Integers

(i) Gain of ₹ 90

The opposite of "gaining" something (an increase) is "losing" it (a decrease).

Hence, opposite of "gain of ₹ 90" is "loss of ₹ 90".

(ii) 70 m above sea-level

"Above" and "below" are opposite directions on a vertical axis.

Hence, opposite of "70 m above sea-level" is "70 m below sea-level".

(iii) 200 m due west

"West" and "east" are opposite directions on a horizontal axis.

Hence, opposite of "200 m due west" is "200 m due east".

(iv) 25% increase

"Increase" means to get larger, while "decrease" means to get smaller.

Hence, opposite of "25% increase" is "25% decrease".

(v) 10°C below freezing point

"Below" and "above" are opposite directions on a temperature scale.

Hence, opposite of “10°C below the freezing point” is “10°C above the freezing point”.

(vi) 30 m to the left

"Left" and "right" are opposite directions.

Hence, opposite of 30 m to the left is 30 m to the right.

(vii) Below average

"Below" and "above" are used to describe values relative to a reference point (the average).

Hence, opposite of below average is above average.

(viii) Gaining 1 kg of weight

"Gaining" weight is the opposite of "losing" weight.

Hence, opposite of gaining 1 kg of weight is losing 1 kg of weight.

(ix) Depositing ₹ 500 in a bank

"Depositing" money increases a bank balance, while "withdrawing" decreases it.

Hence, opposite of depositing ₹ 500 in a bank is withdrawing ₹ 500 from a bank.

(x) +45

The opposite of a positive number is its negative counterpart.

Hence, opposite of +45 is -45.

(xi) -50

The opposite of a negative number is its positive counterpart.

Hence, opposite of -50 is +50.

(i) An increase of 8

"Increase" means to get larger, while "decrease" means to get smaller.

Hence, an increase of 8 can be indicated using a negative sign as a decrease of -8.

(ii) Loss of ₹ 300

A loss is a negative change. This is represented by -₹ 300.

Hence, loss of ₹ 300 is -₹ 300.

(iii) Spending ₹ 725

Spending money decreases a total amount, so it is a negative change. This is represented by -₹ 725.

Hence, spending ₹ 725 is -₹ 725.

(iv) 53 m below sea-level

"Below" is a negative direction. This is represented by -53 m.

Hence, 53 m below sea-level is -53 m.

(v) 20°C below freezing point

"Below" is a negative value on a temperature scale. This is represented by -20°C.

Hence, 20°C below freezing point is -20°C.

(vi) 4 above average

4 above average is a positive change. Its opposite, 4 below average, is represented by -4.

Hence, negative sign of 4 above average is -4.

(i) Integers between 3 and 10

4, 5, 6, 7, 8, 9.

(ii) Integers between -3 and 6

-2, -1, 0, 1, 2, 3, 4, 5.

(iii) Integers between -7 and 0

-6, -5, -4, -3, -2, -1.

(iv) Integers between -12 and -8

-11, -10, -9.

(v) Integers between -20 and -14

-19, -18, -17, -16, -15.

(vi) Integers between -8 and 4

-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3.

Seven integers, each greater than -6 are -5, -4, -3, -2, -1, 0, 1.

(i) -5 ............... 3

A negative number is always less than a positive number.

-5 < 3

(ii) 0 ............... -4

0 is to the right of -4 on the number line.

0 > -4

(iii) -8 ............... -14

On a number line, -8 is to the right of -14, making it the greater number.

-8 > -14

(iv) -23 ............... -16

-16 is closer to zero than -23, so -16 is greater.

-23 < -16

(v) 15 ............... -15

A negative number is always less than a positive number.

15 > -15

(vi) -1 ............... -12

-1 is closer to zero than -12, so -1 is greater.

-1 > -12

(vii) 9 ............... -1

A negative number is always less than a positive number.

9 > -1

(viii) -9 ............... 1

A negative number is always less than a positive number.

(-9) < 1

(ix) -9 ............... -1

-1 is closer to zero than -9, so -1 is greater.

(-9) < (-1)

(i) |-17| - |8|

= 17 - 8

= 9

Hence, |-17| - |8| = 9.

(ii) |-14| + |5|

= 14 + 5

= 19

Hence, |-14| + |5| = 19.

(iii) |-9| + |-6|

= 9 + 6

= 15

Hence, |-9| + |-6| = 15.

(iv) |-23| - |-12|

= 23 - 12

= 11

Hence, |-23| - |-12| = 11.

(v) |-8| + |-6| - |-10|

= 8 + 6 - 10

= 14 - 10

= 4

Hence, |-8| + |-6| - |-10| = 4.

(vi) |-6| + |-1| - |-5|

= 6 + 1 - 5

= 7 - 5

= 2

Hence, |-6| + |-1| - |-5| = 2.

(i) 14, -7, -13, 0, 6, -2, 30

Ascending order: -13 < -7 < -2 < 0 < 6 < 14 < 30.

(ii) -18, 14, -25, 32, -5, 2

Ascending order: -25 < -18 < -5 < 2 < 14 < 32.

(iii) -9, -24, 26, -2, -11, 18, 9

Ascending order: -24 < -11 < -9 < -2 < 9 < 18 < 26.

(iv) 37, -28, -15, 7, 23, -19, -6

Ascending order: -28 < -19 < -15 < -6 < 7 < 23 < 37.

(i) -16, -21, 8, 0, -7, 14

Descending order: 14 > 8 > 0 > -7 > -16 > -21.

(ii) -4, 31, -41, -23, 6, 0, -11

Descending order: 31 > 6 > 0 > -4 > -11 > -23 > -41.

(iii) -60, -9, -36, -8, 5, 15, 0

Descending order: 15 > 5 > 0 > -8 > -9 > -36 > -60.

(iv) -10, -43, -2, 1, -14, 6, 22

Descending order: 22 > 6 > 1 > -2 > -10 > -14 > -43.

(v) -8, -53, 53, -100, 62, 16, -82

Descending order: 62 > 53 > 16 > -8 > -53 > -82 > -100.

(i) All integers from -5 to 3.

(ii) All negative integers greater than -6.

(iii) All non-negative integers less than 6.

(iv) All even positive integers less than 12.

(v) All odd negative integers greater than -8.

(i) 16 + 21

= 37

Hence, 16 + 21 = 37.

(ii) (-16) + 21

= -16 + 21

= 5

Hence, (-16) + 21 = 5.

(iii) 16 + (-21)

= 16 - 21

= -5

Hence, 16 + (-21) = -5.

(iv) (-25) + (-18)

= -25 - 18

= -43

Hence, (-25) + (-18) = -43.

(v) (-32) + (-47)

= -32 - 47

= -79

Hence, (-32) + (-47) = -79.

(vi) 54 + (-89)

= 54 - 89

= -35

Hence, 54 + (-89) = -35.

(vii) (-38) + 45

= -38 + 45

= 7

Hence, (-38) + 45 = 7.

(viii) 96 + (-103)

= 96 - 103

= -7

Hence, 96 + (-103) = -7.

(ix) (-150) + (-15)

= -150 - 15

= -165

Hence, (-150) + (-15) = -165.

(i) 34

Let x be additive inverse of 34.

⇒ 34 + x = 0

⇒ x = 0 - 34

⇒ x = -34

Hence, -34 is the additive inverse of 34.

(ii) -58

Let x be additive inverse of -58.

⇒ -58 + x = 0

⇒ x = 0 - (-58)

⇒ x = 58

Hence, 58 is the additive inverse of -58.

(iii) 0

Let x be additive inverse of 0.

⇒ 0 + x = 0

⇒ x = 0 - 0

⇒ x = 0

Hence, 0 is the additive inverse of 0.

(iv) -1

Let x be additive inverse of -1.

⇒ -1 + x = 0

⇒ x = 0 - (-1)

⇒ x = 1

Hence, 1 is the additive inverse of -1.

(v) 170

Let x be additive inverse of 170.

⇒ 170 + x = 0

⇒ x = 0 - 170

⇒ x = -170

Hence, -170 is the additive inverse of 170.

(i) 101

Successor of 101 = 101 + 1 = 102

Hence, 102 is the successor of 101.

(ii) -47

Successor of -47 = (-47) + 1 = -46

Hence, -46 is the successor of -47.

(iii) -1

Successor of -1 = (-1) + 1 = 0

Hence, 0 is the successor of -1.

(iv) -80

Successor of -80 = (-80) + 1 = -79

Hence, -79 is the successor of -80.

(v) -301

Successor of -301 = (-301) + 1 = -300

Hence, -300 is the successor of -301.

(i) 40

Predecessor of 40 = 40 - 1 = 39

Hence, 39 is the predecessor of 40.

(ii) -32

Predecessor of -32 = -32 - 1 = -33

Hence, -33 is the predecessor of -32.

(iii) -70

Predecessor of -70 = -70 - 1 = -71

Hence, -71 is the predecessor of -70.

(iv) -91

Predecessor of -91 = -91 - 1 = -92

Hence, -92 is the predecessor of -91.

(v) 0

Predecessor of 0 = 0 - 1 = -1

Hence, -1 is the predecessor of 0.

(i) (15) - (21)

= 15 - 21

= -6

Hence, (15) - (21) = -6.

(ii) (-29) - (9)

= -29 - 9

= -38

Hence, (-29) - (9) = -38.

(iii) 70 - (-8)

= 70 + 8

= 78

Hence, 70 - (-8) = 78.

(iv) 24 - (-24)

= 24 + 24

= 48

Hence, 24 - (-24) = 48.

(v) (-36) - (64)

= -36 - 64

= -100

Hence, (-36) - (64) = -100.

(vi) 0 - (-20)

= 0 + 20

= 20

Hence, 0 - (-20) = 20.

(vii) (-63) - (-7)

= -63 + 7

= -56

Hence, (-63) - (-7) = -56.

(viii) (-80) - (-20)

= -80 + 20

= -60

Hence, (-80) - (-20) = -60.

(ix) (-12) - (-71)

= -12 + 71

= 59

Hence, (-12) - (-71) = 59.

(i) -180 from 180

= 180 - (-180)

= 180 + 180

= 360.

Hence, final result = 360.

(ii) 75 from -75

= -75 - 75

= -150

Hence, final result = -150.

(iii) -630 from -70

= -70 - (-630)

= -70 + 630

= 560

Hence, final result = 560.

(iv) -95 from 0

= 0 - (-95)

= 0 + 95

= 95

Hence, final result = 95.

(v) -90 from -1

= -1 - (-90)

= -1 + 90

= 89

Hence, final result = 89.

(vi) -16 from -25

= -25 - (-16)

= -25 + 16

= -9

Hence, final result = -9.

(i) (+23) + (...............) = 0

Let x be the number in blanks,

⇒ 23 + x = 0

⇒ x = 0 - 23

⇒ x = -23

(+23) + (-23) = 0

(ii) (-13) + (...............) = + 20

Let x be the number in blanks,

⇒ -13 + x = 20

⇒ x = 20 - (-13)

⇒ x = 20 + 13

⇒ x = 33

(-13) + (33) = + 20

(iii) (-14) + (...............) = -34

Let x be the number in blanks,

⇒ -14 + x = -34

⇒ x = -34 + 14

⇒ x = -20

(-14) + (-20) = -34

(iv) (-20) + (...............) = -8

Let x be the number in blanks,

⇒ -20 + x = -8

⇒ x = -8 + 20

⇒ x = 12

(-20) + (12) = -8

(v) (-30) - (...............) = -14

Let x be the number in blanks,

⇒ -30 - x = -14

⇒ -x = -14 + 30

⇒ -x = 16

⇒ x = -16

(-30) - (-16) = -14

(vi) (...............) - (-25) = 2

Let x be the number in blanks,

⇒ x + 25 = 2

⇒ x = 2 - 25

⇒ x = -23

(-23) - (-25) = 2

(i) 5 + 4 = 9

Starting from 0 on the number line, move 5 steps to the right and from there move 4 steps to the right to reach 9 as shown below.

(ii) 4 + (-6) = 4 - 6 = -2

Starting from 0 on the number line, move 4 steps to the right and from there move 6 steps to the left to reach -2 as shown below.

(iii) (-4) + 8 = -4 + 8 = 4

Starting from 0 on the number line, move 4 steps to the left and from there move 8 steps to the right to reach 4 as shown below.

(iv) (-5) + 3 = -2

Starting from 0 on the number line, move 5 steps to the left and from there move 3 steps to the right to reach -2 as shown below.

(v) (-3) + (-5) = -3 - 5 = -8

Starting from 0 on the number line, move 3 steps to the left and from there move 5 steps to the left to reach -8 as shown below.

(vi) (-6) + (-3) = -6 - 3 = -9

Starting from 0 on the number line, move 6 steps to the left and from there move 3 steps to the left to reach -9 as shown below.

(vii) 5 - (-2) = 5 + 2 = 7

Starting from 0 on the number line, move 5 steps to the right and from there move 2 steps to the right to reach 7 as shown below.

(viii) (-4) - 5 = -4 - 5 = -9

Starting from 0 on the number line, move 4 steps to the left and from there move 5 steps to the left to reach -9 as shown below.

(ix) 4 - (-4) = 4 + 4 = 8

Starting from 0 on the number line, move 4 steps to the right and from there move 4 steps to the right to reach 8 as shown below.

(i) The sum of two integers is always an integer.

True

Explanation:

The set of integers is closed under addition. This means that when you add any two integers, the result will always be another integer. For example, 5 + (−3) = 2, and 2 is an integer.

(ii) The difference of two integers is always an integer.

True

Explanation:

The set of integers is also closed under subtraction. When you subtract one integer from another, the result is always an integer. For example, 3−8=−5, and -5 is an integer.

(iii) The absolute value of every integer is a positive integer.

True

Explanation:

The absolute value of an integer is its distance from zero, which is always non-negative.

(iv) 0 is the smallest positive integer.

False

Explanation:

The set of positive integers starts with 1. Zero is an integer that is neither positive nor negative. The smallest positive integer is 1.

(v) Every whole number is an integer.

True

Explanation:

The set of whole numbers is {0, 1, 2, 3, ...}. The set of integers is {..., -2, -1, 0, 1, 2, ...}. All the numbers in the set of whole numbers are also included in the set of integers.

(vi) The sum of two integers can never be zero.

False

Explanation:

The sum of an integer and its additive inverse (its opposite) is always zero. For example, the sum of 5 and -5 is 0.

Let the number be x.

∴ 15 + x = -15

⇒ x = -15 - 15

⇒ x = -30

Hence, -30 should be added to 15 to get -15.

Let the number be x.

∴ -3 - x = 18

⇒ -x = 18 + 3

⇒ -x = 21

⇒ x = -21

Hence, -21 should be subtracted from (-3) to get +18.

Given, sum of two integers = -23

One of the numbers = 12

Let x be the number.

∴ 12 + x = -23

⇒ x = -23 - 12

⇒ x = -35

Hence, the other number = -35.

Given,

Amit lost in the first game = -70 points

Amit lost in the second game = -50 points

Amit lost in the third game = -35 points

Amit gained in the fourth game = +60 points

Amit gained in the fifth game = +80 points

Net loss or gain = (-70) + (-50) + (-35) + 60 + 80

= -70 - 50 - 35 + 60 + 80

= -120 - 35 + 60 + 80

= -155 + 60 + 80

= -95 + 80

= -15

Hence, Amit had a net loss of 15 points.

Initial temperature: 2° C

Final temperature: −3° C

Difference = Initial temperature − Final temperature

= 2° C − (−3° C)

= 2° C + 3° C

= 5° C

Hence, the temperature fell by 5° C.

Given, travelled east: +100 km

Travelled west: −130 km

Final position = (+ 100) km + (− 130) km

= 100 − 130

= −30 km

The negative sign indicates that the final position is to the west of Delhi. The distance from Delhi is the absolute value of this number.

Distance = ∣−30∣ km

Distance = 30 km

Hence, the car was finally 30 km to the west of Delhi.

Given, 6 more than -3

⇒ (-3) + 6

⇒ -3 + 6

⇒ 3

Hence, option 3 is the correct option.

Given, 5 less than -8

⇒ (-8) - 5

⇒ -8 - 5

⇒ -13

Hence, option 2 is the correct option.

Given, 8 + |-15|

⇒ 8 + 15

⇒ 23

Hence, option 4 is the correct option.

Given, (-36) + (7)

⇒ -36 + 7

⇒ -29

Hence, option 1 is the correct option.

Given, (-28) - (-6)

⇒ -28 + 6

⇒ -22

Hence, option 2 is the correct option.

The successor of -16 = -16 + 1

⇒ -15.

Hence, option 3 is the correct option.

The predecessor of -13 = -13 - 1

⇒ -14

Hence, option 1 is the correct option.

Let the additive inverse of 5 be x.

∴ x + 5 = 0

⇒ x = 0 - 5

⇒ x = -5

Hence, option 2 is the correct option.

Given, 12 - (-8)

⇒ 12 + 8

⇒ 20

Hence, option 3 is the correct option.

Given, subtracting -7 from 0

⇒ 0 - (-7)

⇒ 0 + 7

⇒ 7

Hence, option 2 is the correct option.

Given, (-8) + (-9) + 13 + (-14)

⇒ -8 - 9 + 13 - 14

⇒ -8 + 4 - 14

⇒ -4 - 14

⇒ -18

Hence, option 1 is the correct option.

Given, sum of two integers = -22.

One number = 8

Let the other number be x.

∴ 8 + x = -22

⇒ x = -22 - 8

⇒ x = -30

Hence, option 3 is the correct option.

Let the number be x.

∴ 8 + x = -15

⇒ x = -15 - 8

⇒ x = -23

Hence, option 3 is the correct option.

Let the number be x.

∴ (-3) - x = 15

⇒ -x = 15 + 3

⇒ -x = 18

⇒ x = -18.

Hence, option 4 is the correct option.

Position of P = -10

Position of Q = 10

Change in moving from point P to point Q = Final position - Initial position

= Q - P = 10 - (-10) = 20.

Hence, option 1 is the correct option.

Position of R = 15

Position of P = -10

Change = Final position - Initial position

= P - R = (-10) - (15) = -25.

Hence, option 2 is the correct option.

Change = Final position - Initial position = S - P

We need,

S - (-10) = 15

S + 10 = 15

S = 15 - 10 = 5.

Hence, option 1 is the correct option.

Change = Final position - Initial position = T - R = -30

T = -30 + R

T = -30 + 15 = -15.

Hence, option 3 is the correct option.

(i) -36 + 8 = -28.

(ii) -36 - 8 = -44.

(iii) -46 - (-6) = -40.

(iv) 53 - (-10) = 63.

(v) 0 - (-13) = 13.

(vi) 9 + 1 - 61 = -51.

False

Reason

The number 0 is an integer, but it is neither positive nor negative.

Positive integers are numbers greater than 0 (1, 2, 3, ...).

Negative integers are numbers less than 0 (-1, -2, -3, ...).

Zero (0) is the integer that separates the positive and negative integers.

True

Reason

The freezing point of water is 0°C. "10°C below freezing point" means 10 degrees less than 0°C, which is expressed as −10°C.

True

Reason

The negative of a negative number is a positive number. Therefore, −(−15) is equal to +15.

True

Reason

The predecessor of a number is the number immediately before it.

Predecessor of (−9) = −9 − 1 = −10.

True

Reason

The expression "subtracting -40 from 40" can be written as:

⇒ 40 − (−40)

⇒ 40 + 40

⇒ 80

True

Reason

The successor of a number is the number that comes directly after it. To find the successor, you add 1 to the number.

Successor of (−1) = −1 + 1 = 0.

False

Reason

The absolute value of −15 is 15.

⇒ ∣−15∣ + 10

⇒ 15 + 10

⇒ 25

The reason states that for any integer a, its additive inverse is −a.

This is a fundamental definition in mathematics. The sum of an integer and its additive inverse is always zero [a + (−a) = 0].

∴ Reason (R) is true.

Let the unknown integer be x.

According to the Reason (R), its additive inverse is −x.

The assertion states that the integer x is 2 less than its additive inverse. We can write this as an equation:

⇒ x = (−x) − 2

To solve for x, we can add x to both sides of the equation:

⇒ x + x = −2

⇒ 2x = −2

Dividing both sides by 2 gives:

⇒ x = −1

The integer is indeed −1.

∴ Assertion (A) is true.

∴ Both A and R are true and R is the correct explanation of A.

Hence, option 1 is the correct option.

According to Assertion: All whole numbers are integers.

By definition, whole numbers are the set {0, 1, 2, 3, ...}. Integers are the set {..., -3, -2, -1, 0, 1, 2, 3, ...}. Since all whole numbers are included in the set of integers.

∴ Assertion (A) is true.

According to Reason: On the number line, all the integers on the left of 0 are negative.

This is a fundamental property of the number line; numbers to the left of zero are negative, and numbers to the right are positive.

∴ Reason (R) is true.

Both A and R are true but R is not the correct explanation of A.

Hence, option 2 is the correct option.