Integers

Write all integers between:

-9 and -3

Answer

All integers between -9 and -3 are:

-8, -7, -6, -5, -4

Write all integers between:

-33 and -26

Answer

All integers between -33 and -26 are:

-32, -31, -30, -29, -28, -27

Write all integers between:

-5 and 2

Answer

All integers between -5 and 2 are:

-4, -3, -2, -1, 0, 1

Write all integers between:

-1 and 6

Answer

All integers between -1 and 6 are:

0, 1, 2, 3, 4, 5

Compare the integers :

-16 and -61

Answer

(16 < 61) ⇒ (–16 > –61) $\hspace{2cm}$ [∵ a < b ⇒ -a > -b]

Compare the integers :

-39 and 13

Answer

We know that every negative integer is less than every positive integer.

∴ –39 < 13

Compare the integers :

-236 and -362

Answer

(236 < 362) ⇒ (–236 > –362) $\hspace{1cm}$ [∵ a < b ⇒ -a > -b]

Compare the integers :

-2 and 0

Answer

We know that every negative integer is less than 0.

∴ –2 < 0

Evaluate :

20 - | -11 |

Answer

We have

20 - | -11 | = 20 - 11 = 9 $\hspace{2cm}$ [∵ |-11| = 11]

Evaluate :

| -23 | - | -16 |

Answer

We have

| -23 | - | -16 | = 23 - 16 = 7 $\hspace{1.5cm}$ [∵ |-23| = 23 and |-16| = 16]

Evaluate :

| -137 | + | 13 |

Answer

We have

| -137 | + | 13 | = 137 + 13 = 150 $\hspace{0.5cm}$ [∵ |-137| = 137 and |13| = 13]

Evaluate :

6 - | -4 |

Answer

We have

6 - |-4| = 6 - 4 = 2 $\hspace{2.7cm}$ [∵ |-4| = 4 ]

Arrange the following integers in ascending order :

-9, 11, -23, 41, -37, 0, -5

Answer

Given negative integers are −9, −23, −37, −5.
In ascending order they are −37 < −23 < −9 < −5.

Given positive integers are 11, 41, 0.
In ascending order they are 0 < 11 < 41.

Hence, all the given integers in ascending order are:
−37 < −23 < −9 < −5 < 0 < 11 < 41.

i.e., −37, −23, −9, −5, 0, 11, 41.

Arrange the following integers in ascending order :

-31, 19, -50, -8, -23, 3, 8

Answer

Given negative integers are −31, −50, −8, −23.
In ascending order they are −50 < −31 < −23 < −8.

Given positive integers are 19, 3, 8.
In ascending order they are 3 < 8 < 19.

Hence, all the given integers in ascending order are:
−50 < −31 < −23 < −8 < 3 < 8 < 19.

i.e., −50, −31, −23, −8, 3, 8, 19.

Arrange the following integers in ascending order :

-2, 12, -43, 31, 7, -35, -10

Answer

Given negative integers are −2, −43, −35, −10.
In ascending order they are −43 < −35 < −10 < −2.

Given positive integers are 12, 31, 7.
In ascending order they are 7 < 12 < 31.

Hence, all the given integers in ascending order are:
−43 < −35 < −10 < −2 < 7 < 12 < 31.

i.e., −43, −35, −10, −2, 7, 12, 31.

Arrange the following integers in descending order:

-24, 16, -40, -5, -13, 6, -1

Answer

Given positive integers are 16, 6.
In descending order they are 16 > 6.

Given negative integers are −24, −40, −5, −13, −1.
In descending order they are −1 > −5 > −13 > −24 > −40.

Hence, all the given integers in descending order are:
16 > 6 > −1 > −5 > −13 > −24 > −40.

i.e., 16, 6, −1, −5, −13, −24, −40.

Arrange the following integers in descending order:

0, -7, 19, -23, -3, 8, 46

Answer

Given positive integers are 19, 8, 46.
In descending order they are 46 > 19 > 8.

Given negative integers are −7, −23, −3.
In descending order they are −3 > −7 > −23.

Also, 0 is less than positive integers and greater than negative integers.

Hence, all the given integers in descending order are:
46 > 19 > 8 > 0 > −3 > −7 > −23.

i.e., 46, 19, 8, 0, −3, −7, −23.

Arrange the following integers in descending order:

-31, -13, -49, 4, 37, -9, -52

Answer

Given positive integers are 4, 37.
In descending order they are 37 > 4.

Given negative integers are −31, −13, −49, −9, −52.
In descending order they are −9 > −13 > −31 > −49 > −52.

Hence, all the given integers in descending order are:
37 > 4 > −9 > −13 > −31 > −49 > −52.

i.e., 37, 4, −9, −13, −31, −49, −52.

Fill in the blanks:

(i) 0 is greater than every ............... integer.

(ii) Every negative integer is less than every ............... integer .

(iii) For any two positive integers a and b, if a < b, then -a ............... -b .

(iv) Modulus of a negative integer is always ............... .

(v) The largest negative integer is ............... .

(vi) The smallest positive integer is ............... .

Answer

(i) 0 is greater than every negative integer.

(ii) Every negative integer is less than every positive integer.

(iii) For any two positive integers a and b, if a < b, then −a > −b.

(iv) Modulus of a negative integer is always positive.

(v) The largest negative integer is −1.

(vi) The smallest positive integer is 1.

Add the following integers :

64 and 36

Answer

We have

64 + 36 = 100

Add the following integers :

73 and -37

Answer

We have

73 + (-37)

Since one integer is positive and the other is negative, we subtract their values and keep the sign of the greater number (73).

73 + (-37) = 73 - 37 = 36

Add the following integers :

-26 and -45

Answer

We have

-26 + (-45)

Since both integers are negative, we add their values and keep the negative sign.

-26 + (-45) = -26 - 45 = -71

Add the following integers :

-51 and 25

Answer

We have

−51 + 25

Since one integer is negative and the other is positive, we subtract their values and keep the sign of the greater number (−51).

−51 + 25 = −(51 − 25) = −26.

Add the following integers :

100 and -32

Answer

We have

100 + (−32)

Since one integer is positive and the other is negative, we subtract their values and keep the sign of the greater number (100).

100 + (−32) = 100 − 32 = 68.

Add the following integers :

0 and -21

Answer

We have

0 + (-21)

Zero is the additive identity. Adding 0 to any integer does not change its value.

∴ 0 + (-21) = -21

Find the additive inverse of :

23

Answer

Since 23 + (−23) = 0

∴ the additive inverse of 23 is -23

Find the additive inverse of :

-33

Answer

Since (−33) + 33 = 0

∴ the additive inverse of −33 is 33

Find the additive inverse of :

-1

Answer

Since (−1) + 1 = 0

∴ the additive inverse of −1 is 1

Find the additive inverse of :

-476

Answer

Since (−476) + 476 = 0

∴ the additive inverse of −476 is 476

Evaluate :

6 - 24

Answer

We have

6 - 24 = 6 + (-24) = -18

Evaluate :

18 - (-8)

Answer

We have

18 - (-8) = 18 + 8 = 26 $\hspace{2cm}$ [∵ a - (-b) = a + b]

Evaluate :

(-16) - (-5)

Answer

We have

(-16) - (-5) = -16 + 5 = -11

Evaluate :

(-20) - 6

Answer

We have

(-20) - 6

(-20) + (-6) = -26

Evaluate :

(-1) - (-19)

Answer

We have

(-1) - (-19) = (-1) + 19 = 18

Evaluate :

8 - (-23)

Answer

We have

8 - (-23) = 8 + 23 = 31 $\hspace{2cm}$ [∵ a - (-b) = a + b]

Verify the following :

(-14) + 9 = 9 + (-14)

Answer

We have

(-14) + 9 = 9 + (-14)

LHS = (−14) + 9
= −5

RHS = 9 + (−14)
= −5

Since LHS = RHS,
∴ (-14) + 9 = 9 + (-14)

Verify the following :

(-8) + (-12) = (-12) + (-8)

Answer

We have

(-8) + (-12) = (-12) + (-8)

LHS = (−8) + (−12)
= −20

RHS = (−12) + (−8)
= −20

Since LHS = RHS,
∴ (-8) + (-12) = (-12) + (-8)

Verify the following :

(-6) + [(-8) + 12] = [(-6) + (-8)] + 12

Answer

We have

(-6) + [(-8) + 12] = [(-6) + (-8)] + 12

LHS = (−6) + [−8 + 12]
= (−6) + 4
= −2

RHS = [(−6) + (−8)] + 12
= (−14) + 12
= −2

Since LHS = RHS,
∴ (-6) + [(-8) + 12] = [(-6) + (-8)] + 12

Verify the following :

[(-9) + (-7)] + (-14) = (-9) + [(-7) + (-14)]

Answer

We have

[(-9) + (-7)] + (-14) = (-9) + [(-7) + (-14)]

LHS = [−9 + (−7)] + (−14)
= (−16) + (−14)
= −30

RHS = (−9) + [−7 + (−14)]
= (−9) + (−21)
= −30

Since LHS = RHS,
∴ [(-9) + (-7)] + (-14) = (-9) + [(-7) + (-14)]

Fill in the blanks:

(i) 8 + ............... = 0

(ii) (-10) + ............... = -10

(iii) (-6) + (-8) = (-8) + ...............

(iv) (-6) + ............... = -14

(v) (-11) + ............... = (-7)

(vi) (-9) + ............... = -1

Answer

(i) 8 + (-8) = 0

(ii) (-10) + 0 = -10

(iii) (-6) + (-8) = (-8) + (-6)

(iv) (-6) + (-8) = -14

(v) (-11) + 4 = (-7)

(vi) (-9) + 8 = -1

Subtract the sum of -137 and -43 from the sum of -103 and 27.

Answer

Calculate the sum of -137 and -43:

(-137) + (-43) = -180

Calculate the sum of -103 and 27:

(-103) + 27 = -76

Subtract the first sum from the second sum:

(-180) - (-76)

= -180 + 76

= -104

∴ The answer is -104

Subtract -29 from -53 and add -16 to the result.

Answer

Subtract -29 from -53:
-53 - (-29) = -53 + 29 = -24

Add -16 to the result: -24 + (-16) = -40

Final Answer is -40

The sum of two integers is 43. If one of them is -27, find the other.

Answer

Let the two integers be x and y.

Given that x + y = 43 and x = -27.

Substitute x in the below equation

x + y = 43
-27 + y = 23 $\hspace{2cm}$(Substituting x)
y = 23 + 27 $\hspace{2.2cm}$(Solve for y)
y = 23 + 27
y = 70

∴ The other integer is 70

Find the successor of :

30

Answer

The successor of 30 is:

30 + 1 = 31

Find the successor of :

-70

Answer

The successor of -70 is:

-70 + 1 = -69

Find the successor of :

-206

Answer

The successor of -206 is:

-206 + 1 = -205

Find the successor of :

-1

Answer

The successor of -1 is:

-1 + 1 = 0

Find the predecessor of:

60

Answer

The predecessor of 60 is:

60 - 1 = 59

Find the predecessor of:

-351

Answer

The predecessor of -351 is:

-351 - 1 = -352

Find the predecessor of:

0

Answer

The predecessor of 0 is:

0 - 1 = -1

Find the predecessor of:

-99

Answer

The predecessor of -99 is:

-99 - 1 = -100

Find the product :

18 x 4

Answer

We have

18 x 4 = 72

Find the product :

(-25) x 6

Answer

We have

(-25) x 6 = -150 $\hspace{2.6cm}$[∵ Negative x Positive = Negative]

Find the product :

(-30) x 7

Answer

We have

(-30) x 7 = -(30 x 7) = -210 $\hspace{2.7cm}$[∵ Negative x Positive = Negative]

Find the product :

8 x (-15)

Answer

We have

8 x (-15) = -(8 x 15) = -120 $\hspace{2.7cm}$[∵ Positive x Negative = Negative]

Find the product :

20 x (-10)

Answer

We have

20 x (-10) = -(20 x 10) = -200 $\hspace{2.2cm}$[∵ Positive x Negative = Negative]

Find the product :

(-12) x (-15)

Answer

We have

(-12) x (-15) = +(12 x 15) = 180 $\hspace{2cm}$[∵ Negative x Negative = Positive]

Find the product :

(-8) x (-13)

Answer

We have

(-8) x (-13) = +(8 x 13) = 104 $\hspace{2cm}$[∵ Negative x Negative = Positive]

Find the product :

(-20) x (-1)

Answer

We have

(-20) x (-1) = +(20 x 1) = 20 $\hspace{2cm}$[∵ Negative x Negative = Positive]

Find the product :

(-9) x 0

Answer

We have

(-9) x 0 = -(9 x 0) = 0 $\hspace{2.9cm}$[∵ Anything multiplied with 0 becomes 0]

Find the product :

0 x (-11)

Answer

We have

0 x (-11) = -(0 x 11) = 0 $\hspace{2.7cm}$[∵ Anything multiplied with 0 becomes 0]

Find the product :

{(-9) x 8} x (-5)

Answer

We have

{(-9) x 8} x (-5)

First multiply inside the brace:
(-9) x 8 = -72

Now multiply the result with last number:
(-72) x (-5) = +(72 x 5) = 360

∴ {(-9) x 8} x (-5) = 360

Find the product :

{(-10) x (-5)} x 6

Answer

We have

{(-10) x (-5)} x 6

First multiply inside the brace:
(-10) x (-5) = 50

Now multiply the result with last number:
50 x 6 = 300

∴ {(-10) x (-5) x 6} = 300

Find the product :

{(-12) x (-15)} x (-2)

Answer

We have

{(-12) x (-15)} x (-2)

First multiply inside the brace:
(-12) x (-15) = 180

Now multiply:
180 x (-2) = -(180 x 2) = -360

∴ {(-12) x (-15)} x (-2) = -360

Find the product :

(-8) x {(-5) x (-3)}

Answer

We have

(-8) x {(-5) x (-3)}

First multiply inside the brace:
(-5) x (-3) = 15

Now multiply the result with the first number:
(-8) x 15 = -(8 x 15) = -120

∴ (-8) x {(-5) x (-3)} = -120

Find the product :

(-11) x {(-8) x 5}

Answer

We have

(-11) x {(-8) x 5}

First multiply inside the brace:
(-8) x 5 = -40

Now multiply the result with the first number:
(-11) x (-40) = +(11 x 40) = 440

∴ (-11) x {(-8) x 5} = 440

Verify the following :

(-14) x (-8) = (-8) x (-14)

Answer

We have

(-14) x (-8) = (-8) x (-14)

LHS = (-14) x (-8) = 112
RHS = (-8) x (-14) = 112

Since LHS = RHS,

∴ (-14) x (-8) = (-8) x (-14) $\hspace{1.5cm}$[a x b = b x a ⇒ Commutative Property of multiplication.]

Verify the following :

{(-7) x 5} x (-6) = (-7) x {5 x (-6)}

Answer

We have

{(-7) x 5} x (-6) = (-7) x {5 x (-6)}

LHS = {(-7) x 5} x (-6)
= (-35) x (-6)
= 210

RHS:(-7) x {5 x (-6)}
= (-7) x (-30)
= 210

Since LHS = RHS,

∴ {(-7) x 5} x (-6) = (-7) x {5 x (-6)} $\hspace{1.5cm}$[{a x b} x c = a x {b x c} ⇒ Associative Property of multiplication.]

Verify the following :

(-10) x {(-7) + (-9)} = {(-10) x (-7)} + {(-10) x (-9)}

Answer

We have

(-10) x {(-7) + (-9)} = {(-10) x (-7)} + {(-10) x (-9)}

LHS = (-10) x {(-7) + (-9)}
=(-10) x {-16}
= 160
RHS = {(-10) x (-7)} + {(-10) x (-9)}
= 70 + 90
= 160

Since LHS = RHS,

∴ (-10) x {(-7) + (-9)} = {(-10) x (-7)} + {(-10) x (-9)} $\hspace{1.5cm}$[a x {b + c} = {a x b} + {a x c} ⇒ Distributive Property of multiplication over addition.]

Find the quotient :

28 ÷ (-7)

Answer

We have

28 ÷ (-7) = $\dfrac {+28}{-7}$ = -4

Find the quotient :

(-65) ÷ 13

Answer

We have

(-65) ÷ 13 = $\dfrac {-65}{+13}$ = -5

Find the quotient :

(-66) ÷ (-6)

Answer

We have

(-66) ÷ (-6) = $\dfrac {-66}{-6}$ = 11

Find the quotient :

(-9) ÷ (-1)

Answer

We have

(-9) ÷ (-1) = $\dfrac {-9}{-1}$ = 9

Find the quotient :

0 ÷ (-2)

Answer

We have

0 ÷ (-2) = $\dfrac {0}{-2}$ = 0

Find the quotient :

(-12) ÷ (-12)

Answer

We have

(-12) ÷ (-12) = $\dfrac {-12}{-12}$ = 1

Write all even integers between

(i) (-4) and 11

(ii) (-13) and (-7)

Answer

We have

(i) All even integers between -4 and 11 are:
-2, 0, 2, 4, 6, 8 and 10

(ii) All even integers between -13 and -7 are:
-12, -10 and -8

Write all odd integers between

(i) (-1) and 7

(ii) (-20) and (-14)

Answer

(i) All odd integers between -1 and 7 are:
1, 3 and 5

(ii) All odd integers between -20 and -14 are:
-19, -17 and -15

Write five consecutive even integers succeeding -21.

Answer

The required consecutive even integers succeeding -21 are:
-20, -18, -16, -14 and -12

Write five consecutive odd integers preceding -36.

Answer

The required consecutive odd integers preceding -36 are:
-37, -39, -41, -43 and -45

The product of two integers is -120. If one number is 15, find the other.

Answer

Let the two integers be p and q.

Given that p x q = -120 and p = 15

Substitute p in the below equation:

p x q = -120

15 x q = −120 $\hspace{2cm}$(Substitute p)

q = $\dfrac{-120}{15}$ $\hspace{2.7cm}$(Solve for q)

q = -8

∴ The other number is -8.

Simplify:

5 {(-6) + (12 ÷ 4)}

Answer

Given expression:

5 {(-6) + (12 ÷ 4)}
= 5 {(-6) + 3} $\hspace{2cm}$ [Simplifying ( )]
= 5 x -3 $\hspace{3cm}$[Simplifying { }]
= -15

The answer is -15

Simplify:

10 - {56 ÷ (-16 + 9)}

Answer

Given expression:

10 - {56 ÷ (-16 + 9)}
= 10 - {56 ÷ (-7)} $\hspace{2cm}$ [Simplifying ( )]
= 10 - (-8) $\hspace{3.1cm}$[Simplifying { }]
= 10 + 8 $\hspace{3.4cm}$[Removing ( )]
= 18

The answer is 18

Simplify:

(-6) - {(-28) ÷ (-7)}

Answer

Given expression:

(-6) - {(-28) ÷ (-7)}
= (-6) - {4} $\hspace{3cm}$ [Simplifying ÷]
= (-6) - 4 $\hspace{3.3cm}$[Removing { }]
= -6 - 4 $\hspace{3.5cm}$[Removing ( )]
= -10

The answer is -10

Simplify:

$8 - \text{\textbraceleft}12 - (9 - \overline{7 - 4})\text{\textbraceright}$

Answer

Given expression:

$8 - \text{\textbraceleft}12 - (9 - \overline{7 - 4})\text{\textbraceright}$

$= 8 - \text{\textbraceleft}12 - (9 - 3)\text{\textbraceright} \hspace{2cm}\text{[Removing 'bar']} \\[1em] = 8 - \text{\textbraceleft}12 - 6\text{\textbraceright} \hspace{2.5cm}\text{[Removing ( )]} \\[1em] = 8 - 6 = 2 \hspace{3cm}\text{[Removing \text{\textbraceleft} \text{\textbraceright}]}$

The answer is 2

Simplify:

$11 - [(-8) - \text{\textbraceleft}10 - (9 - \overline{7 - 4})\text{\textbraceright}]$

Answer

Given expression:

$11 - [(-8) - \text{\textbraceleft}10 - (9 - \overline{7 - 4})\text{\textbraceright}]$

$= 11 - [(-8) - \text{\textbraceleft}10 - (9 - 3)\text{\textbraceright}] \hspace{1.8cm}\text{[Removing 'bar']} \\[1em] = 11 - [(-8) - \text{\textbraceleft}10 - 6\text{\textbraceright}] \hspace{2.5cm}\text{[Simplifying ( )]} \\[1em] = 11 - [(-8) - 4] \hspace{3.7cm}\text{[Simplifying \text{\textbraceleft} \text{\textbraceright}]} \\[1em] = 11 - [-12] \hspace{4cm}\text{[Removing ( )]} \\[1em] = 11 + 12 = 23 \hspace{3.5cm}\text{[Removing [ ]]}$

The answer is 23

Simplify:

$(-9) - [(-5) - \text{\textbraceleft}11 - (8 - \overline{5 - 2})\text{\textbraceright}]$

Answer

Given expression:

$(-9) - [(-5) - \text{\textbraceleft}11 - (8 - \overline{5 - 2})\text{\textbraceright}]$

$= (-9) - [(-5) - \text{\textbraceleft}11 - (8 - 3)\text{\textbraceright}] \hspace{1.8cm}\text{[Removing 'bar']} \\[1em] = (-9) - [(-5) - \text{\textbraceleft}11 - 5\text{\textbraceright}] \hspace{2.5cm}\text{[Simplifying ( )]} \\[1em] = (-9) - [(-5) - 6] \hspace{3.7cm}\text{[Simplifying \text{\textbraceleft} \text{\textbraceright}]} \\[1em] = (-9) - [-11] \hspace{4cm}\text{[Removing ( )]} \\[1em] = -9 + 11 = 2 \hspace{3.9cm}\text{[Removing [ ]]}$

The answer is 2

Simplify:

$10 - [8 - \text{\textbraceleft}11 + 30 ÷ (4 - \overline{5 - 7})\text{\textbraceright}]$

Answer

Given expression:

$10 - [8 - \text{\textbraceleft}11 + 30 ÷ (4 - \overline{5 - 7})\text{\textbraceright}]$

$= 10 - [8 - \text{\textbraceleft}11 + 30 ÷ (4 - (-2))\text{\textbraceright}] \hspace{2cm}\text{[Removing 'bar']} \\[1em] = 10 - [8 - \text{\textbraceleft}11 + 30 ÷ (6)\text{\textbraceright}] \hspace{3cm}\text{[Simplifying ( )]} \\[1em] = 10 - [8 - \text{\textbraceleft}11 + 5\text{\textbraceright}] \hspace{3.9cm}\text{[Simplifying ÷]} \\[1em] = 10 - [8 - 16] \hspace{4.9cm}\text{[Simplifying \text{\textbraceleft} \text{\textbraceright}]} \\[1em] = 10 - [-8] \hspace{5cm}\text{[Simplifying [ ]]} \\[1em] = 10 + 8 = 18 \hspace{4.5cm}\text{[Removing [ ]]}$

The answer is 18

By how much does 5 exceed -5?

Answer

To find how much one number exceeds another, we subtract the smaller number from the larger one:

5 - (-5)
= 5 + 5
= 10

The answer is 10

How much -7 less than -1?

Answer

We need to find the difference between −1 and −7.

So we calculate:

(-1) - (-7)
= -1 + 7
= 6

The answer is 6

What must be subtracted from 7 to get -6?

Answer

Let the required number be x.

7 - x = -6
To solve for x, we rearrange:
x = 7 - (-6)
x = 7 + 6
x = 13

The answer is 13

What must be subtracted from -1 to get -19?

Answer

Let the required number be x

(-1) - x = -19
Rearrange to solve for x:
x = -1 - (-19)
x = -1 + 19
x = 18

The answer is 18

The sum of two integers is -11. If one of them is 5, then find the other.

Answer

Let the two integers be p and q.

Given,

p = 5

p + q = -11

Substitute p in the below equation:

p + q = -11
5 + q = -11 $\hspace{2cm}$[Substituting p]
q = -11 - 5 $\hspace{2.1cm}$[Solve for q]
q = -(11 + 5)
q = -16

The answer is -16

Add the product of (-13) and (-17) to the quotient of (-187) and 11.

Answer

Calculate the product of (-13) and (-17):

(-13) x (-17) = 221 $\hspace{5cm}$(Negative x Negative = Positive)

Calculate the quotient of (-187) and 11:

(-187) ÷ 11 = -17 $\hspace{5cm}$(Negative ÷ Positive = Negative)

Now add them together:

221 + (-17)
= 221 - 17 = 204

The answer is 204

A shopkeeper bought a pen for ₹75, a book for ₹240 and a pencil box for ₹46. He sold the pen for ₹81, book for ₹255 and the pencil box for ₹40. What was his gain or loss?

Answer

Given:

Cost Price (CP) of pen = ₹75
Cost Price (CP) of book = ₹240
Cost Price (CP) of pencil box = ₹46
∴ The total Cost Price(CP) = 75 + 240 + 46 = ₹361

Selling Price of pen = ₹81
Selling Price of book = ₹255
Selling Price of pencil box = ₹40
∴ The total Selling Price(SP) = 81 + 255 + 40 = ₹376

Since SP is greater than CP, it is a gain.

Gain = SP - CP
Gain = 376 - 361 = 15 $\hspace{1cm}$[Substituting the values of total SP and total CP]

∴ The shopkeeper gains ₹15

A man starts from his home and drives 169 km to the East and then 192 km to the West. How far is he from his home finally and in which direction?

Answer

Let East be positive direction so West will be negative direction.

Given:

Distance travelled towards East = 169 km
Distance travelled towards West = -192 km

Total displacement = Distance travelled towards East + Distance travelled towards West

By substituting the values, we get

Total displacement = 169 + (-192)

= 169 - 192 = -23

The negative sign indicates that he is in West direction.

∴ He is 23 km to the West of his home.

During a month, the average day temperature of a desert region was 43°C and the average night temperature was -6°C. Find the difference between the average day temperature and the average night temperature.

Answer

Given:

Average day temperature = 43°C
Average night temperature = -6°C

Difference = Average day temperature - Average night temperature

By substituting the values, we get

Difference = 43 - (-6)

= 43 + 6 = 49

∴ The difference between the average day and the average night temperature = 49°C

Radhey bought 4 pairs of jeans at ₹1256 each . How much did he pay for the jeans in all?

Answer

Given:

Number of pairs of jeans = 4
Cost of 1 pair of jeans = ₹1256

Total cost = Number of pairs x Cost of 1 pair

By substituting the values, we get

Total cost = 4 x 1256 = ₹5024

∴ The total amount he paid is ₹5024

Mathew had no money to pay for his house rent of ₹14240. His 5 friends decided to contribute equally to pay for the rent. How much will each friend pay?

Answer

Given:

Total house rent = ₹14240
Number of friends = 5

Amount paid by each friend = Total house rent ÷ Number of friends

By substituting the values, we get:

Amount paid by each friend = 14240 ÷ 5 = 2848

∴ Each friend contributes ₹2848

A class of 36 students contributed ₹540 each for the picnic. Out of these, 7 students decided not to go. They were returned ₹490 each. What amount was spent on the picnic?

Answer

Given:

Total number of students = 36
Contribution per student = ₹540

Total amount collected = 36 x 540 = 19440

Number of students who did not go = 7
Amount returned per student = ₹490

Total amount returned = 7 x 490 = 3430

Amount spent on picnic = Total amount collected - Total amount returned

By substituting the values, we get:

Amount spent on picnic = 19440 - 3430 = 16010

∴ The amount spent on the picnic = ₹16010

The absolute value of 5 is

1. -5
2. $\dfrac{1}{5}$
3. 0
4. 5

Answer

Absolute value of 5 = |5| = 5

Hence, option 4 is the correct option.

The additive inverse of -7 is

1. 7
2. $\dfrac{1}{7}$
3. 1
4. 0

Answer

Since -7 + 7 = 0

∴ The additive inverse of -7 is 7

Hence, option 1 is the correct option.

The sum of two integers is -8. If one of them is 5, then the other is

1. 3
2. 13
3. -3
4. -13

Answer

Given:

Sum of two integers = -8
One integer = 5
Let the other integer be x

According to the question, the equation can be written as:

5 + x = -8
x = -8 - 5 = -(8 + 5) $\hspace{2cm}$[Solve for x]
x = -13

∴ The other integer is -13

Hence, option 4 is the correct option.

The successor of -41 is

1. -42
2. 41
3. 40
4. -40

Answer

To find the successor, add 1 to the given integer

-41 = -41 + 1 = -40

∴ The successor of -41 is -40

Hence, option 4 is the correct option.

The value of (-7) x 6 + (-7) x 14 is

1. 42
2. -84
3. -140
4. -196

Answer

Given expression:

(-7) x 6 + (-7) x 14

= -7 x (6 + 14) $\hspace{2cm}$[a x (b + c) = (a x b) + (a x c) ⇒ Distributive property]

=-7 x 20 $\hspace{3cm}$[Simplifying ( )]

= -140

∴ The value of (-7) x 6 + (-7) x 14 is -140

Hence, option 3 is the correct option.

Fill in the blanks :

(i) The absolute value of any integer can never be ............... .

(ii) The sum of any integer and its additive inverse is always ............... .

(iii) a x (b + c) = (a x b) + (a x c) is ............... property of multiplication over addition.

(iv) The product of two integers having unlike signs is always a ............... integer.

(v) The additive inverse of an integer a has the ............... sign as a.

Answer

(i) The absolute value of any integer can never be negative.

(ii) The sum of any integer and its additive inverse is always 0.

(iii) a x (b + c) = (a x b) + (a x c) is distributive property of multiplication over addition.

(iv) The product of two integers having unlike signs is always a negative integer.

(v) The additive inverse of an integer a has the opposite sign as a.

State True or False :

For any integer a, the multiplicative inverse is 1.

Answer

False

Reason

The multiplicative inverse of an integer a is $\dfrac{1}{a}$ not a (except when a = 1)

State True or False :

The sum of two integers having unlike signs is always a positive integer.

Answer

False

Reason

The sign of the sum depends on the absolute value of the integers. If the negative integer has a greater absolute value, the sum will be negative. If the positive integer has a greater absolute value, the sum will be positive.

State True or False :

The multiplicative inverse of an integer is never an integer.

Answer

False

Reason

The multiplicative inverse of 1 is 1 and of −1 is −1, which are integers. However, for all other integers, the inverse is a fraction. Hence, it is not true for all integers.

State True or False :

Any integer multiplied to its multiplicative inverse always gives the multiplicative identity.

Answer

True

Reason

By definition, $a \times \dfrac{1}{a} = 1 \text{ for } (a \neq 0).$ Since 1 is the multiplicative identity, this statement is correct.

State True or False :

Every integer has a multiplicative inverse.

Answer

False

Reason

0 does not have a multiplicative inverse because division by zero is not defined.

State True or False :

The quotient of two integers with unlike signs is always negative.

Answer

True

Reason

The rules for division are the same as multiplication. When you divide a positive integer by a negative one (or vice versa), the result is always negative.

Aditya went to Australia during the holidays of his children. They visited the coral reef and they were all excited to go for scuba diving. Aditya dived 36 feet to reach the brain coral as his instructor had directed him. He then rose by 19 feet to travel over a ridge.

(1) What is the depth of the ridge ?

1. 57 feet
2. 19 feet
3. 17 feet
4. 45 feet

(2) Aditya again dived 58 feet to reach the base of the reef. What is the depth of the reef ?

1. 32 feet
2. 41 feet
3. 58 feet
4. 75 feet

(3) In order to see an underwater cave, Aditya rose 26 feet. What is his location with respect to sea-level ?

1. 49 feet
2. 101 feet
3. 26 feet
4. 9 feet

(4) If a whale moving at a depth of 21 feet from the sea level passed over Aditya, what was its distance from Aditya ?

1. 64 feet
2. 49 feet
3. 53 feet
4. 28 feet

Answer

(1) Given:

Initial depth = -36 feet $\hspace{2cm}$(36 feet below sea level)

He rises 19 feet = +19

Depth = -36 + 19 = -17 $\hspace{2cm}$(17 feet below sea level)

Depth = 17 feet

Hence, option 3 is the correct option.

(2) Starting position = -17 feet $\hspace{2cm}$(from the previous step)

Again he dives 58 feet = -58 feet $\hspace{2cm}$(Given)

Depth = -17 + (-58)
= -17 - 58
= -(17 + 58) = -75 feet

Depth = 75 feet

Hence, option 4 is the correct option.

(3) Starting position = -75 feet $\hspace{2cm}$(from the previous step)

He rises by 26 feet = +26 $\hspace{3cm}$(Given)

Location = -75 + 26 = -49

Location = 49 feet below the sea level

Hence, option 1 is the correct option.

(4) Given:

Whale's position: -21 feet
Aditya's position: -49 feet (from the previous step)

Distance of Whale from Aditya = -21 - (-49)

= -21 + 49 = 28

Distance of Whale from Aditya = 28 feet

Hence, option 4 is the correct option.

Tanvi had a keen interest to invest in share market. So, she took lessons from an investment coach. The coach told her to watch the fluctuations in the value of share over a period of time. She found that on April 1, the price of a share of XYZ company was ₹2552.

(1) On April 2, the price of this share changed by gaining ₹37. What was the price of each share of XYZ on April 2 ?

1. ₹2515
2. ₹2589
3. ₹2562
4. ₹2573

(2) On April 3, the price changed by losing ₹16 by 12 pm and then again losing ₹8 by the end of the day. What was the price of each share of XYZ by the end of the day on April 3 ?

1. ₹2528
2. ₹2565
3. ₹2576
4. ₹2603

(3) On April 4, the price of XYZ company's share changed by gaining ₹11 by 12 pm and then again gaining ₹14 by the end of the day. What was the price of each share of XYZ by the end of the day on April 4?

1. ₹2540
2. ₹2564
3. ₹2590
4. ₹2614

(4) On April 5, the price of the share changed by losing ₹13 by 12 pm and then gaining ₹4 by the end of the day. What was the price of each share of XYZ by the end of the day on April 5 ?

1. ₹2599
2. ₹2607
3. ₹2612
4. ₹2581

Answer

(1) Price on April 2

Given:

Opening price (April 1) = ₹2552

Change = gain of ₹37

Price on April 2 = ₹2552 + ₹37 = ₹2589

Hence, option 2 is the correct option.

(2) Price by the end of April 3

Given:

Change 1 = Loss of ₹16
Change 2 = Loss of ₹8
Total loss = ₹16 + ₹8 = ₹24

Price on April 2 = ₹2589 $\hspace{2cm}$(from the previous step)

Price by the end of April 3 = Price on April 2 - Total loss

Substituting the values, we get:

Price by the end of April 3 = ₹2589 - ₹24 = ₹2565

Hence, option 2 is the correct option.

(3) Price by the end of April 4

Given:

Change 1 = Gain of ₹11
Change 2 = Gain of ₹14
Total Gain = ₹11 + ₹14 = ₹25

Price on April 3 = ₹2565 $\hspace{2cm}$(from the previous step)

Price by the end of April 4 = Price on April 3 + Total Gain

Substituting the values, we get:

Price by the end of April 4 = ₹2565 + ₹25 = ₹2590

Hence, option 3 is the correct option.

(4) Price by the end of April 5

Given:

Change 1 = Loss of ₹13
Change 2 = Gain of ₹4

Price on April 4 = ₹2590 $\hspace{2cm}$(from the previous step)

Price by the end of April 5 = ₹2590 - ₹13 + ₹4 = ₹2581

Hence, option 4 is the correct option.

Assertion: Difference of two negative integers cannot be a positive integer.

Reason: For any two integers a and b, a - b = a + (additive inverse of b)

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is false but Reason (R) is true.

Explanation

The assertion is false because the difference of two negative integers can be positive.

Example:

(-3) - (-7) = -3 + 7 = 4
which is positive.

The reason is true because subtraction of integers is defined as adding the additive inverse:

a - b = a + (-b)

Hence, option 4 is the correct option.

Assertion: Product of three negative integers and a positive integer is negative.

Reason: (-1) x (-1) x (-1) x ............... n times = -1, if n is odd.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation

Product of three negative integers:

(negative x negative x negative) = (negative)

So, product of three negative integers is negative.

Multiplying this by a positive integer keeps it negative:

(negative result x positive) = (negative)

The reason correctly explains that an odd number of negative integers gives a negative result.

Hence, option 1 is the correct option.