RS Aggarwal 2019 2020 Solutions for Class 7 Maths Chapter 1 - Integers

Question 1:

Question 2:

(i) 15 + (−8) = 7

(ii) (−16) + 9 = −7

(iii) (−7) + (−23) = −30

(iv) (−32) + 47 = 15

(v) 53 + (−26) = 27

(vi) (−48) + (−36) = −84

Question 3:

(i) 153 + (−302) = −149

(ii)  1005 + (−277) = 728

(iii) (−2035) + 297 = −1738

(iv)  (−489) + (−324) = −813

(v)  (−1000) + 438 = −562

(vi) (−238) + 500 = 262

Question 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

Question 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

Question 6:

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

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

Question 7:

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

Question 8:

(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

Question 9:

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

Question 10:

36 − (−64) = 36 + 64 = 100

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

Here, 100 $\ne$ −100

Thus, they are not equal.

Question 11:

(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]

Question 12:

Here, (a − b) = −9 − (−6) = −3

Similarly, (b − a) = −6 − (−9) = 3

∴ (a−b) ≠ (b−a)

Question 13:

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

53 + a = −16
⇒ a = −16 − 53 = −69

∴ The other integer is −69.

Question 14:

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

∴ The other integer is 96.

Question 15:

We have:

a − (−6) = 4
⇒ a = 4 + (−6) = −2

∴ a = −2

Question 16:

(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.

Question 1:

(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

Question 2:

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

Question 3:

(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

Question 4:

(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

Question 5:

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.

Question 6:

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.

Question 7:

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

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

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

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

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

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

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

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

Question 8:

(i) (−6) × (x) = 6
$x = \frac{6}{-6} = \frac{-6}{6}= -1$

Thus, x = (−1)

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

Question 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 $\times$ 5 + 6 $\times$ (−2)
= 20 + (−12) =8

(ii) Reenu's score = 5 $\times$ 5 + 5 $\times$ (−2)
= 25 − 10 = 15

(iii) Heena's score = 2 $\times$ 5 + 5 $\times$ (−2)
= 10 − 10 = 0

Question 1:

(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 $\times$ (−1) = −a, which is not the multiplicative inverse of a.
(v) True. a $\times$b = b $\times$a
(vi) True. (a $\times$b) $\times$c = a $\times$ (b $\times$c)
(vii) False. Every non-zero integer a has a multiplicative inverse $\frac{1}{a}$, which is not an integer.

Question 2:

(i) 65 $÷$ (−13) = $\frac{65}{-13} =$−5

(ii) (−84) $÷$ 12 = $\frac{-84}{12} =$ −7

(iii) (−76) $÷$ 19 = $\frac{-76}{19} =$−4

(iv) (−132) $÷$ 12 = $\frac{-132}{12} =$−11

(v) (−150) $÷$ 25 = $\frac{-150}{25} =$−6

(vi) (−72) $÷$ (−18) = $\frac{-72}{-18} = 4$

(vii)  (−105) $÷$ (−21) = $\frac{-105}{-21 } =$5

(viii) (−36) $÷$ (−1) = $\frac{-36}{-1} =$36

(ix) 0 $÷$ (−31) =  $\frac{0}{-31 } =$0

(x)  (−63) $÷$ 63 = $\frac{-63}{63} =$−1

(xi)  (−23) $÷$ (−23) = $\frac{-23}{-23} =$1

(xii) (−8) $÷$ 1 =  $\frac{-8}{1} =$−8

Question 3:

(i)
72 ÷ (x) = −4
 $$\begin{gathered} \Rightarrow \frac{72}{x} = -4 \\ \Rightarrow x = \frac{72}{-4} = -18 \end{gathered}$$

(ii)
−36 ÷ (x) = −4
$$\begin{gathered} \Rightarrow \frac{-36}{x} = -4 \\ \Rightarrow x = \frac{-36}{-4 } = 9 \end{gathered}$$

(iii)
(x) ÷ (−4) = 24
$$\begin{gathered} \Rightarrow \frac{x}{-4} = 24 \\ \Rightarrow x = 24\times (-4) = -96 \end{gathered}$$

(iv) 
(x) ÷ 25 = 0
$$\begin{gathered} \Rightarrow \frac{x}{25} = 0 \\ \Rightarrow x = 25\times 0 = 0 \end{gathered}$$

(v)
(x) ÷ (−1) = 36
$$\begin{gathered} \Rightarrow \frac{x}{-1} = 36 \\ \Rightarrow x = 36\times (-1) = -36 \end{gathered}$$

(vi)
(x) ÷ 1 = −37
$$\begin{gathered} \Rightarrow \frac{x}{1}= -37 \\ \Rightarrow x = -37\times 1 = -37 \end{gathered}$$

(vii)
39 ÷ (x) = −1
$$\begin{gathered} \Rightarrow \frac{39}{x} = -1 \\ \Rightarrow x = -1\times 39 = -39 \end{gathered}$$

(viii) 
1 ÷ (x) = −1
$$\begin{gathered} \Rightarrow \frac{1}{x}= -1 \\ \Rightarrow x = -1\times 1 = -1 \end{gathered}$$

(ix)
−1 ÷ (x) = −1
$$\begin{gathered} \Rightarrow \frac{-1}{x} = -1 \\ \Rightarrow x = \frac{-1}{-1}= 1 \end{gathered}$$

Question 1:

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

(iii) False (F). $\frac{-5}{-1} = 5$

(iv)  True (T). $\frac{-8}{1}= -8$

(v)  False (F). $\frac{-1}{-1 }= 1$

(vi) True (T). $\frac{-9}{-1} = 9$

Question 2:

(c) 14
Given:
6 − (−8)
= 6 + 8
= 14

Question 3:

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

Question 4:

(d) 5
We can see that

−3 + 5 = 2

Hence, 2 exceeds −3 by 5.

Question 5:

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

Question 6:

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

Hence, −6 is four (4) less than −2.

Question 7:

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

Question 8:

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

Question 9:

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

Question 10:

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

Question 11:

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

Question 12:

(a) 4
(−36) ÷ (−9) = 4

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

Question 13:

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

Question 14:

(c) not defined

Dividing any integer by zero is not defined.

Question 15:

(b) −11 < −8

Negative integers decrease with increasing magnitudes.

Question 16:

(b) 9

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

Question 17:

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

Hence, the other integer is −10.

Question 18:

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

Hence, the other integer is 22.

Question 19:

(c) 6

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

Question 20:

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

Question 21:

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

Question 22:

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

Question 23:

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

Question 24:

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

Question 1:

(b) 90

$$\begin{gathered} x ÷ (-18) = -5 \\ \Rightarrow \frac{x}{-18} = -5 \\ \therefore x = -18 \times -5 = 90 \end{gathered}$$

Question 2:

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

Hence, the other integer is 55.

Question 3:

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

Hence, the value of p is −5.

Question 4:

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

∴ 144 + (−22) = 122

Question 5:

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.

Question 6:

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

$a÷(-7)=12$
⇒ a = $-\frac{7}{12}$

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

Question 7:

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

(vi)  $(-1{)}^{25} = -1$

Question 8:

(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.

Question 9:

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

∴ The other integer is −8.

Question 10:

(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.

Question 11:

(b)108

(108) ÷ (−18) = −6

Question 12:

(a) 370
We have:

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

Question 13:

(c) −250

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

Question 14:

(b) −3

(−9) − (−6)
= (−9) + 6
= −3

Question 15:

(b) −6

−8 − (−6) = 2

Hence, −8 is −6 less than −2.

Question 16:

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