Rd Sharma 2020 2021 for Class 7 Maths Chapter 1 - Integers

Answer:

 On applying the BODMAS rule, we get:

   (− 15) + 4 ÷ (5 − 3)

= (− 15) + 4 ÷ 2  (On simplifying brackets)

= (− 15) + 2        (On performing division)

= − 13

Answer:

On applying the BODMAS rule, we get:

(− 40) × (− 1) + (− 28) ÷ 7

=  40 + (− 4)                           (On performing division and multiplication)

=  36 

Answer:

On applying the BODMAS rule, we get:

(− 3) + (− 8) ÷ (− 4) − 2 × (− 2)

=  (− 3) + 2 + 4                (On performing division and multiplication)

= (− 3) + 6                       (On performing addition)

= 3                                  (On performing subtraction)

Answer:

On applying the BODMAS rule, we get:

(−3) ​× (−4) ÷ (−2) + (−1)

= (−3) × 2 + (−1)                    (On performing division)

=  −6 − 1                                (On performing multiplication)

= −7                                       (On performing addition)

Answer:

On applying the BODMAS rule, we get:

3 − (5 − 6 ÷ 3)

= 3 − (5 − 2)     (On performing division)

= 3 − 3              (On performing subtraction)

= 0

Answer:

On applying the BODMAS rule, we get:

−25 + 14 ÷ (5 − 3)

= −25 + 14 ÷ 2       (On simplifying brackets)

= −25 + 7               (On performing division)

= −18

Answer:

On applying the BODMAS rule, we get:

​25 $-$ 1/2 {5+4-( 3 + 2 − $\overline{1+3}$ )}
 $$\begin{gathered} = 25- \frac{1}{2}\{9- (3+2-4\left)\right\} [\mathrm{Removing} \mathrm{vinculum}] \\ = 25- \frac{1}{2}\{9-(5-4\left)\right\} [\mathrm{Performing} \mathrm{addition}] \\ =25- \frac{1}{2}\left\{8\right\} [\mathrm{Performing} \mathrm{subtraction}] \\ = 25- 4 \\ = 21 \end{gathered}$$

Answer:

On applying the BODMAS rule, we get:

27 $-$ [38 $-$ {46 $-$ (15 $-$11)}]    (On simplifying vinculum)

= 27 $-$ [38$-$ {46 $-$ 4}]              (On simplifying parentheses)

= 27 $-$ [38 $-$ 42]                       (On simplifying braces)

= 27 $-$ ($-$4) = 31

Answer:

On applying the BODMAS rule, we get:

36 $-$ [18 $-$ { 14 $-$ (15 $-$ 4 ÷ 2 ​× 2)}]
= 36 − [18 − {14 − (15 − 2 × 2)}]                (On performing division)
= 36 $-$ [18 $-$ {14 $-$ (15 $-$ 4)}]                   (On performing multiplication)
= 36 $-$ [18 $-$ {14 $-$ 11}]                            (On simplifying parentheses)
= 36 $-$ [18 $-$ 3]                                          (On simplifying braces)
= 36 $-$ 15
= 21

Answer:

On applying the BODMAS rule, we get:

    45 $-$ [38 $-$ { 60 ÷  3 $-$ (6 $-$ 9 ÷ 3) ÷ 3}]
=  45 $-$ [38 $-$ {60 ÷ 3 $-$ (6 $-$ 3) ÷ 3}]           (On performing division)
=  45 $-$ [38 $-$ {60 ÷ 3 $-$ 3 ÷ 3}]                    (On simplifying parentheses)
=  45 $-$ [38 $-$ {60 ÷ 3 $-$ 1}]                          (On performing division)
=  45 $-$ [38 $-$ {20 $-$ 1}]                                (On performing division)
=  45 $-$ [38 $-$ 19]                                          (On performing subtraction)
=  45 $-$ 19   
= 26

Answer:

On applying the BODMAS rule, we get:

   23 $-$ [23 $-$ {23 $-$ (23 $-$ $\overline{23-23}$)}]
= 23 $-$ [23 $-$ {23 $-$ (23 $-$ 0}]            (On simplifying vinculum)
= 23 $-$ [23 $-$ {23 $-$ 23}]                    (On simplifying parentheses)
= 23 $-$ [23 $-$ 0]                                  (On simplifying braces)
= 23 $-$ 23 = 0

Answer:

On applying the BODMAS rule, we get:

   2550 $-$ [510 $-$ {270 $-$ (90 $-$ $\overline{80+70}$)}]    
= 2550 $-$ [510 $-$ {270 $-$ (90 $-$ 150)}]        (On simplifying vinculum)
= 2550 $-$ [510 $-$ { 270 $-$ ($-$ 60)}]              (On simplifying parentheses)
= 2550 $-$ [510 $-$ 330]                                 (On simplifying braces)
= 2550 $-$ 180
= 2370

Answer:

On applying the BODMAS rule, we get:
 
     4 + $\frac{1}{5}$[{ $-$ 10 × ( 25 $-$ $\overline{13-3}$)} ÷ ($-$5)]
=  4 + $\frac{1}{5}$[{$-$ 10 × (25 $-$ 10)} ÷ ($-$ 5)]        (On simplifying vinculum)
=  4 + $\frac{1}{5}$[{$-$ 10 × 15} ÷ ($-$5 )]                   (On simplifying parentheses)
= 4 + $\frac{1}{5}$[30]                                               (On simplifying braces)
= 4 + 6
= 10

Answer:

On applying the BODMAS rule, we get:
 
$$\begin{gathered} 22-\frac{1}{4}\{-5-(-48)÷(-16\left)\right\} \\ = 22- \frac{1}{4}\{-5-3\} [\mathrm{Performing} \mathrm{division}] \\ = 22-\frac{1}{4}\{-8\} \\ =22-(-2) [\mathrm{Removing} \mathrm{braces}] \\ = 22+2 = 24 \end{gathered}$$

Answer:

On applying the BODMAS rule, we get:

    63 $-$ ($-$ 3) {$-$ 2 $-$ $\overline{8-3}$} ÷ 3 {5 + ($-$ 2) ($-$1)}
=  63 $-$ ($-$ 3) {$-$ 2 $-$ 5} ÷ 3 {5 + 2}       (On simplifying vinculum)
= 63 $-$ ($-$ 3) ($-$ 7 ) ÷ 3 × 7                     (On simplifying braces)
= 63 $-$ ($21÷21$)
= 63 $-$ 1
= 62

Answer:

On applying the BODMAS rule, we get:

   [29 $-$ ($-$ 2) {6 $-$ (7 $-$ 3)}] ÷ [3 × { $-$ 3) × ($-$ 2)}]
= [29 $-$ ($-$ 2) {6 $-$ 4}] ÷ [3 × { 5 + 6}]                       (On simplifying parentheses)
= [29 $-$ ($-$ 2) (2)] ÷ [3 × 11]                                        (On performing subtraction and addition)
= [29 + 4] $÷$ 33                                                           (On performing multiplication)
= 33 $÷$ 33
= 1

Answer:

(i) 9 (2 + 5)
(ii) 12 ÷ (1 + 3)
(iii) 20 ÷ (7 $-$ 2) 
(iv) (2 × 3 ) $-$ 8 
(v) 40 ÷ {(9 + 10) + 1}
(vi) 2 × {(19 $-$ 6) $-$ 1}

Answer:

The number of integers in the given product is even.

∴ (−1) × (−1) × (−1) × (−1) × ... 500 times

= 1 × 1 × 1 × 1 × ... 500 times

= 1

Hence, the correct answer is option (b).

Answer:

(−1) + (−1) + (−1) + (−1) + ... 500 times

= −(1 + 1 + 1 + 1 + ... 500 times)

= −500

Hence, the correct answer is option (d).

Answer:

We know that, for every integer a, there exists integer −a such that

a + (−a) = 0 = −a + a

Here, −a is the additive inverse of a and a is the additive inverse of −a.

Now, 7 + (−7) = 0 = −7 + 7

∴ 7 is the additive inverse of −7.

Hence, the correct answer is option (c). 

Answer:

The modulus (or absolute value) of an integer is its numerical value regardless of its sign. The absolute value of an integer is always non-negative.

It is given that,

Modulus of x = | x | = 9

Now, | −9 | = 9 and | 9 | = 9

∴ x = −9 or x = 9

⇒ x = ± 9

Hence, the correct answer is option (c).

Answer:

Difference between 5 and −4 = 5 − (−4) = 5 + 4 = 9

Thus, 5 exceed −4 by 9.

Hence, the correct answer is option (c).

Answer:

Difference between −3 and −7 = (−3) − (−7) = −3 + 7 = 4

Thus, −7 is less than −3 by 4.

Hence, the correct answer is option (a).

Answer:

Sum of two integers = 24

One of the integers = −19

∴ Other integer = Sum of two integers − One of the integers

                         = 24 − (−19)

                         = 24 + 19

                         = 43

Hence, the correct answer is option (a).

Answer:

Let x be subtracted from −6 to obtain −14.

∴  −6 − x = −14

Putting x = 8, we get

LHS = −6 − 8 = −6 + (−8) = −14 = RHS

Thus, 8 must be subtracted from −6 to obtain −14.

Hence, the correct answer is option (a).

Answer:

Let x be divided by 6 to get −18.

$$\begin{gathered} \therefore x÷6=-18 \\ \Rightarrow \frac{x}{6}=-18 \end{gathered}$$

Putting x = −108, we get

LHS = $\frac{-108}{6}=-\frac{\left|-108\right|}{6}=-\frac{108}{6}=-18$ = RHS

Thus, −108 should be divided by 6 to get −18.

Hence, the correct answer is (c).

Answer:

We know that if a and b are two negative integers, then the integer with greater absolute value is less than the integer with smaller absolute value.

Absolute value of −12 = | −12 | = 12

Absolute value of −9 = | −9 | = 9

∴ −12 < −9

Also,

(−12) + 9 = −3 < 0

and  (−12) × 9 = −(12 × 9) = −108 < 0 

Hence, the correct answer is option (b).

Answer:

Sum of two integers = −8

One of the integers = 12

∴ Other integer = Sum of two integers − One of the integers

                          = −8 − 12

                          = −8 + (−12)

                          = −20

Hence, the correct answer is option (d).

Answer:

−14 subtracted from −18

= −18 − (−14)

= −18 + 14

= −4

Hence, the correct answer is option (b).

Answer:

(−35) × 2 + (−35) × 8

= (−35) × (2 + 8)                              [a × b + a × c = a × (b + c)]

= (−35) × 10

= −350

Hence, the correct answer is option (a).

Answer:

We know that if a is a non-zero integer, then 0 ÷ a = 0.

∴ x ÷ 29 = 0

⇒ x = 0

Hence, the correct answer is option (c).

Answer:

x = (−10) + (−10) + ... 15 times

   = − (10 + 10 + ... 15 times)

   = −150

y = (−2) × (−2) × (−2) × (−2) × (−2)

   = −(2 × 2 × 2 × 2 × 2)                 (When the number of negative integers in a product is odd, the product is negative)

   = −32

∴ x − y = −150 − (−32) = −150 + 32 = −118

Hence, the correct answer is option (b).

Answer:

a = (−1) × (−1) × (−1) × ... 100 times

Here, the number of integers in the product is even.

∴ a = (−1) × (−1) × (−1) × ... 100 times

      = 1 × 1 × 1 × ... 100 times

      = 1

b = (−1) × (−1) × (−1) × ... 95 times

Here, the number of integers in the product is odd.

∴ b = (−1) × (−1) × (−1) × ... 95 times

      = −(1 × 1 × 1 × ... 95 times)

      = −1

So,

a + b = 1 + (−1) = 0

Hence, the correct answer is option (c).

Answer:

|| 3 − 12| − 4|

= || 3 + (−12)| − 4|

= || −9| − 4|

= |9 − 4|                 (Absolute value of an integer is its numerical value regardless of its sign)

= |5|

= 5

Hence, the correct answer is option (b).

Answer:

a − (−9) = 5            (Given)

⇒ a + 9 = 5  

Putting a = −4, we get

LHS = −4 + 9 = 5 = RHS

∴ a = −4

Hence, the correct answer is option (c).

Answer:

It is given that the sum of two integers is 10, which is a positive integer.

But, we know that the sum of two negative integers is always a negative integer.

So, if the sum of two integers is positive and one of them is negative, then the other has to be positive.

For example,

−2 + 12 = 10

−5 + 15 = 10

Thus, the other integer has to be positive.

Hence, the correct answer is option (b).

Answer:

x = (−1) × (−1) × (−1) × (−1) × ... 25 times

The number of integers in the given product is odd.

∴ x = (−1) × (−1) × (−1) × (−1) × ... 25 times

      = −(1 × 1 × 1 × ... 25 times)

      = −1

y = (−3) × (−3) × (−3)

The number of integers in the given product is odd.

∴ y = (−3) × (−3) × (−3)

      = −(3 × 3 × 3)

      = −27

So,

xy = (−1) × (−27) = 27               (Product of two negative integers is always positive)

Hence, the correct answer is option (b).

Answer:

(i) 12 × 7 = 84

(ii) (−15) × 8 =  −120

(iii) (−25) × (−9) =  225

(iv) 125 × (−8) =  −1000

Answer:

(i) 3 × (−8) × 5 = $-3\times (8\times 5)$ = $-$120

(ii) 9 × (−3) × (−6) = $9\times (3\times 6)$= 162

(iii) (−2) × 36 × (−5) = $36 \times (2\times 5)$ = 360

(iv) (−2) × (−4) × (−6) × (−8) =  $(2\times 4\times 6\times 8)$ = 384

Answer:

(i) 1487 × 327 + (−487) × 327 = $327 (1487-487) = 327 \times 1000 =327000$

(ii) 28945 × 99 − (−28945) = $28945 (99 -(-1\left)\right) = 28945 (99+1) = 2894500$

Answer:

Answer:

The integer, whose product with −1 is the given number, can be found by multiplying the given number by −1.

Thus, we have:

(i) 58 × (−1) =  −58

(ii) 0 ​× (−1) = $- (0\times 1)$= 0

(iii) (−225) ​× (−1) =  225

Answer:

Negative numbers, when multiplied even number of times, give a positive number. However, when multiplied odd number of times, they give a negative number. Therefore, we have:

(i) (negative) 8 times × (positive)  1 time = $\mathrm{positive} \times \mathrm{positive}$ = positive integer

(ii) (negative) 21 times  × (positive) 3 times = $\mathrm{negative} \times \mathrm{positive}$ = negative integer

(iii) (negative) 199 times × (positive) 10 times = $\mathrm{negative} \times \mathrm{positive}$= negative integer

Answer:

(i) ( 8 + 9) × 10 = 170   >   8 + 90 = 98

(ii) (8 − 9) ​× 10 = −10  >  8 − 90 = − 82

(iii) {(−2) − 5 } ​× (−6) = −7 ​× (−6) = 42  >   (−2) − 5 × (−6)  = ( −2 ) −  (−30)  = −2 + 30 = 28

Answer:

(i) a × (−1) = −30  

When multiplied by a negative integer, a gives a negative integer. Hence, a should be a positive integer.

a = 30

(ii) a × (−1) = 30   

​When multiplied by a negative integer, a gives a positive integer. Hence, a should be a negative integer​.

a = −30

Answer:

(i)
LHS = 19 ​× { 7 + (−3) } = 19 × {4} =  76
   
RHS =  19 × 7 + 19 × (−3) = 133 + (−57) = 76

Because LHS is equal to RHS, the equation is verified.

(ii)
LHS = (−23) {(−5) + 19} = (−23) { 14} = −322

RHS = (−23) × (−5) + (−23) × 19 = 115 + (−437) = −322

Because LHS is equal to RHS, the equation is verified.

Answer:

(i) True. Product of two integers with opposite signs give a negative integer.

(ii) True. Negative integers, when multiplied odd number of times, give a negative integer.

(iii) False. Product of two integers, one of them being a negative integer, is not necessarily positive. For example, (−1) × 2 = −2

(iv) False. For two non-zero integers a and b, their product is not necessarily greater than either a or b. For example, if a = 2 and  b = −2, then, a × b = −4, which is less than both 2 and −2.

(v) False. Product of a negative integer and a positive integer can never be zero.

(vi) True. If a > 1, then, $\mathrm{a}\times \mathrm{b} \ne \mathrm{b}\times \mathrm{a} \ne \mathrm{b}$

Answer:

(i) 102  ÷ 17 = $\frac{\left|102\right|}{\left|17\right|} = \frac{102}{17} =$6
(ii) −85  ÷  5 = $-\frac{\left|-85\right|}{\left|5\right|} = -\frac{85}{5} =$ − 17
(iii) −161  ÷ ( − 23)  = $\frac{\left|-161\right|}{\left|-23\right|} =\frac{161}{23}=$ 7
(iv) 76   ÷  − 19 =  $-\frac{\left|76\right|}{\left|-19 \right|} = -\frac{76}{19} =$− 4
(v) 17654   ÷  (− 17654) =  $- \frac{\left|17654\right|}{\left|-17654\right|}= -\frac{17654}{17654}=$− 1
(vi) (−729)  ÷  (− 27) = $\frac{\left|-729\right|}{\left|-27\right|} = \frac{729}{27}=$  27
(vii) 21590   ÷  − 10 = $-\frac{\left|21590\right|}{\left|-10\right|} = -\frac{21590}{10}=$− 2159
(viii) 0 ​ ÷ ​(−135) = $-\frac{\left|0\right|}{\left|-135\right|}= -\frac{0}{135}= 0$

Answer:

(i) 296  ÷ $-$148 = $-\frac{\left|296\right|}{\left|-148\right|}= -\frac{\left|296\right|}{\left|148\right|}=-\frac{296}{148}= -2$
∴ $296 ÷ (-2) = -148$

(ii) − 88  ÷ 11 = $-\frac{\left|-88\right|}{\left|11\right|}=-\frac{\left|88\right|}{\left|11\right|}=-\frac{88}{11}= -8$
∴ $-88 ÷ -8 = 11$

(iii) 84  ÷ 12 = $\frac{\left|84\right|}{\left|12 \right|}= \frac{84}{12}=7$
∴ $84÷7 = 12$

(iv) $25\times (-5) = -125$

∴ $-125÷-5 = 25$

(v) $156\times (-2) = -312$

∴ $-312÷156 = -2$

(vi) $567\times (-1) = -567$

∴ $-567 ÷ 567 = -1$

Answer:

(i)
$$\begin{gathered} \mathrm{LHS} =\frac{\left|0\right|}{\left|4\right|}=\frac{0}{4}=0 \\ = \mathrm{RHS} \end{gathered}$$

Because LHS is equal to RHS, the equation is true.

(ii)
$$\begin{gathered} \mathrm{LHS} = -\frac{\left|0\right|}{\left|-7\right|}=-\frac{0}{7}=-0=0 \\ = \mathrm{RHS} \end{gathered}$$

Because LHS is equal to RHS, the equation is true.

(iii)
$$\begin{gathered} \mathrm{LHS} = -\frac{\left|-15\right|}{\left|0\right|}=-\frac{15}{0}= \mathrm{Not} \mathrm{defined} \\ \ne \mathrm{RHS} \end{gathered}$$
Because LHS is not equal to RHS, the equation is false.

(iv)
$$\begin{gathered} \mathrm{LHS} = \frac{\left|0\right|}{\left|0\right|}= \frac{0}{0}= \mathrm{Not} \mathrm{Defined} \\ \ne \mathrm{RHS} \end{gathered}$$

Because LHS is not equal to RHS, the equation is false.

(v)
$$\begin{gathered} \mathrm{LHS} =\frac{\left|-8\right|}{\left|-1\right|}= \frac{8}{1}=8 \\ \ne \mathrm{RHS} \end{gathered}$$
Because LHS and RHS are not equal, the equation is false.

(vi)
$$\begin{gathered} \mathrm{LHS} = \frac{\left|-8\right|}{\left|-2\right|} = \frac{8}{2}=4 \\ = \mathrm{RHS} \end{gathered}$$

Because LHS is equal to RHS, the equation is true.

Answer:

On applying the BODMAS rule, we get:

36  ÷  6  + 3

= 6 + 3  (On performing division)

 = 9

Answer:

On applying the BODMAS rule, we get:

24 + 15 ÷ 3

= 24 + 5  (On performing division)

= 29

Answer:

On applying the DMAS rule, we get:

120 − 20 ÷ 4

= 120 − 5 (On performing division)

= 115

Answer:

On applying the DMAS rule, we get:

32 − ( 3 ​× 5 ) + 4

= 32 − 15 + 4    (On performing multiplication)

=  36 − 15         (On performing addition)

= 21                  (On performing ​subtraction)

Answer:

On applying the DMAS rule, we get:

3 $-$ ( 5 $-$ 6 ÷ 3)

= 3 $-$ ( 5 − 2)       (On performing division)

= 3 $-$ 3                (On performing subtraction)

= 0

Answer:

On applying the DMAS rule, we get:

21 − 12 ÷ 3 × 2

= 21 − 4 × 2         (On performing division)

= 21 − 8               (On performing multiplication)

= 13                      (On performing subtraction)

Answer:

On applying the DMAS rule, we get:

16 + 8 ÷ 4 − 2 ​× 3

= 16 + 2 − 6            (On performing division and multiplication)

= 18 − 6

= 12

Answer:

On applying the DMAS rule, we get:

28 $-$ 5 × 6 + 2

= 28 $-$ 30 + 2 (On performing multiplication)

= 30 $-$ 30       (On performing addition)

= 0                 (On performing subtraction)

Answer:

On applying the DMAS rule, we get:

(− 20) ​× (− 1) + (−28) ÷ 7

=  20 + (− 4)   (On performing division and multiplication)

= 20 − 4 

= 16

Answer:

On applying the DMAS rule, we get:

(− 2) + (− 8) ÷ (− 4)

= (− 2) + 2  (On performing division)

= 0             (On performing addition)