Integers

Some integers are marked on the following number line:

(i) Write these integers in ascending order.

(ii) Write these integers in descending order.

(iii) Few dots have been marked on the above number line. Write appropriate integer at each dot.

Answer

The integers marked on the number line are: −18, −9, −4, 0, 3, 8, 12.

(i) The integers in ascending order are:

−18, −9, −4, 0, 3, 8, 12

(ii) The integers in descending order are:

12, 8, 3, 0, −4, −9, −18

(iii) The dots marked on the number line represent the following integers:

-17, −14, −11, -7, -5, −3, 1, 4, 7, 9, 11

Arrange 7, -5, 4, 0 and -4 in ascending order and mark them on a number line to check your answer.

Answer

Given negative integers are −5, −4.
In ascending order they are −5 < −4.

Given positive integers are 7, 4.
In ascending order they are 4 < 7.

Also, 0 lies between negative integers and positive integers.

Hence, the given integers in ascending order are:
−5 < −4 < 0 < 4 < 7.

i.e., −5, −4, 0, 4, 7.

On the number line, marking these integers:

−5 lies to the left of −4, which lies to the left of 0, which lies to the left of 4, which lies to the left of 7. This confirms our ascending order.

In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Rohit's scores in five successive rounds were 15, -3, -7, 12 and 8, what was his total at the end?

Answer

Rohit's scores in five successive rounds = 15, −3, −7, 12 and 8

Total score = 15 + (−3) + (−7) + 12 + 8

Grouping positives and negatives:

= (15 + 12 + 8) + ((−3) + (−7))

= 35 + (−10)

= 35 − 10

= 25

∴ Rohit's total score at the end = 25

Ruchi deposited ₹ 4370 in her account on Monday and then withdrew ₹ 2875 on Tuesday. Next day she deposited ₹ 1550. What was her balance on Thursday?

Answer

Amount deposited on Monday = ₹ 4370
Amount withdrawn on Tuesday = ₹ 2875 (denoted by −2875)
Amount deposited on Wednesday = ₹ 1550

Balance on Thursday = ₹ 4370 + ₹ (−2875) + ₹ 1550

= ₹ 4370 − ₹ 2875 + ₹ 1550

= ₹ 1495 + ₹ 1550

= ₹ 3045

∴ Ruchi's balance on Thursday = ₹ 3045

Evaluate the following:

(i) |-13| - |9|

(ii) |13 - 5| - |-9|

(iii) |35 - 21| - |8 - 3|

Answer

(i) |−13| − |9|

= 13 − 9 $\hspace{2cm}$ [∵ |−13| = 13 and |9| = 9]

= 4

(ii) |13 − 5| − |−9|

= |8| − |−9|

= 8 − 9 $\hspace{2cm}$ [∵ |8| = 8 and |−9| = 9]

= −1

(iii) |35 − 21| − |8 − 3|

= |14| − |5|

= 14 − 5 $\hspace{2cm}$ [∵ |14| = 14 and |5| = 5]

= 9

Arrange the following integers in ascending order:

-39, 35, -102, 0, -51, -5, -6, 7

Answer

Given negative integers are −39, −102, −51, −5, −6.
In ascending order they are −102 < −51 < −39 < −6 < −5.

Given positive integers are 35, 7.
In ascending order they are 7 < 35.

Also, 0 lies between negative integers and positive integers.

Hence, all the given integers in ascending order are:
−102 < −51 < −39 < −6 < −5 < 0 < 7 < 35.

i.e., −102, −51, −39, −6, −5, 0, 7, 35.

Arrange the following integers in descending order:

-31, 139, -203, -97, 0, 4208

Answer

Given positive integers are 139, 4208.
In descending order they are 4208 > 139.

Given negative integers are −31, −203, −97.
In descending order they are −31 > −97 > −203.

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

Hence, all the given integers in descending order are:
4208 > 139 > 0 > −31 > −97 > −203.

i.e., 4208, 139, 0, −31, −97, −203.

State whether each of the following statement is true or false:

(i) 0 is the successor of -1 in integers

(ii) 0 has no predecessor in integers

(iii) -2 is the predecessor of -1

(iv) 0 is greater than every negative integer.

Answer

(i) True
Reason — In the set of integers, the successor is found by adding 1. Since −1 + 1 = 0, 0 is indeed the successor of -1.

(ii) False
Reason — Unlike the set of whole numbers or natural numbers, every integer has a predecessor. The predecessor of 0 is -1 (found by calculating 0 - 1).

(iii) True
Reason — The predecessor of a number is the integer immediately to its left on the number line, or the value found by subtracting 1.

Since -1 - 1 = -2, -2 is the predecessor of -1.

(iv) True
Reason — On a standard horizontal number line, 0 is located to the right of all negative integers, making it greater than any negative value.

Use the sign >, < or = in the box to make the following statements true:

(i) (-11) + (-7) $\boxed{\phantom{653}}$ (-11) - (-7)

(ii) 23 - 41 + 11 $\boxed{\phantom{653}}$ 23 - 41 - 11

(iii) 40 - (-39) + (-5) $\boxed{\phantom{653}}$ 40 + (-39) - (-5)

(iv) (-3) + 13 - (15) $\boxed{\phantom{653}}$ 25 - (-2) + (-33)

Answer

(i) LHS = (−11) + (−7) = −(11 + 7) = −18
RHS = (−11) − (−7) = −11 + 7 = −4

Since −18 < −4,

∴ (-11) + (-7) $\boxed{ \lt }$ (-11) - (-7)

(ii) LHS = 23 − 41 + 11 = 34 − 41 = −7
RHS = 23 − 41 − 11 = 23 − 52 = −29

Since −7 > −29,

∴ 23 - 41 + 11 $\boxed{ \gt }$ 23 - 41 - 11

(iii) LHS = 40 − (−39) + (−5) = 40 + 39 − 5 = 74
RHS = 40 + (−39) − (−5) = 40 − 39 + 5 = 6

Since 74 > 6,

∴ 40 - (-39) + (-5) $\boxed{ \gt }$ 40 + (-39) - (-5)

(iv) LHS = (−3) + 13 − 15 = 10 − 15 = −5
RHS = 25 − (−2) + (−33) = 25 + 2 − 33 = −6

Since −5 > −6,

∴ (-3) + 13 - (15) $\boxed{ \gt }$ 25 - (-2) + (-33)

Write a pair of integers whose:

(i) sum is -3

(ii) difference is -5

(iii) difference is 4

Answer

(i) Sum is -3

One such pair is -5 and 2

∵ -5 + (2) = −3

(ii) Difference is -5

One such pair is -2 and 3

∵ (−2) - (3) = −2 - 3 = −5

(iii) Difference is 4

One such pair is -7 and -11

∵ (−7) − (−11) = −7 + 11 = 4

In a quiz, team A scored -30, 20, 0 and team B scored 20, 0, -30 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?

Answer

Total score of Team A = (−30) + 20 + 0

= −10 + 0

= −10

Total score of Team B = 20 + 0 + (−30)

= 20 − 30

= −10

Since both teams scored −10,

∴ Both teams scored equal.

Yes, we can add integers in any order. This is because addition of integers is commutative as well as associative.

Find the sum of integers -72, 237, 84, 72, -184, -37

Answer

Sum = (−72) + 237 + 84 + 72 + (−184) + (−37)

Grouping suitable integers:

= [(−72) + 72] + [237 + 84] + [(−184) + (−37)]

= 0 + 321 + (−221)

= 321 − 221

= 100

∴ The required sum = 100.

Write two integers which are smaller than -3, but their difference is greater than -3

Answer

Let the two integers be −5 and −4.

Both −5 and −4 are smaller than −3.

Now, their difference:

(−5) − (−4) = −5 + 4 = -1

Since -1 > −3,

∴ The two integers are −5 and −4.

Find the following products:

(i) 7 × (-35)

(ii) (-13) × (-15)

(iii) (-12) × (-11) × (-10)

(iv) (-13) × 0 × (-24)

(v) (-1) × (-2) × (-3) × 4

(vi) (-3) × (-6) × (-2) × (-1)

Answer

(i) 7 × (-35)

= −(7 × 35)

= −245 $\hspace{1cm}$ [∵ Positive × Negative = Negative]

(ii) (−13) × (−15)

= +(13 × 15)

= 195 $\hspace{1cm}$ [∵ Negative × Negative = Positive]

(iii) (−12) × (−11) × (−10)

= [(−12) × (−11)] × (−10)

= 132 × (−10)

= −1320 $\hspace{1cm}$ [∵ Positive × Negative = Negative]

(iv) (−13) × 0 × (−24)

= 0 $\hspace{1cm}$ [∵ Anything multiplied with 0 becomes 0]

(v) (−1) × (−2) × (−3) × 4

= [(−1) × (−2)] × [(−3) × 4]

= 2 × (−12)

= −24 $\hspace{1cm}$ [∵ Positive × Negative = Negative]

(vi) (−3) × (−6) × (−2) × (−1)

= [(−3) × (−6)] × [(−2) × (−1)]

= 18 × 2 $\hspace{1cm}$ [∵ Negative × Negative = Positive]

= 36

Verify the following:

(i) 37 × [6 + (-3)] = 37 × 6 + 37 × (-3)

(ii) (-21) × [(-6) + (-4)] = (-21) × (-6) + (-21) × (-4)

Answer

(i) 37 × [6 + (−3)] = 37 × 6 + 37 × (−3)

LHS = 37 × [6 + (−3)]
= 37 × 3
= 111

RHS = 37 × 6 + 37 × (−3)
= 222 + (−111)
= 222 − 111
= 111

Since LHS = RHS,

∴ 37 × [6 + (−3)] = 37 × 6 + 37 × (−3) $\hspace{1cm}$ [Distributive Property]

(ii) (−21) × [(−6) + (−4)] = (−21) × (−6) + (−21) × (−4)

LHS = (−21) × [(−6) + (−4)]
= (−21) × (−10)
= 210

RHS = (−21) × (−6) + (−21) × (−4)
= 126 + 84
= 210

Since LHS = RHS,

∴ (−21) × [(−6) + (−4)] = (−21) × (−6) + (−21) × (−4) $\hspace{1cm}$ [Distributive Property]

Using suitable properties, evaluate the following:

(i) 8 × 53 × (-125)

(ii) (-8) × (-2) × 3 × (-5)

(iii) (-6) × 2 × (-8) × 5

(iv) 15 × (-25) × (-4) × (-10)

(v) 26 × (-48) + (-48) × (-36)

(vi) 724 × (-56) + (-724) × 44

(vii) (-47) × 102

(viii) (-39) × (-97)

Answer

(i) 8 × 53 × (−125)

= 53 × [8 × (−125)] $\hspace{1cm}$ [Using commutative and associative property]

= 53 × (−1000)

= −53000

(ii) (−8) × (−2) × 3 × (−5)

= [(−8) × (3)] × [(-2) × (−5)] $\hspace{1cm}$ [Using associative property]

= -24 × (10)

= −240

(iii) (−6) × 2 × (−8) × 5

= [(−6) × (−8)] × [2 × 5] $\hspace{1cm}$ [Using commutative and associative property]

= 48 × 10

= 480

(iv) 15 × (−25) × (−4) × (−10)

= 15 × [(−25) × (−4)] × (−10) $\hspace{1cm}$ [Using associative property]

= 15 × 100 × (−10)

= 1500 × (−10)

= −15000

(v) 26 × (−48) + (−48) × (−36)

= (−48) × 26 + (−48) × (−36) $\hspace{1cm}$ [Using commutative property]

= (−48) × [26 + (−36)] $\hspace{1cm}$ [Using distributive property]

= (−48) × (−10)

= 480

(vi) 724 × (−56) + (−724) × 44

= 724 × (−56) + 724 × (−44) $\hspace{1cm}$ [∵ (−724) × 44 = 724 × (−44)]

= 724 × [(−56) + (−44)] $\hspace{1cm}$ [Using distributive property]

= 724 × (−100)

= −72400

(vii) (−47) × 102

= (−47) × (100 + 2)

= (−47) × 100 + (−47) × 2 $\hspace{1cm}$ [Using distributive property]

= −4700 + (−94)

= −4700 − 94

= −4794

(viii) (−39) × (−97)

= (−39) × [(−100) + 3]

= (−39) × (−100) + (−39) × 3 $\hspace{1cm}$ [Using distributive property]

= 3900 + (−117)

= 3900 − 117

= 3783

Fill in the blanks to make the following true statements:

(i) (-4) × ... = 44

(ii) 7 × ... = -42

(iii) ... × (-13) = 143

(iv) (-5) × ... = 0

Answer

(i) (−4) × (−11) = 44

∵ 44 ÷ (-4) = -11

(ii) 7 × (−6) = −42

∵ -42 ÷ 7 = -6

(iii) (−11) × (−13) = 143

∵ 143 ÷ (-13) = -11

(iv) (−5) × 0 = 0

∵ Anything multiplied with 0 becomes 0.

A certain freezing process requires that room temperature be lowered from 32°C at the rate of 5°C every hour. What will be the room temperature 8 hours after the freezing process begins?

Answer

Initial room temperature = 32°C

Rate of decrease in temperature = 5°C per hour

Total decrease in temperature in 8 hours = 8 × 5°C = 40°C

Since the temperature is decreasing, change in temperature = −40°C

Room temperature after 8 hours = 32°C + (−40°C)

= 32° C − 40° C

= −8° C

∴ The room temperature 8 hours after the freezing process begins = −8° C

In a class test containing 10 questions, 5 marks are awarded for every correct answer and 2 marks are deducted for every incorrect answer and 0 for questions not attempted.

(i) Rohit gets four correct and six incorrect answers. What is his score?

(ii) Seema gets 5 correct and 5 incorrect answers. What is her score?

(iii) Ritu attempted 7 questions and gets only 2 correct answers. What is her score?

Answer

Marks given for each correct answer = 5

Marks deducted for each incorrect answer = −2

(i) Rohit's score:

Number of correct answers = 4

∴ Marks for correct answers: 4 x 5 = 20

Number of incorrect answers = 6

∴ Marks for incorrect answers: 6 x (-2) = -12

Rohit's score = Marks for correct answers + Marks for incorrect answers

= 20 + (−12)

= 20 − 12

= 8

∴ Rohit's score = 8

(ii) Seema's score:

Number of correct answers = 5

∴ Marks for correct answers: 5 x 5 = 25

Number of incorrect answers = 5

∴ Marks for incorrect answers: 5 x (-2) = -10

Seema's score = Marks for correct answers + Marks for incorrect answers

= 25 + (−10)

= 25 − 10

= 15

∴ Seema's score = 15

(iii) Ritu's score:

Ritu attempted 7 questions out of which 2 were correct.

∴ Marks for correct answers: 2 x 5 = 10

Number of incorrect answers = 7 − 2 = 5

∴ Marks for incorrect answers: 5 x (-2) = -10

Ritu's score = Marks for correct answers + Marks for incorrect answers

= 10 + (−10)

= 10 − 10

= 0

∴ Ritu's score = 0

Evaluate the following:

(i) (-36) ÷ (-9)

(ii) 150 ÷ (-25)

(iii) (-270) ÷ 27

(iv) (-59) ÷ 59

(v) 0 ÷ (-17)

(vi) (-784) ÷ (-56)

Answer

(i) (−36) ÷ (−9)

= $\dfrac{-36}{-9}$

= 4

(ii) 150 ÷ (−25)

= $\dfrac{150}{-25}$

= −6

(iii) (−270) ÷ 27

= $\dfrac{-270}{27}$

= −10

(iv) (−59) ÷ 59

= $\dfrac{-59}{59}$

= −1

(v) 0 ÷ (−17)

= $\dfrac{0}{-17}$

= 0 $\hspace{1cm}$ [∵ 0 divided by any non-zero integer is 0]

(vi) (−784) ÷ (−56)

= $\dfrac{-784}{-56}$

= 14

Evaluate the following:

(i) 13 ÷ [(-2) + 1]

(ii) (-47) ÷ [(-45) + (-2)]

(iii) [(-6) + 5] ÷ [(-2) + 1]

(iv) [(-48) ÷ (-6)] ÷ (-2)

Answer

(i) 13 ÷ [(−2) + 1]

= 13 ÷ (−1)

= −13

(ii) (−47) ÷ [(−45) + (−2)]

= (−47) ÷ (−47)

= 1

(iii) [(−6) + 5] ÷ [(−2) + 1]

= (−1) ÷ (−1)

= 1

(iv) [(−48) ÷ (−6)] ÷ (−2)

= 8 ÷ (−2)

= −4

Verify that (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) for a = -225, b = 15 and c = -3

Answer

Given: a = −225, b = 15 and c = −3

LHS = (a ÷ b) ÷ c

= [(−225) ÷ 15] ÷ (−3)

= (−15) ÷ (−3)

= 5

RHS = a ÷ (b ÷ c)

= (−225) ÷ [15 ÷ (−3)]

= (−225) ÷ (−5)

= 45

Since 5 ≠ 45,

∴ (a ÷ b) ÷ c ≠ a ÷ (b ÷ c) for a = −225, b = 15 and c = −3

This shows that division is not associative for integers.

Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for

(i) a = -10, b = 1 and c = 1

(ii) a = 12, b = 1 and c = -2

Answer

(i) Given: a = −10, b = 1 and c = 1

LHS = a ÷ (b + c)

= (−10) ÷ (1 + 1)

= (−10) ÷ 2

= −5

RHS = (a ÷ b) + (a ÷ c)

= [(−10) ÷ 1] + [(−10) ÷ 1]

= (−10) + (−10)

= −20

Since −5 ≠ −20,

∴ a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for a = −10, b = 1 and c = 1

(ii) Given: a = 12, b = 1 and c = −2

LHS = a ÷ (b + c)

= 12 ÷ [1 + (−2)]

= 12 ÷ (−1)

= −12

RHS = (a ÷ b) + (a ÷ c)

= (12 ÷ 1) + [12 ÷ (−2)]

= 12 + (−6)

= 12 − 6

= 6

Since −12 ≠ 6,

∴ a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for a = 12, b = 1 and c = −2

Fill in the blanks to make the following statements true:

(i) 239 ÷ ...... = 1

(ii) (-85) ÷ ...... = -1

(iii) (-213) ÷ ...... = 1

(iv) (-43) ÷ ...... = 43

(v) ...... ÷ (-21) = 4

(vi) (-66) ÷ ...... = -3

Answer

(i) 239 ÷ 239 = 1

(ii) (−85) ÷ 85 = −1

(iii) (−213) ÷ (−213) = 1

(iv) (−43) ÷ (−1) = 43

(v) (−84) ÷ (−21) = 4

(vi) (−66) ÷ 22 = −3

Explanation

(i) Any non-zero number divided by itself equals 1.

(ii) To result in a negative quotient, a negative number must be divided by a positive number.

(iii) A negative number divided by the same negative number equals 1.

(iv) Dividing a negative number by -1 changes its sign to positive.

(v) Multiply the quotient and divisor: 4 x (-21) = -84.

(vi) Divide the dividend by the quotient: (-66) ÷ (-3) = 22

In a competition 3 marks are given for every correct answer and (-2) marks are given for every incorrect answer and no marks for not attempting any question.

(i) Sachin scored 24 marks. If he got 14 correct answers, how many questions has he attempted incorrectly?

(ii) Nalini scores (-7) marks in this competition, though she has got 9 correct answers. How many questions she has attempted incorrectly?

Answer

Marks given for each correct answer = 3

Marks given for each incorrect answer = −2

(i) Sachin's total score = 24

Sachin's correct answers = 14

Marks for correct answers = 14 x 3 = 42

Marks obtained for incorrect answers = Total score − Marks for correct answers

= 24 − 42

= −18

Number of incorrect answers = Marks obtained for incorrect answers ÷ Marks given for each incorrect answer

= (−18) ÷ (−2)

= 18 ÷ 2

= 9

∴ Sachin attempted 9 questions incorrectly.

(ii) Nalini's total score = −7

Marks scored for 9 correct answers = 9 × 3 = 27

Marks obtained for incorrect answers = Total score − Marks for correct answers

= −7 − 27

= −34

Number of incorrect answers = Marks obtained for incorrect answers ÷ Marks given for each incorrect answer

= (−34) ÷ (−2)

= 34 ÷ 2

= 17

∴ Nalini attempted 17 questions incorrectly.

An elevator descends into a mine shaft at the rate of 6 m/min. If the descend starts from 10 m above the ground level, how long will it take to reach the shaft 350 m below the ground level?

Answer

Starting position of elevator = 10 m above ground level = +10 m

Final position of elevator = 350 m below ground level = −350 m

Total distance to be descended = 10 − (−350)

= 10 + 350

= 360 m

Rate of descent = 6 m/min

Time taken = $\dfrac{\text{Total distance}}{\text{Rate of descent}}$

= $\dfrac{360 \text{ m}}{6 \text{ m/min}}$

= 60 min

= 1 hour

∴ The elevator will take 1 hour (or 60 minutes) to reach the shaft.

7 - 8 ÷ (-2) + 3 × (-4)

Answer

Given expression:

7 − 8 ÷ (−2) + 3 × (−4)

= 7 − (−4) + (−12) $\hspace{2cm}$ [Performing ÷ and ×]

= 7 + 4 − 12 $\hspace{3cm}$ [Removing brackets]

= 11 − 12

= −1

Hence, 7 - 8 ÷ (-2) + 3 × (-4) = -1.

9 - {7 - 24 ÷ (8 + 6 × 2 - 16)}

Answer

Given expression:

9 − {7 − 24 ÷ (8 + 6 × 2 − 16)}

= 9 − {7 − 24 ÷ (8 + 12 − 16)} $\hspace{2cm}$ [Simplifying × inside ( )]

= 9 − {7 − 24 ÷ 4} $\hspace{3.7cm}$ [Simplifying ( )]

= 9 − {7 − 6} $\hspace{4.8cm}$ [Simplifying ÷]

= 9 − 1 $\hspace{6cm}$ [Simplifying { }]

= 8

Hence, 9 - {7 - 24 ÷ (8 + 6 × 2 - 16)} = 8.

-11 - [-6 - {3 - 5(8 ÷ 4 - 1)}]

Answer

Given expression:

−11 − [−6 − {3 − 5(8 ÷ 4 − 1)}]

= −11 − [−6 − {3 − 5(2 − 1)}] $\hspace{2cm}$ [Simplifying ÷ inside ( )]

= −11 − [−6 − {3 − 5(1)}] $\hspace{2.7cm}$ [Simplifying ( )]

= −11 − [−6 − {3 − 5}] $\hspace{3.5cm}$ [Simplifying ×]

= −11 − [−6 − (−2)] $\hspace{3.7cm}$ [Simplifying { }]

= −11 − [−6 + 2] $\hspace{4.5cm}$ [Removing ( )]

= −11 − [−4] $\hspace{5.4cm}$ [Simplifying [ ]]

= −11 + 4 $\hspace{6cm}$ [Removing [ ]]

= −7

Hence, -11 - [-6 - {3 - 5(8 ÷ 4 - 1)}] = −7

(-3) × (-12) ÷ (-4) + 3 × 6

Answer

Given expression:

(−3) × (−12) ÷ (−4) + 3 × 6

= 36 ÷ (−4) + 3 × 6 $\hspace{2cm}$ [Simplifying × from left to right]

= −9 + 18 $\hspace{3.5cm}$ [Simplifying ÷ and ×]

= 9

Hence, (-3) × (-12) ÷ (-4) + 3 × 6 = 9

14 ÷ (3 of 2 - 3 + 4) - 9(5 - 3)

Answer

Given expression:

14 ÷ (3 of 2 − 3 + 4) − 9(5 − 3)

= 14 ÷ (6 − 3 + 4) − 9(2) $\hspace{2cm}$ [Simplifying 'of' inside ( ) and simplifying ( )]

= 14 ÷ 7 − 9 × 2 $\hspace{3.5cm}$ [Simplifying ( )]

= 2 − 18 $\hspace{5cm}$ [Simplifying ÷ and ×]

= −16

Hence, 14 ÷ (3 of 2 - 3 + 4) - 9(5 - 3) = −16

Fill in the blanks:

(i) ... is the greatest negative integer.

(ii) ((-10) + 3) + (-12) = (-10) + (3 + ....)

(iii) The product of three negative integers and the product of two positive integers is a .... integer.

(iv) The division of any integer by zero is .....

(v) The integer whose product with (-1) is 22 is ....

(vi) (-15) × ... = 120

(vii) .... ÷ (-6) = -12

(viii) (-10) × ((-15) + 33) = .... × (-15) + (-10) × 33

(ix) .... ÷ (-25) = 0

(x) ((-8) × (-13)) × 27 = (-8) × ((.....) × 27)

(xi) 13 × (-6) = - (..... × .....) = .....

(xii) (-a) + b = b + additive inverse of .....

(xiii) When -25 is divided by ..... the quotient is 5

(xiv) There are .... pairs of integers satisfying a + b = -1

(xv) The value of the expression ((-60) ÷ 12) ÷ (-5) is ....

Answer

(i) -1 is the greatest negative integer.

(ii) ((-10) + 3) + (-12) = (-10) + (3 + -12)

(iii) The product of three negative integers and the product of two positive integers is a negative integer.

(iv) The division of any integer by zero is not defined

(v) The integer whose product with (-1) is 22 is -22

(vi) (-15) × -8 = 120

(vii) 72 ÷ (-6) = -12

(viii) (-10) × ((-15) + 33) = -10 × (-15) + (-10) × 33

(ix) 0 ÷ (-25) = 0

(x) ((-8) × (-13)) × 27 = (-8) × ((-13) × 27)

(xi) 13 × (-6) = - (13 × 6) = -78

(xii) (-a) + b = b + additive inverse of a

(xiii) When -25 is divided by -5 the quotient is 5

(xiv) There are infinitely many pairs of integers satisfying a + b = -1

(xv) The value of the expression ((-60) ÷ 12) ÷ (-5) is 1

State whether the following statements are true (T) or false (F):

(i) For every integer a, |a| is either positive or zero.

(ii) The difference of two negative integers cannot be a positive integer.

(iii) We can write a pair of integers whose sum is not an integer.

(iv) If we divide an integer by (-1), then the result is the additive inverse of the integer.

(v) 1 is the additive identity of integers.

(vi) (-17) × 6 is a whole number.

(vii) (-5) × (-8) × 0 is a positive integer.

(viii) (-237) × 0 is same as 0 × (-89)

(ix) Closure property holds for subtraction of integers.

(x) Commutative property does not hold for subtraction of integers.

(xi) Associative property holds for subtraction of integers.

(xii) Closure property holds for division of integers.

(xiii) Commutative property does not hold for division of integers.

(xiv) Multiplication fact (-8) × (-12) = 96 is same as division fact 96 ÷ (-12) = -8

(xv) [(-32) ÷ 8] ÷ 2 = (-32) ÷ (8 ÷ 2)

(xvi) For every integer a, a ÷ a = 1

(xvii) The successor of 0 × (-10) is 1 × (-10)

Answer

(i) True
Reason — The absolute value of an integer is always non-negative, i.e., either positive or zero.

(ii) False
Reason — For example, (−2) − (−5) = −2 + 5 = 3, which is a positive integer.

(iii) False
Reason — Sum of two integers is always an integer (closure property of addition).

(iv) True
Reason — For any integer a, a ÷ (−1) = −a, which is the additive inverse of a.

(v) False
Reason — 0 is the additive identity of integers, not 1. (1 is the multiplicative identity.)

(vi) False
Reason — (−17) × 6 = −102, which is a negative integer, not a whole number.

(vii) False
Reason — (−5) × (−8) × 0 = 40 × 0 = 0, which is neither positive nor negative.

(viii) True
Reason — (−237) × 0 = 0 and 0 × (−89) = 0. Both are equal to 0.

(ix) True
Reason — The difference of any two integers is always an integer.

(x) True
Reason — For integers a and b, a − b ≠ b − a in general. For example, 5 − 3 = 2 but 3 − 5 = −2.

(xi) False
Reason — Subtraction is not associative for integers. For example, (5 − 3) − 2 = 0 but 5 − (3 − 2) = 4.

(xii) False
Reason — Division of two integers may not always yield an integer. For example, 1 ÷ 2 is not an integer.

(xiii) True
Reason — Division is not commutative for integers. For example, 12 ÷ (−3) = −4 but (−3) ÷ 12 is not an integer.

(xiv) True
Reason — From (−8) × (−12) = 96, we get 96 ÷ (−12) = −8. They are corresponding multiplication and division facts.

(xv) False
Reason — LHS = [(−32) ÷ 8] ÷ 2 = (−4) ÷ 2 = −2
RHS = (−32) ÷ (8 ÷ 2) = (−32) ÷ 4 = −8
Since −2 ≠ −8, the statement is false. (Division is not associative.)

(xvi) False
Reason — When a = 0, 0 ÷ 0 is not defined. The statement holds only for non-zero integers.

(xvii) False
Reason — 0 × (−10) = 0, and successor of 0 is 1.
But 1 × (−10) = −10.
Since 1 ≠ −10, the statement is false.

State whether the following statements are true or false. Justify your answer.

(i) The sum of a positive integer and a negative integer is always a positive integer.

(ii) The sum of two integers is always greater than their difference.

(iii) For any two integers a and b, the inequality -a < b is always true.

(iv) The product of two integers is always greater than the sum of the integers.

Answer

(i) False
Reason — The sign of the sum depends on the absolute values of the integers. For example, 3 + (−5) = −2, which is negative, not positive.

(ii) False
Reason — For example, take a = 5 and b = −3.
Sum = 5 + (−3) = 2
Difference = 5 − (−3) = 5 + 3 = 8
Here the sum (2) is less than the difference (8).

(iii) False
Reason — For example, take a = −5 and b = −10.
Then −a = 5 and b = −10.
Here −a > b (i.e., 5 > −10), so the inequality −a < b does not hold.

(iv) False
Reason — For example, take a = 1 and b = 2.
Sum = 1 + 2 = 3
Product = 1 × 2 = 2
Here the product (2) is less than the sum (3).

If the integers 10, -7, 5, 3, -4 and 0 are marked on the number line, then the integer which lies on the extreme left is

1. 10

2. 0

3. -7

4. -4

Answer

The integer which is the smallest will lie on the extreme left of the number line.

Among 10, −7, 5, 3, −4 and 0, the smallest integer is −7.

∴ The integer that lies on the extreme left is −7.

Hence, option 3 is the correct option.

On the number line, the value of (-3) × 3 lies on the right hand side of

1. -10

2. -6

3. 0

4. 9

Answer

(−3) × 3 = −9

On the number line, −9 lies on the right of −10.

(−9 lies to the left of −6, 0 and 9.)

Hence, option 1 is the correct option.

The value of 5 ÷ (-1) does not lie between

1. 0 and -10

2. 0 and 10

3. -3 and -10

4. -7 and 7

Answer

5 ÷ (−1) = −5

Now we check each option:

1. 0 and −10: −5 lies between 0 and −10.
2. 0 and 10: −5 does not lie between 0 and 10 (since −5 is negative)
3. −3 and −10: −5 lies between −3 and −10.
4. −7 and 7: −5 lies between −7 and 7.

So, −5 does not lie between 0 and 10.

Hence, option 2 is the correct option.

The next number in the pattern -62, -37, -12, ... is

1. 25

2. 0

3. 13

4. -13

Answer

Observing the pattern:

−62 + 25 = −37
−37 + 25 = −12
−12 + 25 = 13

So, the next number is 13.

Hence, option 3 is the correct option.

Multiplication of integers satisfies the property of

1. closure

2. commutativity

3. associativity

4. all of these

Answer

Multiplication of integers satisfies all three properties:

• Closure: a × b is an integer
• Commutativity: a × b = b × a
• Associativity: (a × b) × c = a × (b × c)

Hence, option 4 is the correct option.

The number of integers between -20 and -10 are

1. 8

2. 9

3. 10

4. 11

Answer

Integers between −20 and −10 are:

−19, −18, −17, −16, −15, −14, −13, −12, −11

So, there are 9 integers between −20 and −10.

Hence, option 2 is the correct option.

If the sum of two integers is -10 and one of them is 2, then the other is

1. 8

2. -8

3. 12

4. -12

Answer

Let the other integer be x.

Given: 2 + x = −10

x = −10 − 2

x = −12

∴ The other integer is −12.

Hence, option 4 is the correct option.

The integer that must be subtracted from -5 to obtain -12 is

1. 7

2. -7

3. 17

4. -17

Answer

Let the required integer be x.

Given:

−5 − x = −12

x = −5 − (−12)

x = −5 + 12

x = 7

∴ The required integer is 7.

Hence, option 1 is the correct option.

Which of the following is not the additive inverse of a?

1. -(-a)

2. -a

3. a ÷ (-1)

4. a × (-1)

Answer

The additive inverse of a is −a.

1. −(−a) = a (not the additive inverse of a)
2. −a (this is the additive inverse of a)
3. a ÷ (−1) = −a (this is the additive inverse of a)
4. a × (−1) = −a (this is the additive inverse of a)

So, −(−a) is not the additive inverse of a.

Hence, option 1 is the correct option.

0 ÷ (-10) is equal to

1. 0

2. -1

3. -10

4. none of these

Answer

0 ÷ (−10) = $\dfrac{0}{-10}$ = 0

[∵ 0 divided by any non-zero integer is 0]

Hence, option 1 is the correct option.

(-33) × 102 + (-33) × (-2) is equal to

1. 3300

2. -3300

3. 3432

4. -3432

Answer

(−33) × 102 + (−33) × (−2)

= (−33) × [102 + (−2)] $\hspace{1cm}$ [Using distributive property]

= (-33) x [102 - 2]

= (−33) × 100

= −3300

Hence, option 2 is the correct option.

101 × (-1) + 0 ÷ (-1) is equal to

1. -101

2. 101

3. -102

4. 102

Answer

101 × (−1) + 0 ÷ (−1)

= −101 + 0

= −101

Hence, option 1 is the correct option.

(-3) × 5 is not equal to

1. 3 × (-5)

2. -(3 × 5)

3. (-3) × (-5)

4. 5 × (-3)

Answer

(−3) × 5 = −15

1. 3 × (−5) = −15 (equal)

2. −(3 × 5) = −15 (equal)

3. (−3) × (−5) = 15 (not equal)

4. 5 × (−3) = −15 (equal)

So, (−3) × 5 is not equal to (−3) × (−5).

Hence, option 3 is the correct option.

If a and b are two integers, then which of the following may not be an integer?

1. a + b

2. a - b

3. a × b

4. a ÷ b

Answer

Integers are closed under addition, subtraction and multiplication, but not under division.

For example, 1 ÷ 2 = $\dfrac{1}{2}$, which is not an integer.

So, a ÷ b may not be an integer.

Hence, option 4 is the correct option.

For a non-zero integer a, which of the following is not defined?

1. a ÷ 0

2. 0 ÷ a

3. a ÷ 1

4. a ÷ a

Answer

For a non-zero integer a:

1. a ÷ 0 is not defined (division by zero is undefined)

2. 0 ÷ a = 0 (defined)

3. a ÷ 1 = a (defined)

4. a ÷ a = 1 (defined)

Hence, option 1 is the correct option.

Statement I: 0 is the multiplicative identity for integers.

Statement II: The number 0 is an integer.

Answer

Let us first consider Statement I.

The multiplicative identity for integers is 1 (since a × 1 = a for any integer a), not 0.

So, Statement I is false.

Now let us consider Statement II.

The number 0 is indeed an integer (it is neither positive nor negative).

So, Statement II is true.

Thus, Statement I is false but Statement II is true.

Hence, option 2 is the correct option.

Statement I: When we multiply the absolute values of two non-zero integers, we always get a positive integer.

Statement II: |-a| = |a|, for all integers a.

Answer

Let us first consider Statement I.

The absolute value of any non-zero integer is a positive integer. So, the product of two positive integers is also a positive integer.

So, Statement I is true.

Now let us consider Statement II.

By the definition of absolute value, |−a| = |a| for all integers a.

For example, |−5| = 5 = |5|.

So, Statement II is true.

Thus, both Statement I and Statement II are true.

Hence, option 3 is the correct option.

Statement I: Multiplying two integers with different signs results in a positive integer.

Statement II: If a, b and c are integers, then a × (b × c) = (a × b) × (a × c).

Answer

Let us first consider Statement I.

Multiplying two integers with different signs results in a negative integer, not a positive integer.

For example, 3 × (−5) = −15, which is negative.

So, Statement I is false.

Now let us consider Statement II.

The associative property of multiplication is a × (b × c) = (a × b) × c, not (a × b) × (a × c).

So, Statement II is false.

Thus, both Statement I and Statement II are false.

Hence, option 4 is the correct option.

Statement I: If a is any non-zero integer, then 0 ÷ a = 0 and a ÷ a = 1.

Statement II: All the four operations - addition, subtraction, multiplication and division follow the property of closure for all integers.

Answer

Let us first consider Statement I.

For any non-zero integer a:

• 0 ÷ a = 0 (since a × 0 = 0)
• a ÷ a = 1 (since a × 1 = a)

So, Statement I is true.

Now let us consider Statement II.

Integers are closed under addition, subtraction and multiplication. However, integers are not closed under division.

For example, 1 ÷ 2 = $\dfrac{1}{2}$, which is not an integer.

So, Statement II is false.

Thus, Statement I is true but Statement II is false.

Hence, option 1 is the correct option.

Evaluate the following:

(i) (-7) × (-9) × (-11)

(ii) (-5) × 7 × (-6) × (-8)

(iii) (-1024) ÷ 32

(iv) (-216) ÷ (-12)

Answer

(i) (−7) × (−9) × (−11)

= [(−7) × (−9)] × (−11)

= 63 × (−11)

= −693

(ii) (−5) × 7 × (−6) × (−8)

= [(−5) × 7] × [(−6) × (−8)]

= (−35) × 48

= −1680

(iii) (−1024) ÷ 32

= $\dfrac{-1024}{32}$

= −(1024 ÷ 32)

= −32

(iv) (−216) ÷ (−12)

= $\dfrac{-216}{-12}$

= +(216 ÷ 12)

= 18

What will be the sign of the product if we multiply 39 negative integers and 98 positive integers?

Answer

Product of 39 negative integers:

Since 39 is an odd number, the product of 39 negative integers will be a negative integer.

Product of 98 positive integers:

The product of 98 positive integers will be a positive integer.

Now, product = (negative integer) × (positive integer) = negative integer

∴ The sign of the resulting product will be negative.

Use the sign >, < or = in the box to make the following statements true:

(i) (-15) + 38 $\boxed{\phantom{653}}$ 27 + (-50)

(ii) (-13) × 0 × (-5) $\boxed{\phantom{653}}$ (-7) × (-6) × 14

(iii) (-18) ÷ (-3) $\boxed{\phantom{653}}$ (-10) + (-15) + 31

(iv) (-5) × (-7) × (-10) $\boxed{\phantom{653}}$ (-1400) ÷ (-4)

Answer

(i) LHS = (−15) + 38 = 23

RHS = 27 + (−50) = −23

Since 23 > −23,

∴ (−15) + 38 $\boxed{\gt}$ 27 + (−50)

(ii) LHS = (−13) × 0 × (−5) = 0

RHS = (−7) × (−6) × 14 = 42 × 14 = 588

Since 0 < 588,

∴ (−13) × 0 × (−5) $\boxed{\lt}$ (−7) × (−6) × 14

(iii) LHS = (−18) ÷ (−3) = 6

RHS = (−10) + (−15) + 31 = −25 + 31 = 6

Since 6 = 6,

∴ (−18) ÷ (−3) $\boxed{=}$ (−10) + (−15) + 31

(iv) LHS = (−5) × (−7) × (−10) = 35 × (−10) = −350

RHS = (−1400) ÷ (−4) = 350

Since −350 < 350,

∴ (−5) × (−7) × (−10) $\boxed{\lt}$ (−1400) ÷ (−4)

Is ((-45) ÷ (-15)) ÷ (-3) = (-45) ÷ ((-15) ÷ (-3))?

Answer

LHS = ((−45) ÷ (−15)) ÷ (−3)

= 3 ÷ (−3)

= −1

RHS = (−45) ÷ ((−15) ÷ (−3))

= (−45) ÷ 5

= −9

Since −1 ≠ −9,

∴ ((−45) ÷ (−15)) ÷ (−3) ≠ (−45) ÷ ((−15) ÷ (−3))

This shows that division is not associative for integers.

Simplify the following:

(i) (-7) + (-6) ÷ 2 - {(-5) × (-4) - (3 - 5)}

(ii) 11 - [7 - {5 - 3 (9 - $\overline{3 - 6}$)}]

Answer

(i) Given expression:

(−7) + (−6) ÷ 2 − {(−5) × (−4) − (3 − 5)}

= (−7) + (−3) − {20 − (−2)} $\hspace{1cm}$ [Simplifying ÷, ×, and ( )]

= (−7) + (−3) − {20 + 2} $\hspace{2cm}$ [Removing ( )]

= (−7) + (−3) − 22 $\hspace{3.4cm}$ [Simplifying { }]

= −7 − 3 − 22

= −32

The answer is −32

(ii) Given expression:

11 - [7 - {5 - 3 (9 - $\overline{3 - 6}$)}]

= 11 - [7 - {5 - 3(9 - (-3))}] $\quad$[Removing 'bar']

= 11 - [7 - {5 - 3(9 + 3)}] $\quad$[Removing ()]

= 11 - [7 - {5 - 3(12)}] $\quad$[Simplifying ()]

= 11 - [7 - {5 - 36}] $\quad$[Simplifying x]

= 11 - [7 - (-31)] $\quad$[Simplifying {}]

= 11 - [7 + 31] $\quad$[Removing ()]

= 11 - 38 $\quad$[Removing []]

= -27

The answer is −27

Arrange the numbers -5 + 8, 3 × (-4), -15 ÷ 5, |-5 - 7|, -7 - 3 × (-2) and -7 - 4 ÷ (-2) in descending order.

Answer

Let us first evaluate each expression:

−5 + 8 = 3

3 × (−4) = −12

−15 ÷ 5 = −3

|−5 − 7| = |−12| = 12

−7 − 3 × (−2) = −7 − (−6) = −7 + 6 = −1

−7 − 4 ÷ (−2) = −7 − (−2) = −7 + 2 = −5

So, the numbers are: 3, −12, −3, 12, −1, −5

Arranging them in descending order:

12 > 3 > −1 > −3 > −5 > −12

∴ |−5 − 7| > −5 + 8 > −7 − 3 × (−2) > −15 ÷ 5 > −7 − 4 ÷ (−2) > 3 × (−4)

|−5 − 7|, −5 + 8, −7 − 3 × (−2), −15 ÷ 5, −7 − 4 ÷ (−2), 3 × (−4)

Write a pair of integers whose product is -12 and there lies seven integers between them.

Answer

We need a pair of integers whose product is −12 and there are seven integers between them.

If there are seven integers between two integers a and b, then their absolute difference is 8.

The factor pairs of −12 are: (1, −12), (−1, 12), (2, −6), (−2, 6), (3, −4), (−3, 4)

Checking the absolute difference of each pair:

• |1 − (−12)| = 13
• |(−1) − 12| = 13
• |2 − (−6)| = 8
• |(−2) − 6| = 8
• |3 − (−4)| = 7
• |(−3) − 4| = 7

Here only two factors have the absolute difference 8.

So, the required pair is 2 and −6 or −2 and 6.

Verification: 2 × (−6) = −12

The seven integers lying between 2 and −6 are: −5, −4, −3, −2, −1, 0, 1.

A shopkeeper earns a profit of ₹2 by selling a pen and incurs a loss of 50 paise per pencil and loss of 15 paise per eraser while selling pencils and erasers of old stock. On a particular day, he earns a profit of ₹10. If he sold 10 pens and the number of pencils and erasers he sold are in the ratio 7 : 10, then find the number of pencils and erasers he sold on that day.

Answer

Profit per pen = ₹2 = 200 paise

Loss per pencil = 50 paise (denoted by −50 paise)

Loss per eraser = 15 paise (denoted by −15 paise)

Number of pens sold = 10

Profit from pens = Number of pens sold x Profit per pen

= 10 × 200

= 2000 paise

Total profit on that day = ₹10 = 1000 paise

Given,

Number of pencils and erasers sold are in the ratio 7 : 10.

Let the number of pencils sold = 7x

Let the number of erasers sold = 10x

Loss from pencils = 7x × 50 = 350x paise

Loss from erasers = 10x × 15 = 150x paise

Total loss = 350x + 150x = 500x paise

Net profit = Profit from pens − Total loss

Substituting the values:

1000 = 2000 − 500x

500x = 2000 − 1000

500x = 1000

x = $\dfrac{1000}{500}$

x = 2

∴ Number of pencils sold = 7x = 7 × 2 = 14

∴ Number of erasers sold = 10x = 10 × 2 = 20

∴ The shopkeeper sold 14 pencils and 20 erasers on that day.