16 < -2 < 0 < 16 < 21

Question 10:

i) 36 > 7 > 0 > -3 > -9 > -132
ii) 51 > 0 > -2 > -8 > -53
iii) 36 > 0 > -5 > -71 > -81
iv) 413 > 102 > -7 > -365 > -515

Question 11:

i) 4 more than 6
We want an integer that is 4 more than 6. So, we will start from 6 and proceed 4 steps to the right to obtain 10.

 
ii) 5 more than -6
We want an integer that is 5 more than -6. So, we will start from -6 and proceed 5 steps to the right to obtain -1.



iii) 6 less than 2
We want an integer that is 6 less than 2. So, we will start from 2 and proceed 6 steps to the left to obtain -4.





iv) 2 less than -3
We want an integer that is 2 less than -3. So, we will start from -3 and proceed 2 steps to the left to obtain -5

Question 12:

i) False
This is because 0 is greater than every negative integer.

ii) False
0 is an integer as we know that every whole number is an integer and 0 is a whole number.

iii) True
0 is an integer that is neither positive nor negative. So, the opposite of zero is zero.

iv) False
Since 10 is greater than 6, -10 is smaller than -6.

v) True
This is because an absolute value is a positive number. For example, -2 is an integer, but its absolute value is 2 and it is greater than -2.

vi) True
This is because all negative integers are to the left of 0.

vii) True
This is because natural numbers are positive and every positive integer is greater than every negative integer.

viii) False
This is because the successor of -187 is equal to -186 (-186 + 1). In succession, we move from the left to the right along a number line.

ix) False
This is because the predecessor of -215 is -216 (-216 - 1). To find the predecessor, we move from the right to the left along a number line.

Question 13:

i) The value of |-9| is 9
ii) The value of |-36| is 36
iii) The value of |0| is 0
iv) The value of |15| is 15
v) The value of |-3| is 3
$\therefore$ -|-3| = -3
vi) 7 + |-3|
= 7 + 3          (The value of |-3| is 3)
= 10
vii) |7 - 4|
= |3| 
= 3                 (The value of |3| is 3)

viii) 8 - |-7|
= 8 - 7           (The value of |-7| is 7)
= 1

Question 1:

i) Every negative integer that is to the right of -7 is greater than -7.
So, five negative integers that are greater than -7 are -6, -5, -4, -3, -2 and -1.
ii) Every negative integer that is to the left of -20 is less than -20.
So, five negative integers that are less than -20 are -21, -22, -23, -24 and -25.

Question 2:

i) On the number line, we start from 0 and move 9 steps to the right to reach a point A. Now, starting from A, we move 6 steps to the left to reach point B.


B represents the integer 3.
$\therefore$ 9 + (−6) = 3

(ii) On the number line, we start from 0 and move 3 steps to the left to reach point A. Now, starting from A, we move 7 steps to the right to reach point B.
B represents the integer 4.
$\therefore$ (−3) + 7 = 4

                

(iii) On the number line, we start from 0 and move 8 steps to the right to reach point A. Now, starting from A, we move 8 steps to the left to reach point B.
B represents the integer 0.
$\therefore$ 8 + (−8) = 0


(iv) On the number line, we start from 0 and move 1 step to the left to reach point A. Now, starting from A, we move 3 steps to the left to reach point B.
B represents the integer −4.
$\therefore$ (−1) + (−3) = −4
                               

(v) On the number line, we start from 0 and move 4 steps to the left to reach point A. Now, starting from A, we move 7 steps to the left to reach point B.
B represents the integer −11.
$\therefore$ (−4) + (−7) = −11



(vi) On the number line, we start from 0 and move 2 steps to the left to reach point A. Now, starting from A, we move 8 steps to the left to reach point B.
B represents the integer −10.
$\therefore$ (−2) + (−8) = −10


(vii) On the number line, we start from 0 and move 3 steps to the right to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 4 steps to the left to reach point C.
C represents the integer −3.
$\therefore$ 3 + (−2) + (−4) = −3



(viii) On the number line, we start from 0 and move 1 step to the left to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 3 steps to the left to reach point C.
C represents the integer −6.
$\therefore$(−1) + (−2) + (−3) = −6

             

(ix) On the number line, we start from 0 and move 5 steps to the right to reach point A. Now, starting from A, we move 2 steps to the left to reach point B. Again, starting from B, we move 6 steps to the left to reach point C.
C represents the integer −3.
$\therefore$ 5 + (−2) + (−6) = −3

              

Question 3:

(i)
(−3) + (−9)
= −3 − 9
= −12

(ii)
(−7) + (−8)
= −7 − 8
= −15

(iii)
(−9) + 16
= −9 + 16
= 7

(iv)
(−13) + 25 
= −13 + 25
= 12

(v)
8 + (−17)
= 8 − 17
= −9

(v)
2 + (−12)
= 2 − 12
= −10

Question 4:

(i) 
$\begin{array}{l}-\text{365} \\ \frac{-\text{87}}{-\text{452}}\end{array}$
−365
−365  −87 −365  −87 -365 and -87 are both negative integers. So, we add 365 and 87, and put the negative sign before the sum. 

(ii) 
$\begin{array}{l}-\text{687}\\ \frac{-\text{73}}{-\text{760}}\end{array}$
-687 and -73 are both negative integers. So, we add 365 and 87, and put the negative sign before the sum. 

(iii)
 $\begin{array}{l}-\text{1065}\\ \frac{-\text{987}}{-2052}\end{array}$
-1065 and -987 are both negative integers. So, we add 1065 and 987, and put the negative sign before the sum.

(iv)
 $\begin{array}{l}-\text{3596}\\ \frac{-\text{1089}}{-4685}\end{array}$
-3596 and -1089 are both negative integers. So, we add 3596 and 1089, and put the negative sign before the sum.

Question 5:

i)
$$\begin{gathered} \begin{array}{l}-206 \\ \frac{+\mathrm{98}}{-108} \\ \mathrm{Since} \mathrm{we} \mathrm{are} \mathrm{adding} \mathrm{a} \mathrm{negative} \mathrm{number} \mathrm{with} \mathrm{a} \mathrm{positive} \mathrm{number},\\ \mathrm{we} \mathrm{shall} \mathrm{subtract} \mathrm{the} \mathrm{smaller} \mathrm{number}, \mathrm{i}.\mathrm{e}. 98 \mathrm{from} \mathrm{the} \mathrm{greater} \mathrm{number}, \mathrm{i}.\mathrm{e}. 206\\ 206 - 98 = 108 \\ \mathrm{Since} \mathrm{the} \mathrm{greater} \mathrm{number} \mathrm{is} \mathrm{negative}, \mathrm{the} \mathrm{sign} \mathrm{of} \mathrm{the} \mathrm{result} \mathrm{will} \mathrm{be} \mathrm{negative}. \\ \mathrm{So}, \mathrm{the} \mathrm{answer} \mathrm{will} \mathrm{be} -108\end{array} \end{gathered}$$

ii) 
$\begin{array}{l}178 \\ \frac{\text{ -69}}{\text{ 1}0\text{9}} \end{array}$
$$\begin{gathered} \begin{array}{l}\text{Since we are adding a negative number with a positive number,}\\ \text{we shall subtract the smaller number, i.e. 69, from the greater number, i.e. 178}\\ \text{178 - 69 = 109 }\\ \mathrm{Since the greater number is positive, the sign} \text{of the result will be positive.}\end{array} \\ \mathrm{So, }\mathrm{the} \mathrm{answer will be 109} \end{gathered}$$

(iii) 

$\begin{array}{l} 312 \\ \text{ -}\frac{103 }{\underset{}{209}} \\ \end{array}$
Since we are adding a negative number with a positive number,
we shall subtract the smaller number, i.e. -103, from the greater number, i.e. 312
312 - 103 = 209
Since the greater number is positive, the sign of the result will be positive.
So, the answer will be 209

$\begin{array}{l}\left(iv\right)\\ -493\\ \frac{+289}{-204}\end{array}$

Since we are adding a negative number with a positive number,
we shall subtract the smaller number, i.e. 289, from the greater number, i.e. 493.
493 - 289 = 204
Since the greater number is negative, the sign of the result will be negative.
So, the answer will be -204

Question 6:

$\begin{array}{l}\left(\text{i}\right)\text{ 137 and }-\text{354}\\ -\text{354}\\ \frac{+137}{-217}\\ \\ \left(\text{ii}\right)\text{ 1}00\text{1 and }-\text{13}\\ 1001\\ \frac{-13}{988}\\ \\ \text{(iii)} -\text{3}0\text{57 and 199}\\ -\text{3}0\text{57}\\ \frac{\text{ 199}}{-2858}\\ \\ \left(\text{iv}\right)-\text{36 and 1}0\text{27}\\ \text{3}0\text{57}\\ \frac{-\text{36}}{3021}\\ \\ \left(\text{v}\right)-\text{389 and }-\text{1}0\text{32}\\ -\text{1}0\text{32}\\ \frac{-\text{389}}{-1421}\\ \\ \left(\text{vi}\right)-\text{36 and 1}00\\ \text{1}0\text{0}\\ \frac{-\text{36}}{64}\\ \\ \text{(vii) 3}00\text{2 and }-\text{888}\\ \text{3}00\text{2}\\ \frac{-\text{888}}{2114}\end{array}$
(viii) −18, + 25 and −37
 25 + (−18) + (−37)
= 25 – (18 + 37)
= 25 – 55
= –30
 
(ix) −312, 39 and 192
  39 + 192 + (−312)
= 39 + 192 - 312
= 231 −312
= −81
(x) −51, −203, 36 and −28
  36 + (−51) + (−203) + (−28)
= 36 − (51 + 203 + 28)
= 36 – 282
= −246

Question 7:

(i) −57 + 57 = 0
So, the additive inverse of −57 is 57.

(ii) 183 − 183 = 0
So, the additive inverse of 183 is −183.

(iii) 0 + 0 = 0
So, the additive inverse of 0 is 0.

(iv) −1001 + 1001 = 0
So, the additive inverse of​ −1001 is 1001.

(v) 2054 − 2054 = 0
So, the additive inverse of​ 2054 is −2054

Question 8:

(i) The successor of 201:
201 + 1 = 202
(ii) The successor of 70:
70 + 1 = 71
(iii) The successor of −5:
−5 + 1 = −4
(iv) The successor of −99:
−99 + 1 = −98
(v) The successor of −500:
−500 + 1 = −499

Question 9:

(i) The predecessor of 120:
120 − 1 = 119
(ii) The predecessor of 79:
79 − 1 = 78
(iii) The predecessor of −8:
−8 − 1 = −9
(iv) The predecessor of −141:
−141 − 1 = −142
​(v) The predecessor of −300:
−300 − 1 = −301

Question 10:

(i) (−7) + (−9) + 12 + (−16)
  = 12 − (7 + 9 + 16)
  = 12 − 32
  = −20

(ii)  37 + (−23) + (−65) + 9 + (−12)
      = 37 + 9 − (23 + 65 + 12) 
      = 46-100
      = −54

​(iii) (−145) + 79 + (−265) + (−41) + 2
     = 79 +2 − ( 145 + 265 + 41)
     = 81 − 451
     = −370

(iv) 1056 + (−798) + (−38) + 44 + (−1)
     = 1056 + 44 − (798 + 38 + 1)
     = 1100 − 837
     = −263

Question 11:

Let the distance covered in the direction of north be positive and that in the direction of south be negative.

Distance travelled to the north of Patna = 60 km
Distance travelled to the south of Patna = -90 km
Total distance travelled by the car = 60 + (​-90)
                                                   = -30 km
The car was 30 km south of Patna.

Question 12:

Total cost price  = Price of pencils + Price of pens
                         = 30 + 90 + 25
                         = Rs 145

Total amount sold = Price of pen + Price of pencils
                           = 20 + 70
                           = 90
Selling price - costing price = 90 $-$ 145
                                          = $-$55 
The negative sign implies loss.
Hence, his net loss was Rs 55. 

Question 13:

(i) True
For example: - 2 + (-1) = -3
 
(ii) False
It can be negative or positive.
For example: -2 + 3 = 1 gives a positive integer, but -5 + 2 = -3 gives a negative integer.

(iii) True
For example: 100 + (-100) = 0

(iv) False
For example: (-5) + 2 + 3 = 0

(v) False
|-5| = 5  and | -3 | = 3, 5 > 3

(vi) False
|8 − 5| = 3
|8| + |−5| = 8 + 5
= 13
$\therefore$|8 − 5|$\ne$|8| + |−5|

Question 1:

(i) a + 6 = 0
 => a = 0 − 6
=> a = − 6

(ii) 5 + a = 0
=> a = 0 − 5

(iii) a + (−4) = 0
=> a = 0 − (−4)
=> a = 4

(iv) −8 + a = 0
   => a = 0 + 8
  => a = 8

Question 2:

(i) −34 − 18
 = −52

(ii) 25 − (−15)
 = 25 + 15
 = 40
(iii) −28 from −43
= −43 − (−28)
= −43 + 28
​= −15

(iv) 68 from −37
 = −37 − 68
 = −105
​(v)  219 from 0
 =  0 − 219 
 = −219

(vi) −92 from 0
= 0 − (−92)
= 0 + 92
= 92

(vii) −135 from −250
= −250 − (−135)
​= −250 + 135
= −115

(viii) −2768 from −287
= −287 − (−2768)
​= 2768 −​ 287
= 2481

(ix) 6240 from −271
= −271 − (6240)
= −271 − 6240
= −6511

(x) −3012 from 6250
= 6250 − (−3012)
= 6250 + 3012
​= 9262

Question 3:

Sum of −1050 and 813:
−1050 + 813
= −237
Subtracting the sum of −1050 and 813 from −23:
−23 − (−237)
= −23 +237
= 214

Question 4:

Sum of 138 and −250:
138 + (−250)
= 138 − 250
= −112
Sum of 136 and −272:
= 136 + (−272)
= 136 − 272
 = −136
Subtracting the sum of −250 and 138 from the sum of 136 and −272:
−136 − (−112​)  
= −136 + 112​
= −24
 

Question 5:

Adding 33 and −47:
​33 + (−47)
= 33 − 47
= −14

Subtracting −84 from −14:
−14 − (−84)
= −14 + 84
= 70 

Question 6:

Difference of −8 and −68:
−8 − (−68)
​= −8 + 68
= 60

Adding -36 to 60:
−36 + 60
= 24 

Question 7:

(i) [37 − (−8)] + [11 − (−30)]
= (37 + 8) + (11 + 30)
= 45 + 41
= 86

(ii) [−13 − (−17) + [−22 − (−40)]
=  (−13 +17) + (-22 + 40)
= 4 + 18
= 22

Question 8:

No, they are not equal.

34 − (−72) 
= 34 + 72
​= 106

(−72) − 34
= −72 − 34
= −106

Since 106 is not equal to −106, the two expressions are not equal.

Question 9:

Let the other integer be x.
According to question, we have:
x + 170 =  −13
=> x = −13 − 170
=>  x = −183
Thus, the other integer is −183.

Question 10:

Let the other integer be x.
According to question, we have:
x + (−47) = 65
=> x − 47 = 65
=>  x = 65 + 47
=> x = 112 
Thus, the other integer is 112.

Question 11:

(i) True
An integer added to an integer gives an integer.

(ii) True
An integer subtracted from an integer gives an integer.

iii) False
 −8 − (−7)
 = −8 + 7
 = −1       
Since 14 is greater than 1, −1 is greater than −14.

iv) True
−5 − 2 = −7 
Since 8 is greater than 7, −7 is greater than −8.
                − 7 > −8

​v) False
L.H.S.
(−7) − 3 = −10
R.H.S.
(−3) − (−7)
= (−3) + 7
= 4
$\therefore$ L.H.S. $\ne$ R.H.S.

Question 12:

Let us consider the height above the sea level as positive and that below the sea level as negative.
$\therefore$ Height of point A from sea level = 5700 m
    Depth of point B from sea level = -39600 m
Vertical distance between A and B = Distance of point A from sea level - Distance of point B from sea level  
= 5700 - (​-39600) 
= 45300 m                                                            

Question 1:

Initial temperature of Srinagar at 6 p.m. = 1°C
Final temperature of Srinagar at midnight = −4°C
Change in temperature = Final temperature - Initial temperature
                                      â€‹= (−4 − 1)°C
                                      = −5°C
So, the temperature has changed by −5°C.
The negative sign indicates that the temperature has fallen.
So, the temperature has fallen by 5°C.

Question 2:

(i) 15 by 9
    = 15 × 9
    = 135

(ii) 18 by −7
= –(18 × 7)
      = –126

(iii) 29 by –11
= –(29 × 11)
= –319

(iv) –18 by 13
= –(18 × 13)
       = –234

(v) –56 by 16
= –(56 × 16)
      = –896

(vi) 32 by –21
= –(32 × 21)
= –672

(vii) –57 by 0
= –(57 × 0)
        = 0

(viii) 0 by –31
= –(0 × 31)
= 0

(ix) –12 by –9
      = (–12) × (– 9)
      = 108


(x) (–​746) by (–8)
      = (–746) × (–8)
      = 5968

(xi) 118 by −7
​= 118 × (-7)
= –826

(xii) −238 by −143
= (−238) × (−143)
= 34034


 

Question 3:

(i)  (–2) × 3 × (–4)
   = [(–2) × 3] × (–4)
   = (–6) × (–4)
  = 24

(ii) 2 × (–5) × (–6)
     = [2 × (–5)] × (–6)
     = (–10) × (–6)
     = 60

(iii) (–8) × 3 × 5
     = [(–8) × 3] × 5
     = (–24) × 5
     = –120
(iv) 8 × 7 × (–10)
     = [8 × 7] × (–10)
     = 56 × (–10)
     = –560
(v)  (–3) × (–7) × (–6)
     = [(–3) × (–7)] × (–6)
     = 21 × (–6)
     = –126
(vi) (–8) × (–3) × (–9)
     = [(–8) × (–3)] × (–9)
     = 24 × (–9)
     = –216

Question 4:

(i) 18 × (–27) × 30
            = (–27) × [18 × 30]
            = (–27) × 540
            = –14580

(ii) (–8) × (–63) × 9
            = [(–8) × (–63)] × 9
            = 504 × 9
            = 4536

(iii) (–17) × (–23) × 41
            = [(–17) × (–23)] × 41
            = 391 × 41
            = 16031

(iv) (–51) × (–47) × (–19)
            = [(–51) × (–47)] × (–19)
            = 2397 × (–19)
            = – 45543

Question 5:

(i)
L.H.S.                                             
=18 × [9 + (–7)]                                
= 18 × [9 – 7]                                      
= 18 × 2                                              
= 36 
   R.H.S.
=18 × 9 + 18 × (–7)
= 162 – (18 × 7)  
= 162 – 126
= 36
$\therefore$ L.H.S = R.H.S
Hence, verified.

(ii) (–13) × [(–6) + (–19)] = (–13) × (–6) + (–13) × (–19)
L.H.S.                                                             
= (–13) × [(–6) + (–19)]                                
= (–13) × [–6 – 19]                          
= (–13) × (–25)                                              
= 325
R.H.S.
= (–13) × (–6) + (–13) × (–19)
= 78 + 247
= 325

$\therefore$ L.H.S = R.H.S
Hence, verified.

Question 6:

Question 7:

(i) The product of a positive integer and a negative integer is negative.
            True

(ii) The product of two negative integers is a negative integer.
            False
The product of two negative integers is always a positive integer.

(iii) The product of three negative integers is a negative integer.
            True

(iv) Every integer when multiplied by (–1) gives its multiplicative inverse.
          False

Every integer when multiplied by (1) gives its multiplicative inverse.

Question 1:

(i) (–9) × 6 + (–9) × 4
Solution:
Using the distributive law:
(–9) × 6 + (–9) × 4
= (–9) × (6+9)
= (–9) × 10
= –90

(ii) 8 × (–12) + 7 × (–12)
Solution:
Using the distributive law:
8 × (–12) + 7 × (–12)
= (–12) × (8+7)
= (–12) × 15
= –180

(iii) 30 × (–22) + 30 × (14)
Solution:
Using the distributive law:
30 × (–22) + 30 × (14)
= 30 × [(–22) + 14]
= 30 × [–22 + 14]
= 30 × (–8)
= –240

(iv) (–15) × (–14) + (–15) × (–6)
Solution:
(–15) × (–14) + (–15) × (–6)
Using the distributive law:
= (–15) × [ (–14) + (–6)]
= (–15) × [–14 – 6]
= (–15) × (–20)
= 300

(v) 43 × (–33) + 43 × (–17)
Solution:
43 × (–33) + 43 × (–17)
Using the distributive law:
= (43 ) × [–(33) + (–17)]
= (43 ) × [–33 – 17]
= 43 × (–50)
= –2150

(vi)  (–36) × (72) + (–36) × 28
Solution
(–36) × (72) + (–36) × 28
Using the distributive law:
 = (–36) × (72 + 28 )
 = (–36) × 100
 = –3600 

(vii) (–27) × (–16) + (–27) × (–14)
Solution:
(–27) × (–16) + (–27) × (–14)
Using the distributive law:
= (–27) × [(–16) + (–14)]
= (–27) × [–16 –14]
= (–27) × [–30]
= 810

Question 2:

(i) 85 by –17

= $\frac{-85}{17}$
= –5

(ii) –72 by 18

=$\frac{-72}{18}$
= –4
(iii) –80 by 16

= $\frac{-80}{16}$
= –5
(iv) –121 by 11

=$\frac{-121}{11}$
 = –11

 (v) 108 by –12

=  $\frac{108}{-12}$
 = –9
(vi)  –161 by 23
= $\frac{-161}{23}$
= –7
(vii) –76 by –19

=$\frac{-76}{-19}$
= 4
(viii) –147 by –21

= $\frac{-147}{-21}$
= 7
(ix) –639 by –71

$\begin{array}{l}=\frac{-639}{-71}\\ =9\end{array}$
(x) –639 by –71

$\begin{array}{l}=\frac{-639}{-71}\\ =9\end{array}$
(x) –15625 by –125

$\begin{array}{l}=\frac{-15625}{-125}\\ =125\end{array}$


(xi) 2067 by –1

$\begin{array}{l}=\frac{2067}{-1}\\ =-2067\end{array}$

(xii) 1765 by –1765

$$\begin{gathered} \begin{array}{l}=\frac{1765}{-1765}\\ =-1\times \frac{1765}{1765}\end{array} \\ = -1\times 1 \\ = -1 \end{gathered}$$

(xiii) 0 by –278

 $\begin{array}{l}=\frac{0}{-278}\\ =0\end{array}$

(xiv) 3000 by –100

$\begin{array}{l}=\frac{3000}{-100}\\ =-30\end{array}$

Question 3:

(i) 80 ÷ (–16) = –5
(ii) (–84) ÷ (12) = –7
(iii) (–125) ÷ (–5) = 25
(iv) (0) ÷ (372) = 0
(v) (–186) ÷ 1 = –186
(vi) (–34) ÷ 17 = –2
(vii) (–165) ÷ 165 = –1
(viii) (–73) ÷ –1 = 73
(ix) 1 ÷ (–1) = –1

Question 1:

(i) True
(ii) False
This is because we cannot divide any integer by 0. If we do so, we get the quotient as infinity.
(iii) True
(iv) False
This is because the division of any two negative integers always gives a positive quotient.
(v) True
(vi) True
(vii) True
(viii) True
(ix) False
This is because the division of any two negative integers always gives a positive quotient.

Question 2:

(b) –4 < –3
Since 4 is greater than 3, –4 is less than –3.

Question 3:

(c) –5

2 less than –3 means the following:
= –3 – 2
= –5

Question 4:

c) –1

4 more than –5 means the following:
= –5 + 4
= –1

Question 5:

(a) –9

2 less than −7 means the following:
= −7 − 2
= −9

Question 6:

(b) 10
7 + |-3|
= 7 + (+ 3)   (The absolute value of −3 is 3.)
= 7 + 3 
= 10

Question 7:

(c) –77
(−42) + (−35) 
= −42 − 35
= −77

Question 8:

(b) –31
(−37) + 6
= −37 + 6
= −31

Question 9:

(c) 22
49 + (−27)
= 49 − 27
​= 22 

Question 10:

(c) –17

In succession, we move from the left to the right of the number line.

Question 11:

(b) –17
To find the predecessor of a number, we move from the right to the left of a number line. 

Question 12:

(a) 5
If we add the additive inverse of a number to the number, we get 0.
−5 + 5 = 0

Question 13:

(b) –7
−12 − (−5) 
= −12 + 5
= −7

Question 14:

(b) 13.5 − (−8) 
= 5 + 8
= 13

Question 15:

 (c) –55
Let x be the other integer.
x + 30 = –25
$\Rightarrow$ x = –25 – 30
$\Rightarrow$ x = –55

Question 16:

(a) 25

Let the other integer be x
x + (-5) = 20
$\Rightarrow$x - 5 = 20
$\Rightarrow$x = 25

​

Question 17:

(b) −21

Let the other integer be x.
x + 8 = −13
=> x  = −13 − 8
=> x = −21

Question 18:

 (b) 8

0 − (−8)
= 0 + 8
= 8

Question 19:

(c) 0

8 + (−8) 
= 8 − 8 
= 0

Question 20:

(c) 1

(−6) + 4 − (−3)
= −6 + 4 + 3
= −6 + 7
= 1

Question 21:

(c) 10
6 − (−4) 
= 6 + 4 
= 10

Question 22:

(a) –20
(−7) + (−9) + 12 + (−16)
= −7 − 9 + 12 −16 
= −20

Question 23:

(c) –12
–​4 –​ 8
= –​12

Question 24:

(c) 3

We have:
−6 − (−9)
= −6 + 9
= 3

Question 25:

(c) 15

We have:
 10  − (−5)
​= 10 + 5
= 15 

Question 26:

(b) –54
We have:
(−6) × 9 
= −(6 × 9​)
= −54 

Question 27:

(a) –90

(−9) × 6 + (−9) × 4 
Using distributive law:
 (−9) × (6 + 4)
= (−9) × (10)
= −90

Question 1:

(b) –4

36 ÷ (−9) 

 $$\begin{gathered} \frac{36}{-9} \\ =\frac{36}{9\times (-1)} \\ = \frac{1}{(-1)}\times \frac{36}{9} \\ = -1 \times 4= -4 \end{gathered}$$

Question 2:

The numbers ...–4, –3, –2, –1, 0, 1, 2, 3, 4... are integers.
The group of positive and negative numbers including 0 is called integers.

–5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5

Question 3:

(i) 0, –3
0
This is because 0 is greater than any negative integer.

(ii) –4, –6
–4
Since 6 is greater than 4, –4 is greater than –6.

(iii) –99, 9
9
This is because every positive integer is greater than any negative integer.

(iv) –385, –615
–385
Since 615 is greater than 385, –385 is greater than –615.

Question 4:

We can arrange the given integers in the increasing order in the following manner:
 –36, –18, –5, –1, 0, 1, 8, 16

Question 5:

(i) 9 – |–6|
= 9 – (6)
= 9 – 6
= 3

(ii) 6 + |–4|
= 6 + (4)
= 6 + 4
= 10

(iii) –8 – |–3|
= –8 – 3
= –11

Question 6:

Four integers less than –6 (i.e. four negative integers that lie to the left of –6) are –7, –8, –9 and –10.
Four integers greater than –6 (i.e. four negative integers that lie to the right of –6 ) are –5, –4, –3 and –2.

Question 7:

(i) 8 + (–16)
= 8 – 16
= –8

(ii) (–5) + (–6)
= –5 – 6
= –11

(iii) (–6) × (–8)
       = (6 × 8)
       = 48

(iv) (–36) ÷ 6
    $$\begin{gathered} \frac{-36}{6} \\ = (-1)\times \frac{36}{6} \\ = -6 \end{gathered}$$
(v) 
30 – (–50)
= 30 + 50
= 80

(vi) (–40) ÷ (–10)
$\begin{array}{l}=\frac{-40}{-10}\\ =\frac{(-1)\times 40}{(-1)\times 10}\\ =4\end{array}$

(vii) 8 × (–5)
= –(8 × 5)
= –40

(viii) (–30) – 15
= –30 – 15
= –45

Question 8:

Let the integer be x.
$\therefore$ 34 + x = –12
or x = –12 – 34
or x = –46
Therefore, the other integer is –46.

Question 9:

(i) (–24) × (68) + (–24) × 32
= –(24) × (68+32)
= –24 × 100
= –2400

(ii) (–9) × 18 – (–9) × 8
= –(9 ) × [18 – 8]
= –9 × 10
= –90

(iii) (–147) ÷ (–21)

$\begin{array}{l}=\frac{-147}{-21}\\ =\frac{(-1)\times 147}{(-1)\times 21}\\ =\frac{(-1)}{(-1)}\times \frac{147}{21}\\ =7\end{array}$

(iv) 16 ÷ (–1)

$\begin{array}{l}=\frac{16}{-1}\\ =\frac{16\times (-1)}{(-1)\times (-1)}\\ =16\times (-1)\\ =-16\end{array}$   {Multiplying the numerator and the denominator by (–1)}

Question 10:

(b) −88
The successor of −89 is â€‹−88. The successor of a number lies towards its right on a number line. â€‹ 
−88 lies to the right of ​−89.  

Question 11:

(b) â€‹−100
The predecessor of a number lies to the left of the number.
​​−100 lies to the left of −​99. Hence, ​​−100 is a predecessor of −​99.

Question 12:

(c) â€‹23
Additive inverse of a number added to the number gives 0.
 −23 + 23 = 0
Hence, 23 is the additive inverse of  −23. 

Question 13:

(b) > 


Here, L.H.S. = (−13 + 6) 
                   = −7


R.H.S. = −25 − (−9)
            = −25 + 9
​            = −16
  
−7 > −16

L.H.S. > R.H.S.

Question 14:

 (c) 20 

x + (−8) = 12
=> x − 8 = 12
=> x = 12 + 8
=> x = 20

Question 15:

(c) -2

5 more than (−7) means 5 added to (−7).
   5 + (−7)
= 5 − 7
= −2

Question 16:

(d) −47
Let the number to be added to 16 be x.
x + 16 = (−31)
=> x = (−31)−16
=> x = −47

Question 17:

(d) −70
−36 ​− 34
= −70

Question 18:

(i)
Let the required number be x.
   −23 − x = 15
=> −23 = 15 + x
=> 15 + x = −23
=> x = −15 −23
=> x = −38 

(ii) 
The largest negative integer is -1.
 
(iii) 
The smallest positive integer is 1.

(iv) 
(−8) + (−6) − (−3) 
= (−8) + (−6) +3
= −8 â€‹−6 + 3
= −11

(v) 
The predecessor of −200:
(−200 − 1)
= −201