CBSE Class 7 Mathematics Integers Worksheet Set 01

Question. The value of 5(10 – 9) is            
(A) 5
(B) 5 × 10 – 5 × 9
(C) (A) and (B)
(D) none of these

Answer:        D

Question. The sum of two integers is also an integer, this property of integers is called         
(A) Closure
(B) Commutative
(C) Associative
(D) None of these

Answer:      A 

Question. Reciprocal of 1/7 is:            
(A) 7
(B) 1
(C) –7
(D) 1/7

Answer:    A

Question. For any integer x what is true:            
(A) x/0 is not defined (B) x/1 = a (where x ≠ a)
(C) option
(A) and
(B) both are wrong
(D) option (A) and (B) both are rigth

Answer:    A

Question. Every integer is also a            
(A) natural number
(B) whole number
(C) (A) and
(B) both
(D) none of these

Answer:          D

Question. When 0 is multiplied by any negative integer, their product will be            
(A) a positive integer
(B) a negative integer
(C) zero
(D) none of these

Answer:           C

Question. Every positive integer is greater than            
(A) zero
(B) every negative integer
(C) both (A) and (B)
(D) none of these

Answer:   C

Question. The product of two integers is 12, if one integer is –3 then the other one is:            
(A) +4
(B) –4
(C) 3
(D) –3

Answer:   B

Question. On subtracting (–6) from 0, we get:            
(A) –6
(B) 6
(C) 7
(D) None of these

Answer:   B

Question. The sum of two integers is always            
(A) a natural number
(B) a whole number
(C) an integer
(D) none of these

Answer:          B

Question. The multiplicative identity of integers is            
(A) 0
(B) +1
(C) –1
(D) none of these

Answer:   C

Question. The additive inverse of –6 is:            
(A) 6
(B) 0
(C) –5
(D) –7

Answer:   A

Question. 30 × (–23) + 30 × 14 = ? 
(A) –270
(B) 270
(C) 1110
(D) –1110

Answer:   A

Question. (–8) / 0 = ?  
(A) –8
(B) 0
(C) 8
(D) Not defined

Answer:     D

Question. How much less than –8 is –3?   
(A) –5
(B) 5
(C) 11
(D) –11

Answer:    A

Question. The sum of two integers is 93. If one of them is –59, the other one is: 
(A) 34
(B) –34
(C) 152
(D) –152

Answer:    C

Question. Resolve the brackets and simplify: (28 / 2) / (56 / 8).
(A) 1
(B) 4
(C) 3
(D) 2

Answer:    D

Question. For integers:            
(A) Addition is associative
(B) Addition is commutative
(C) Integer “0” is the identity under addtion
(D) All of the above

Answer:      D

Question. By how much does –3 exceed –5 ?  
(A) –2
(B) 2
(C) 8
(D) –8

Answer:   B

Question. What must be subtracted from –3 to get –9?   
(A) –6
(B) 12
(C) 6
(D) –12

Answer:     C

Question. The additive identity of integers is            
(a) –1
(B) 1
(C) 0
(D) none of these

Answer:    c

Question. The greatest positive integer is            
(A) 0
(B) 100
(C) 999
(D) none of these

Answer:   D

Question. Write the value of
(i) | +22 |
(ii) | –8 |
(iii) | 18 – 8 |
(iv) | – 5 – 4 |
(v) – | 3 – 2 |

Answer:  (i) 22  (ii) 8  (iii) 10  (iv) 9  (v) –1

Question. Arrange the following integers in ascending order
(i) – 20, 13, 4, 0, –5, +5
(ii) +30, –2, 0, –6, –20, 8

Answer: (i) –20, –5, 0, 4, 5, 13   (ii) –20, –6, –2, 0, 8, 30

Question. The temperature at 12 noon was 10°C above zero. At what time the temperature will be 8°C below 0°C, if it decreases at the rate of 2°C ? If it decreases at the rate of 2°C per hour till midnight, what would be the temperature at mid-night ?

Answer: 9 pm, –14ºC

Question. In a class test (+3) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any questions.
(i) Ram scored 20 marks. If he has got 12 correct answers, how many questions has he attempted incorrectly ?
(ii) Mohan scores –5 marks in the test though he has written 7 correct answers. How many questions has he attempted incorrectly ?

Answer: (i) 8    (ii) 13

Question. A water tank has steps inside it. A monkey is sitting on the toppest step (i.e., the first step). The water level is at the ninth step.
(i) He jumps 3 steps down and then jumps 2 steps up. If he continues in this way, in how many jumps will he reach the water level ?
(ii) After drinking water, he wants to go back, for this he jumps 4 steps up and then jumps 2 steps down in every move. In how many jumps will he reach back the top step ?

Answer: (i) 11 times     (ii) 5 times

Question. Which temperature is higher ?
(i) 40°C or –40°C
(ii) –18°C or 12°C
(iii) –2°C or –4°C
(iv) 17°C or 27°C

Answer: (i) 40ºC  (ii) 12ºC  (iii) –2ºC  (iv) 27ºC

Question. Verify and name the property used :
(i) –110 × (–237) = (–237) × (–110)
(ii) (–35 × 4) × (–152) = –35 × [4 × (–152)]

Answer: (i) Commutative property of multiplication.  (ii) Associative property of multiplication

Question. Verify and name the property used :
(i) –117 × 251 + (–117) × 249
(ii) 156 × 273 – 156 × (–73)

Answer: (i) Distributive property of multiplication over addition
              (ii) Distributive property of multiplication over subtraction

Question. A boy has j- 350 in his bank account. He deposits j- 40 everyday for 10 days. What will be the amount in his account at the end of 10 days ?

Answer: j- 750

Question. In a class test containing 10 questions, 5 marks are answered for every correct answer and (–2) marks are awarded for every incorrect answer and 0 for questions not attempted .
(i) Mohan gets four correct and six incorrect answers. What is his score ?
(ii) Reshma gets five correct answer and five incorrect answers. What is her score ?
(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score ?

Answer:  (i) 8 marks (ii) 15 marks (iii) 0 mark

Question. Evaluate the following :           
(i) (– 40) ÷ 10
(ii) 60 ÷ (–6)
(iii) (– 49) ÷ (–7) (iv) (–79) ÷ 79
(v) 13 ÷ [(–4) + 3]
(vi) 0 ÷ (–14) (vii) (– 41) ÷ [(– 40) + (–1)]
(viii) [(– 48) ÷ 12] ÷ 4
(ix) Is [(–7) + (6)] = [(–3) + 2] ?

Answer: (i) –4  (ii) –10  (iii) 7  (iv) –1  (v) –13  (vi) 0  (vii) 1  (viii) –1   (ix) yes

Question. Write down a pair of integers whose :
(i) sum is – 7
(ii) difference is –10
(iii) sum is 0

Answer: (i) –1, –6 or –9, 2       (ii) –11, –1 or 35, –45     (iii) –1, 1 or 20, –20

A. Define

i.Natural number:- The natural numbers include the positive integers (also known as non-negative integers) and a fewexamples include 1, 2, 3, 4, 5, 6, .. ∞. In other words,natural numbers are a set of all the whole numbersexcluding 0.

ii.Whole number :- Whole numbers are a set of numbers including the set of natural numbers (1 to infinity) and the integer'0'. Fraction numbers and negative numbers are not counted in whole numbers.

iii. Prime number:- prime numbers are wholenumbers greater than 1, that have only two factors – 1 and the number itself. Prime numbers are divisible only by the number 1 or itself. For example, 2, 3, 5, 7 and 11 are the first few prime numbers.

Iv. Integers:- An integer is a whole number (not a fractional number) that can be positive, negative, or zero. For example :- 1,2,3,4,0,-1,-2,-3,-4,-5 etc.

B.PROPERTIES OF INTEGERS FOR ADDITION AND SUBTRACTION:-

C. Solve the following:-

1.Evaluate:

(i) 15 + (−8)
Answer. 15 + (−8) = 7 

(ii) (−16) + 9
Answer. (−16) + 9 = −7

(iii) (−7) + (−23)
Answer. (−7) + (−23) = −30

(iv) (−32) + 47
Answer. (−32) + 47 = 15

(v) 53 + (−26)
Answer. 53 + (−26) = 27

(vi) (−48) + (−36)
Answer. (−48) + (−36) = −84

2.Find the sum of:

(i) 153 and − 302
Answer. 153 + (−302) = −149

(ii) 1005 and − 277
Answer. 1005 + (−277) = 728

(iii) − 2035 and 297
Answer. (−2035) + 297 = −1738

(iv) − 489 and − 324
Answer. (−489) + (−324) = −813

(v) − 1000 and 438
Answer. (−1000) + 438 = −562

(vi) − 238 and 500
Answer. (−238) + 500 = 262

3. Find the additive inverse of:

(i) − 83
Answer. Additive inverse of −83 = −(−83) = 83

(ii) 256
Answer. Additive inverse of 256 = −(256) = −256

(iii) 0
Answer. Additive inverse of 0 = −(0) = 0

(iv) – 2001
Answer. Additive inverse of 2001 = −(−2001) = 2001

4. Subtract:

(i) 28 from − 42
Answer. −42 − 28 = (−42) + (−28) = −70

(ii) − 36 from 42
Answer. 42 −(−36) = 42 + 36 = 78

(iii) − 37 from − 53
Answer. -53 - (-37) = (-53) - (-37) = -16

(iv) − 66 from − 34
Answer. -34 - (-66) = -34 + 66 = 32

(v) 318 from 0
Answer. 0 - 318 = -318

(vi) − 153 from − 240
Answer. (-240) - (-153) = -87

(vii) − 64 from 0
Answer. 0 - (-64) = 0 + 64 = 64

(viii) − 56 from 144
Answer. 144 - (-56) = 144 + 56 = 200

5.Subtract the sum of 

- 1032 and 878 from -34.
Answer. Sum of -1032 and 878 = -1032 + 87 = -154
Subtracting the sum from -34, we get -34 - (-154)
= (−34)+ 154 = 120

6. Subtract 

−134 from the sum of 38 and −87
Answer. First, we will calculate the sum of 38 and −87.
38 + (−87) = −49 Now,
subtracting −134 from the sum,
we get: −49 − (−134) =(−49) + 134 = 85

7.Fill in the blanks:

(i) {(−13) + 27} + (−41) = (−13) + {27 + (......)}
Answer. −41 (∵ Associative property)

(ii) (−26) + {(−49) + (−83)} = {(−26) + (−49)} + (......)
Answer. −83 (∵ Associative property)

(iii) 53 + (−37) = (−37) + (......)
Answer. 53 (∵ Commutative property)

(iv) (−68) + (−76) = (......) + (−68)
Answer.−76 (∵ Commutative property)

(v) (−72) + (......) = −72
Answer. 0 (∵ Additive identity)

(vi) − (−83) = ......   
Answer. 83 (∵ Additive inverse)

(vii) (−60) − (......) = − 59
Answer. (−60) − (−59) = −1

(viii) (−31) + (......) = − 40
Answer. (−40) − (−31) = −9

8.Simplify:

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

Q1 What is the value of (−23) –[(−25) – {(−17) –(−19) }
(a) 38
(b) 24
(c) −14
(d) −34

Q2 What should be subtracted from −1963 to obtain −9512?
(a) −11475
(b) −7549
(c) 7549
(d) 11475

Q3 What is the missing value in the expression
{(−267) – (−276) – 15= □ +(−2) + (−1) } ?
(a) −18
(b) −3
(c) 3
(d) 8

Q4 What is the additive inverse of the result obtained by subtraction of −8 from the additive inverse of −13?
(a) −21
(b) −5
(c) 5
(d) 21 

Q5 The temperature of a city is 2°C.Next day the temperature falls by 5°C.What is the temperature of the city the next day?
(a) −5°C
(b) −3°C
(c) 4°C
(d) 7°C

Q6 What is the sum of the additive inverses of −29 and 61?
(a) −90
(b) −32
(c) 90
(d) 32

Q7 On a particular day,the temperature of four places P,Q,R,S are recorded at noon and midnight of the day

At which place is the decrease in temperature the maximum?
(a)P
(b)Q
(c)R
(d)S

Q8 The temperature of a city was found to be 3°C on a particular day .Next day the temperature was found to be −4°C .What is the change in the temperature of the city over the two days?
(a) Decrease by 7°C
(b) Decrease by 1°C
(c) Increase by 7°C
(d) Increase by 1°C

Q9 What is the difference between greatest two-digit positive number and smallest three-digit
negative number?
(a) 199
(b) 110
(c) 1100
(d) 1098

Q10 What is the value of the expression,
(− 216) – (−351)+( −98) –(71)+(28) –(−56) ?
(a) −50
(b) 15
(c) −15
(d) 50

Q11 The temperature in New York city on two consecutive days were 1°C and −6°C.What is the
decrease in the temperature of New York city in two days?
(a) −7°C
(b) 7°C
(c) 5°C
(d) −5°C

INTEGERS (SUBTRACTION AND MULTIPLICATION)

Q1. Find 36 − (−64) and (−64) − 36. Are they equal?
Solution.
36 − (−64) = 36 + 64 = 100
Now, (−64) − 36
= (−64) + (−36) = −100
Here, 100 ≠≠ −100
Thus, they are not equal.

Q2. If a = − 8, b = − 7, c = 6, verify that (a+b) + c = a + (b+c).
Solution.
(a + b) + c = (−8 + (−7)) + 6
= −15 + 6 = −9
a + (b + c) = −8 + (−7 + 6)
= −8 + (−1) = −9
Hence, (a + b) + c = a + (b + c) [i.e., Property of Associativity]

Q3. If a = − 9 and b = − 6, show that (a−b) ≠ (b−a).
Solution.
Here, (a − b) = −9 − (−6) = −3
Similarly, (b − a) = −6 − (−9) = 3
∴ (a−b) ≠ (b−a)

Q4. The sum of two integers is − 16. If one of them is 53, find the other.
Solution. Let the other integer be a.
Then,
we have:
53 + a = −16
⇒ a = −16 − 53 = −69
∴ The other integer is −69.

Q5. The sum of two integers is 65. If one of them is − 31, find the other.
Solution. Let the other integer be a. Then,
−31 + a = 65
⇒ a = 65 − (−31) = 96
∴ The other integer is 96.

Q6. The difference of an integer a and (−6) is 4. Find the value of a.
Solution.
We have:
a − (−6) = 4
⇒ a = 4 + (−6) = −2
∴ a = −2

Q7. Write a pair of integers whose sum gives

(i) zero;
Solution. Consider the integers 8 and −8. Then,
we have:
8 + (−8) = 0

(ii) a negative integer;
Solution. Consider the integers 2 and (−9).
Then,
we have: 2 + (−9)= −7,
which is a negative integer.

(iii) an integer smaller than both the integers;
Solution. Consider the integers −4 and −5.
Then, we have: (−4) + (−5) = −9,
which is smaller than −4 and −5.

(iv) an integer greater than both the integers;
Solution. Consider the integers 2 and 6.
Then,
we have: 2 + 6 = 8,
which is greater than both 2 and 6.

(v) an integer smaller than only one of the integers.
Solution. Consider the integers 7 and −4. Then,
we have: 7 + (−4) = 3,
which is smaller than 7 only.

Q8. For each of the following statements, write (T) for true and (F) for false:

(i) The smallest integer is zero.
Solution. F (false). −3, −90 and −100 are also integers.
We cannot determine the smallest integer, since they are infinite.

(ii) − 10 is greater than − 7
Solution. F (false). −10 is less than −7.

(iii) Zero is larger than every negative integer.
Solution.T (true). All negative integers are less than zero.

(iv) The sum of two negative integers is a negative integer. 
Solution. T (true).

(v) The sum of a negative integer and a positive integer is always a positive integer.
Solution. F (false). Example: −9 + 2 = −7

Q9. Multiply:

(i) 16 by 9
Solution. 16 ×× 9 = 144

(ii) 18 by − 6
Solution. 18 ×× (−6)
= −(18×6)
= -(18×6)
= −108

(iii) 36 by − 11
Solution. 36 ×× (−11)
= − (36×11)
= - (36×11)
= −396

(iv) − 28 by 14
Solution. (−28) ××14
= −(28×14) = -(28×14)
= −392

(v) − 53 by 18
Solution. (−53) ×× 18
= −(53×18) = -(53×18)
= −954

(vi) − 35 by 0
Solution. (−35) ×× 0 = 0

(vii) 0 by − 23
Solution. 0 ×× (−23) = 0

(viii) − 16 by − 12
Solution. (−16) ×× (−12) = 192

(ix) − 105 by − 8
Solution. (−105) ×× (−8) = 840

(x) − 36 by − 50
Solution. (−36) ×× (−50) = 1800

(xi) − 28 by − 1
Solution. (−28) ×× (−1) = 28

(xii) 25 by − 11
Solution. 25 ×× (−11)
= − (25×11)
= - (25×11)
= −275

Q10. Find each of the following products:

(i) 3 × 4 × (−5) 
Solution. 3 × 4 × (−5)
= (12) × (−5)
= −60

(ii) 2 × (−5) × (−6)
Solution. 2 × (−5) × (−6)
= (−10) × (−6)
= 60

(iii) (−5) × (−8) × (−3)
Solution. (−5) × (−8) × (−3)
= (−5) × (24)
= −120

(iv) (−6) × 6 × (−10)
Solution. (−6) × 6 × (−10)
= 6 × (60)
= 360

(v) 7 × (−8) × 3
Solution. 7 × (−8) × 3
= 21 × (−8)
= −168

(vi) (−7) × (−3) × 4
Solution. (−7) × (−3) × 4
= 21 × 4
= 84

Q1) If the integers 12,-6, 8,-5, 7,-4, 3 are marked on the number line, the one that comes on the extreme left is:
a) 12
b) -4
c) -6
d) 3

Q2) The difference in temperatures +50⁰C and -50⁰C is:
a) 100⁰C
b) 0⁰C
c) 50⁰C
d)-100⁰C

Q3) The integer x for which |1-x| =3 is:
a) 3,0
b) 4,2
c) -2,3
d) 4,-2

Q4) The numbers between -30 and -15 are :
a) 14
b) 15
c) 16
d) 17

Q5) Multiplicative identity for any integer a is:
a) 0
b) 1
c) a
d) a/b

Q6) Represent each of the following integers on the number line: 3 -1, 5, -4, 8, -10 

Q7) The sum of two integers is (-11).If one of them is 72 ,find the other number.

Q8) Using quicker grouping, find the product of :

(a) 16x 20 x ( -25)

(b) ( -125) x 17 x (-8)

(c) (-50) x( -41) x12

(d) (-25) x 36 x 80

(e) (-700) x 49 + (-700)

(f) (-562) x 96 + (-562) x 4

Q9) Using suitable properties, find the value of :

(a) -58 x 104

(b) (-279) x 1001

(c) 36 x (-99)

(d) (-73) x 8 + 73 x (-92)

(e) 517 x (-36) + 517 x (-64)

Q10) Divide:

(a) (-525) ÷(-25)

(b) (-512) ÷ 32

(c) 2352÷(-42)

(d) (-3128) ÷34

Q11) Fill in the blanks:

(a) 450 x 0 =______ (b) 450 x (-1)=_______ (c) (-450) x 1=_____

(d) (-450) x (-1)=________ (e) 0 ÷(-20)=___ (f) (-20)÷ 1=_______

(g) (-20)÷ (-1)=______ (h) 20÷ (-1)=________ (i) (-20)÷ (-1)=_____

Q12) The product of two integers is (-840). If one of them is (-70), find the other.

Q13) Simplify the following:

(a) 120÷[18 – {20 – 3(9-5)}]

(b) 64 ÷[16 x (-3) – 2 x {17 – 25 }]

Q14) Find the value of :

(a) |-15|

(b)- |-20 |

(c) -50 +| -2 |

(d) 18 - | -36|

(e) 46 - |-5 |-|6|

Q15) Write : (a) Additive inverse of 0
(b) Additive inverse of 15
(c) Additive inverse of -36.

Q16) By how much does 2 exceed ( -5)?

Q17) The product of three integers is ( -85). If the two integers are ( -17) and ( -5), find the third.

Q18) Raghu has 30 marbles. In 5 games he won 3 marbles each and in 7 games he lost 2 marbles each. Find the marbles he has at the end of the games.

Q19) Name 4 real life situations in which integers can be used.

INTEGERS (DIVISION)

Rules on How to Divide Integers

RULE 1: The product of a positive integer and a negative integer is negative. (+) × (-) = (-)
RULE 2: The product of two positive or two negative integers is always positive. (+)× (+)= (+) (-) × (-) = (+)
RULE 3: When you divide 0 by another number the answer is always 0.
RULE 4 : When you divide by 1, the answer is the same as the number you were dividing.
RULE 5 : Any number divided by the same number is 1.
RULE 6 : When you divide two negative numbers, THE NEGATIVE sign of numerator DENOMINATOR GET cancelled with each other.

QUESTIONS BASED ON ABOVE RULES

Q1. Divide:

(i) 65 by −13.
Solution. 65 ÷(−13) = -(65/13) = −5

(ii) −84 by 12.
Solution. (−84) ÷12 = -(84/12) = −7

(iii) −76 by 19.
Solution. (−76) ÷ 19 = −(76/19) = −4

(iv) −132 by 12
Solution. (−132) ÷ 12 = −(132/12) = −11

(v) −150 by 25.
Solution. (−150) ÷ 25 = −(150/25) = −6

(vi) −72 by −18.
Solution. (−72) ÷(−18) = (-72/-18) = 4.
[(-)sign cancelled with each other in numerator and denominator]

(vii) −105 by −21.
Solution. (−105) ÷ (−21) = (−105/−21) = 5
[(-)sign cancelled with each other in numerator and denominator]

(viii) −36 by −1
Solution. (−36) ÷(−1) = (−36/−1) = 36.
[(-)sign cancelled with each other in numerator and denominator]

(ix) 0 by −31.
Solution. 0 ÷ (−31) = -(0/31) = 0

(x) −63 by 63
Solution. (−63) ÷63 = −(63/63 ) = −1

Q2. Fill in the blanks

(i) 72 ÷ (......) = −4.
(ii) −36 ÷ (......) = −4. (iii) (......) ÷ (−4) = 24
(iv) (......) ÷ 25 = 0. (v) (......) ÷ (−1) = 36.
(vi) (......) ÷ 1 = −37 (vii) 39 ÷ (......) = −1.
(viii) 1 ÷ (......) = −1. (ix) −1 ÷ (......) = −1
Answer.
(i). 72 ÷ (x) = −4 ⇒72÷x = −4. ⇒x = 72÷( −4 )= -(72÷4) = −18
(ii) −36 ÷ (x) = −4 ⇒−36÷ x = −4 ⇒x = (−36)÷( −4 )= 9
(iii). (x) ÷ (−4) = 24 ⇒x÷ (-4 ) = 24⇒x = 24×(−4) = -(24÷4) = -96
(iv) (x) ÷ 25 = 0 ⇒x÷ 25 = 0. ⇒ x = 25×0 = 0
(v). (x) ÷ (−1) = 36 ⇒x = 36×(-1) = -36
(vi). (x) ÷ 1 = −37 ⇒x = -37×1 = -37
(vi) 39 ÷ (x) = −1 ⇒x = -1×39 = -39
(viii) 1 ÷ (x) = −1 ⇒x = -1×1 = -1 (ix). −1 ÷ (x) = −1 ⇒−x = (-1)÷(-1)= 1.

Q3.Write (T) for true and (F) for false for each of the following statements.

(i) 0 ÷ (−4) = 0
(ii) (−6) ÷ 0 = 0
(iii) (−5) ÷ (−1) = −5
(iv) (−8) ÷ 1 = −8
(v) (−1) ÷ (−1) = −1
(vi) (−9) ÷ (−1) = 9
Answer.
 (i) True (T). Dividing zero by any integer gives zero.
(ii) False (F). Division by zero gives an indefinite number.
(iii) False (F). Because (-5)÷(-1) = 5÷1 = 5
(iv) True (T). −8÷1 =-(8÷1)= -8
(v) False (F). (−1)÷(−1 )= 1÷1 = 1.
(vi) True (T). (−9)÷(-1)= 9÷1 = 9

I. Perform the following operations on Integers :-
1. (+5) + (+2)       6. (-7) + (-3)         11. (+9) + (-4)
2. (-5) + (-2)         7. (+7) + (-3)        12. (-9) + (+4)
3. (+5) + (-2)        8. (-7) + (+3)        13. (-7) +(-2)
4. (-5) + (+2)        9. (+9) + (+4)       14. (-7) +(+2)
5. (+7) + (+3)     10. (-9) + (-4)         15. (+7) +(-2)

II. Perform the following operations on Integers :-
1. (+5) - (+2)         6. (-7) - (-3)          11. (+9) - (-4)
2. (-5) - (-2)           7. (+7) - (-3)         12. (-9) - (+4)
3. (+5) - (-2)          8. (-7) - (+3)         13. (-7) - (-2)
4. (-5) - (+2)          9. (+9) - (+4)        14. (-7) - (+2)
5. (+7) - (+3)       10. (-9) - (-4)          15. (+7) -(-2)

III. Perform the following operations on Integers :-
1. (+5) X (+2)                   11. (+9) X (-4)
2. (-5) X (-2)                     12. (-9) X (+4)
3. (+5) X (-2)                    13. (-7) X(-2)
4. (-5) X (+2)                    14. (-7) X (+2)
5. (+7) X (+3)                   15. (+7) X (-2)
6. (-7) X (-3)                     16 . -8 / -2
7. (+7) X (-3)                    17. -8/2
8. (-7) X (+3)                    18. 8/(-2)
9. (+9) X (+4)                   19. 24/(-2)
10. (-9) X (-4)                   20. -18/6

IV. Find

(a) (-7)-8-(-25)

(b) (-13) + 32-8-1

(c) (-7) + (-8) + (-90)

(d) 50-(-40)-(-2)

V. Fill in the blanks with <,> or = sign.

(a) (-3) + (-6) ________ (-3)-(-6)

(b) (-21)-(-10) ________ (-31) + (-11)

(c) 45-(-11) ________ 57+(-4)

(d) (-25)-(-42) _________ (-42)-(-25)

VI. Fill in the blanks using >,< or = sign.

(a) (-8) + (-4) _________ (-8) - (-4)

(b) (-3) + 7 – (19) _________ 15 -- 8 + (-9)

(c) 23 – 41 + 11 _________ 23 –-41 – 11

(d) 39 + (-24) – (15) ______ 36 --(-52) – (-36)

(e) -231 + 79 + 51 ______ -399 + 159 + 81

VII. Fill in the blanks

(a) (-8) + ______=0

(b) 13 + ______=0

(c) 12 + (-12)=______

(d) (-4) + _____=-12

(e)_____-15=-10

VIII. (i) Find each of the following products.

(a) (-20) X (-2) X (-5) X 7

(b) (-1) X (-5) X (-4) X (-6)

(ii) Fill in the blanks

(a) (-3) X ______=27

(b) 5 X _____=-35

(c)_____ X (-8)=(-56)

(d)_____ X (-12)=132

IX. Solve

(a) (-30) / 10 (f)0 / (-12)

(b) 50 / (-5) (g)(-31) / {(-30) + (-1)}

(c) (-36) / (-9) (h)({(-36) + 12} / 3

(d) (-49) / 49 (i){(-6) + 5)} /{(-2 + 1)}

(e) 13 / {(-2) + 1}

X. Fill in the blanks

(a) 369 /______=369

(b) (-75) / ______=-1

(c) (-206) / _____=-1

(d) -87 / _____=87

(e) ______ / 1 = -87

(f) ______/ 48=-1

(g) 20 / _____=-2

(h) ______ /(4)=-3

XI. Solve using suitable properties :-
1) (-5782396) X 9+5782396 X (-1)                   16) (-17) X (-29)
2) 8326429 X (-98) + (-8326429) X (-2)           17) (-27) X (-39)
3) (-7624928) X 997 + (7624928) X (-3)          18) (-37) X (-49)
4) (-98) X (-9) + 98                                          19) 8 X 53 X (-125)
5) (-237) X (-99) + 237                                    20) 8 X 56739 X (-125)
6) (-563) X (-999)+ 563                                   21) (-8) X 32597 X 12
7) 7 X (50-2)                                                    22) 8 X (-63298) X 125
8) 253 X (100-1)                                              23) (-8) X (-53) X 125
9) 567 X (1000-1)                                            24) 8 X (-56739) X (-125)
10) (-41) X 102                                               25) 15 X (-25) X (-4) X (-10)
11) (-34) X 1002                                             26) 23 X (-25) X (-4) X (-10)
12) (-67) X 101                                               27) 627 X (-125) X (-8)
13) (-84) X 99                                                 28) 583 X (-125) X (-5) X 16
14) (-24) X 998                                               29) (-749) X 25 X (-250) X 16
15) (-17) X 9997                                             30) 183X (-25) X 8 X 5

XII. In a class test containing 15 questions, 4 marks are given for every correct answer and (-2) marks for every incorrect answer.(i) Gurpreet attempts all questions but only 9 of her answers are correct. What is her total score? (ii) One of her friends gets only 5 answers correct. What will be her score?

XIII. In a class test containing 10 questions, 5 marks are awarded for every correct answer and (-2) marks are awarded for every incorrect answer and 0 for not attempting any of the questions.
(a) Mohan gets 4 correct answers and 6 incorrect answers. What is his score?
(b) Reshma gets 5 correct and 5 incorrect answers. What is her score?
(c) Heena gets 2 correct and 5 incorrect answers out of 7 questions she attempted. What is her score?

XIV. In a test (+5) marks are given for every correct answer and (-2) for incorrect answer
(i) Radhika answered all the questions and scored 30 marks though she got 10 correct answers
(ii) Jay also answered all the questions and scored (-12) marks though he got 4 correct answers. How many incorrect questions did they attempt?

XV. In a class test + 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) Radhika scored 20 marks if she has got 12 correct answers. How many questions had she attempted incorrectly?
ii) Mohini scored -5 marks in the test though she had got 7 correct answers. How many questions had she attempted incorrectly?

XVI. An elevator descends into the mine shaft @ 6 min per minute. If it descends from 10m above the ground, how long will it take to reach -350m?

XVII. Suppose we represent the distance above the ground, what will be its positive integer and that below the ground by a negative integer, then answer the following: -
(i) An elevator descends into a mine shaft at the rate of 5 m per minute. What will be its position after 1 hour?
(ii) If it begins descending from 15 m above the ground, what will be its position after 45 minutes?

XVIII. A certain freezing process requires that room temperature be lowered from 40º C at the rate of 5º C every hour. What will be the room temperature 10 hours after the process begins? 

XIX. The temperature at the noon was 10°C above zer o. If it decreases @ 2°C per hour until midnight, at what rate would the temperature will be 8°C below zero? What would be the temperature at midnight?

XX.A cement company earns a profit of Rs 8 per bag of white cement sold and a loss of Rs 5 per bag of grey cement sold.
(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
(b) What is the no. of white cement bags it must sell to have neither profit nor loss, if the no. of grey bags sold is 6,400 bags.

Q (21) A shopkeeper earns a profit of Re 1 by selling 1 pen and incurs a loss of 40 paise per pencil while selling pencils of her old stock.
(i) In a particular month she incurs a loss of Rs 5. In this period, she sold 45 pens. How many pencils did she sell in this period?
(ii) In the next month she earns neither profit nor loss. If she sold 70 pens, how many pencils were sold?

Question. –32 + 47
Answer : 15

Question. Find the value of (–23) {(–5) + (25)}
Answer : –460

Question. The sum of two integers is (–16). If one of them is 53, find the other.
Answer : –69

Question. What will be the sum of (–2035) and 285?
Answer : –1750

Question. Subtract –134 from the sum of 37 and –87.
Answer : 84

Question. Find the product of (–25) and (14).
Answer : –350

Question. An elevator descends into a mine shaft at the rate of 6 meter per minute. If the descentent starts from the ground level, how long will it take to reach 240 meter?
Answer : 40 minutes

Question. Find the value of 13 – (15 – 6 3).
Answer : 0

Question. Rohit earns profit of Re. 1 by selling one pen and suffers a loss of 50 paise by selling one pencil. One day Rohit earns Rs. 5. If he sold 25 pens, how many pencils did he sell?
Answer : 40 pencils

Question. Subtract the sum of –1878 and 878 from 2000.
Answer : 3000

Question. Find the sum of (8 + 9) × 10 and 8 + 9 × 10.
Answer : 268

Question. Multiply (–36) by (–50).
Answer : 1800

Question. Divide 324 by (–27).
Answer : –12

Question. What is the difference between [36 – (–64) and (–64) – 36]
Answer : 200

Question. If the sum of two integers is –1600 and one of them is 549, find the other number.
Answer : –2149

Question. Find the product : (–2) × 36 × (–5).
Answer : 360

Question. Multiply : (–2) × (–4) × (–5) × (–8)
Answer : 320

Question. The sum of two integers is 65. If one of them is –35, find the other
Answer : 100

Question. Determine the integer when it is divided by (–9), the answer is 23.
Answer : –207

Question. Find the value of 1569 × 887 – 569 × 887.
Answer : 887000 

Question. The product of two integers is –51. If one integer is –17, find the other.
Answer : 3

Question. Divide (–98) by (–14)
Answer : 7

Question. Which integer if multiplied by (–1) will be 432 ?
Answer : –432

Question. What will be the sum of (–1006) and (–934)?
Answer : –1940

Question. The temperature at 1.00 p.m. was 10°C above zero. If it decreases at the rate of 2°C per hour until 10 p.m., at what time would the temperature be 6°C below zero?
Answer : 9 p.m.

Question 1. Verify \(a - (-b) = a + b\) for the following values of 'a' and 'b'.
(a) \(a = 75\), \(b = 84\)
Answer: LHS = \(a - (-b) = 75 - (-84) = 75 + 84 = 159\)
RHS = \(a + b = 75 + 84 = 159\)
Since LHS = RHS, verified.
In simple words: Subtracting a negative number is the same as adding its positive counterpart.

Exam Tip: Remember the rule negative multiplied by negative gives positive: \(-(-b) = +b\).

Question 1.(b) Verify \(a - (-b) = a + b\) for \(a = 118\), \(b = 125\).
Answer: LHS = \(118 - (-125) = 118 + 125 = 243\)
RHS = \(118 + 125 = 243\)
Since LHS = RHS, verified.
In simple words: Two negative signs next to each other combine to form a positive sign.

Exam Tip: Always rewrite double negative signs as a single positive sign before calculating.

Question 1.(c) Verify \(a - (-b) = a + b\) for \(a = 25\), \(b = 30\).
Answer: LHS = \(25 - (-30) = 25 + 30 = 55\)
RHS = \(25 + 30 = 55\)
Since LHS = RHS, verified.
In simple words: Both sides yield the exact same value.

Exam Tip: Show all substitution steps clearly to avoid sign errors.

Question 2. Write down a pair of integers whose
(a) Sum is \(-3\)
Answer: \(-5\) and \(2\)
\(-5 + 2 = -3\)
In simple words: Adding a positive number to a larger negative number results in a negative sum.

Exam Tip: Multiple pairs can exist for this question; any pair that adds up to the target integer is correct.

Question 2.(b) Write down a pair of integers whose sum is \(0\).
Answer: \(-7\) and \(7\)
\(-7 + 7 = 0\)
In simple words: An integer and its additive inverse always add up to zero.

Exam Tip: Any number and its negative counterpart form a pair with a sum of zero.

Question 2.(c) Write down a pair of integers whose difference is \(2\).
Answer: \(5\) and \(3\)
\(5 - 3 = 2\)
In simple words: The distance between 5 and 3 on the number line is 2 units.

Exam Tip: Ensure the order of subtraction matches the required difference.

Question 2.(d) Write down a pair of integers whose difference is \(-5\).
Answer: \(2\) and \(7\)
\(2 - 7 = -5\)
In simple words: Subtracting a larger number from a smaller number yields a negative difference.

Exam Tip: Pay close attention to the sign of the resulting difference.

Question 3. Verify the following:
a) \((-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)]\)
Answer: LHS = \((-21) \times [-4 - 6] = (-21) \times (-10) = 210\)
RHS = \([(-21) \times (-4)] + [(-21) \times (-6)] = 84 + 126 = 210\)
Since LHS = RHS, verified.
In simple words: This demonstrates the distributive property of multiplication over addition for integers.

Exam Tip: Use the distributive property \(a \times (b + c) = (a \times b) + (a \times c)\) to simplify calculations.

Question 3. b) \(15 \times [6 + (-3)] = [15 \times 6] + [15 \times (-3)]\)
Answer: LHS = \(15 \times [6 - 3] = 15 \times 3 = 45\)
RHS = \([15 \times 6] + [15 \times (-3)] = 90 + (-45) = 90 - 45 = 45\)
Since LHS = RHS, verified.
In simple words: Multiplying a number by a sum gives the same result as multiplying by each addend separately and adding the products.

Exam Tip: Evaluate inside the brackets first when checking LHS.

Question 3. c) \((-15) \times [(-8) + (-6)] = [(-15) \times (-8)] + [(-15) \times (-6)]\)
Answer: LHS = \((-15) \times [-8 - 6] = (-15) \times (-14) = 210\)
RHS = \([(-15) \times (-8)] + [(-15) \times (-6)] = 120 + 90 = 210\)
Since LHS = RHS, verified.
In simple words: The distributive law holds true for negative integers as well.

Exam Tip: Product of two negative integers is always positive.

Question 4. Evaluate:

a) \((-100) \div 5\)
Answer: \((-100) \div 5 = -20\)
In simple words: Dividing a negative integer by a positive integer results in a negative quotient.

Exam Tip: Unlike signs in division always yield a negative result.

Question 4. b) \((-36) \div (-4)\)
Answer: \((-36) \div (-4) = 9\)
In simple words: Dividing a negative integer by another negative integer results in a positive quotient.

Exam Tip: Like signs in division yield a positive result.

Question 4. c) \((-41) \div [(-40) + (-1)]\)
Answer: Denominator = \(-40 - 1 = -41\)
\((-41) \div (-41) = 1\)
In simple words: Any non-zero integer divided by itself equals 1.

Question 4. d) \(0 \div (-18)\)
Answer: \(0 \div (-18) = 0\)
In simple words: Zero divided by any non-zero integer is always zero.

Exam Tip: Do not confuse dividing zero by a number with dividing a number by zero.

Question 4. e) \([(-36) \div 12] \div 3\)
Answer: First solve the bracket: \((-36) \div 12 = -3\)
Then divide by 3: \(-3 \div 3 = -1\)
In simple words: Follow BODMAS/PEMDAS by evaluating the innermost bracket first.

Exam Tip: Division is not associative, so parentheses dictate the order of operation.

Question 4. f) \((-50) \div (50)\)
Answer: \((-50) \div 50 = -1\)
In simple words: A number divided by its opposite gives \(-1\).

Question 4. g) \(60 \div (-6)\)
Answer: \(60 \div (-6) = -10\)
In simple words: Positive divided by negative is negative.

Question 4. h) \((-48) \div -48\)
Answer: \((-48) \div (-48) = 1\)
In simple words: Negative divided by negative is positive.

Question 4. i) \((-13) \div (13)\)
Answer: \((-13) \div 13 = -1\)
In simple words: Negative divided by positive is negative.

Question 5. Do as directed:

1. In a test (+5) marks are given for every correct answer and (-2) marks are given every incorrect answer.
(i) Radhika answered all the questions and scored 30 marks though she got 10 correct answers.
Answer: Marks for 10 correct answers = \(10 \times 5 = 50\)
Let \(x\) be the number of incorrect answers.
Marks obtained for incorrect answers = \(-2x\)
Total score = Marks for correct answers + Marks for incorrect answers
\(50 + (-2x) = 30\)
\(-2x = 30 - 50\)
\(-2x = -20\)
\(x = 10\)
Radhika attempted 10 incorrect answers.
In simple words: Subtract her correct score from her total score to find the deficit caused by incorrect answers.

Exam Tip: Set up a linear equation representing total score to solve for unknown attempts.

Question 5. (ii) Jay also answered all the questions and scored (-12) marks though he got 4 correct answers. How many incorrect answers had they attempted?
Answer: Marks for 4 correct answers = \(4 \times 5 = 20\)
Let \(y\) be the number of incorrect answers.
Total score = Marks for correct answers + Marks for incorrect answers
\(20 + (-2y) = -12\)
\(-2y = -12 - 20\)
\(-2y = -32\)
\(y = 16\)
Jay attempted 16 incorrect answers.
In simple words: Account for positive marks earned and solve for the negative deductions needed to reach \(-12\).

Question 6. In a class test containing 15 questions 4 marks are given for every correct answer and (-2) marks are given for every incorrect answers:
i) Gurupreet attempts all question but only 9 of her answers are correct. What is her total score?
Answer: Number of correct answers = 9, so marks = \(9 \times 4 = 36\)
Total questions = 15, so incorrect answers = \(15 - 9 = 6\)
Marks for incorrect answers = \(6 \times (-2) = -12\)
Total score = \(36 + (-12) = 24\)
Gurupreet's total score is 24.
In simple words: Add the points from correct answers and subtract the penalty points for incorrect answers.

Exam Tip: Find the number of incorrect answers by subtracting correct answers from the total number of questions.

Question 6. ii) One of her friends gets only 5 answers correct. What will be her score?
Answer: Number of correct answers = 5, so marks = \(5 \times 4 = 20\)
Incorrect answers = \(15 - 5 = 10\)
Marks for incorrect answers = \(10 \times (-2) = -20\)
Total score = \(20 + (-20) = 0\)
Her score is 0.
In simple words: Positive points gained equal negative points lost, resulting in a zero score.

Exam Tip: A score of zero means positive marks exactly balance out the negative marks.

Question 7. Write five pairs of integers \((a, b)\) such that \(a \div b = -6\).
Answer: 1. \((-6, 1)\) since \(-6 \div 1 = -6\)
2. \((6, -1)\) since \(6 \div (-1) = -6\)
3. \((-12, 2)\) since \(-12 \div 2 = -6\)
4. \((12, -2)\) since \(12 \div (-2) = -6\)
5. \((-18, 3)\) since \(-18 \div 3 = -6\)
In simple words: Choose any multiple of \(-6\) as the first integer and its corresponding factor as the second.

Exam Tip: One integer in each pair must be negative for the quotient to be negative.

Question 8. Find
i) \((-3) \times (-6) \times (-2) \times (-1)\)
Answer: \((-3) \times (-6) = 18\)
\(18 \times (-2) = -36\)
\(-36 \times (-1) = 36\)
In simple words: An even number of negative signs in multiplication yields a positive product.

Exam Tip: Count the total number of negative signs first: even count means positive, odd count means negative.

Question 8. ii) \((-12) \times (-11) \times (10)\)
Answer: \((-12) \times (-11) = 132\)
\(132 \times 10 = 1320\)
In simple words: Multiply two negative numbers first to get a positive number, then multiply by 10.

Exam Tip: Multiplying by 10 simply appends a zero to the product.

Question 8. iii) \((-320) \times (-1)\)
Answer: \((-320) \times (-1) = 320\)
In simple words: Multiplying an integer by \(-1\) changes its sign.

Exam Tip: \(-1\) acts as the sign-reversing operator in multiplication.

Question 8. iv) \((-18) \times 0 \times (-16)\)
Answer: Any number multiplied by 0 is 0.
\(-18 \times 0 \times (-16) = 0\)
In simple words: The presence of zero in a string of multiplications makes the entire product zero.

Exam Tip: Look out for zero as a factor to instantly determine the product.

Question 8. v) \(9 \times (-5) \times (-3)\)
Answer: \(9 \times (-5) = -45\)
\(-45 \times (-3) = 135\)
In simple words: Two negative factors produce a positive product.

Exam Tip: Perform multiplications sequentially from left to right.

Question 8. vi) \((-41) \times 10\)
Answer: \((-41) \times 10 = -410\)
In simple words: Attach a zero to the end of \(-41\).

Exam Tip: Multiplying by powers of 10 shifts digits accordingly.

Question 8. vii) \((-21) \times (-30)\)
Answer: \(21 \times 30 = 630\)
In simple words: Negative multiplied by negative gives positive.

Question 8. viii) \((-1) \times 225\)
Answer: \((-1) \times 225 = -225\)
In simple words: Multiplying by \(-1\) negates the number.

Exam Tip: Any number multiplied by 1 is the number itself; multiply by \(-1\) to invert the sign.

Question 8. ix) \((-22) \times (-1)\)
Answer: \((-22) \times (-1) = 22\)
In simple words: Inverting a negative number gives a positive number.

Exam Tip: Two negative signs cancel each other out.

Question 8. x) \((-20) \times (-2) \times (-5) \times 7\)
Answer: \((-20) \times (-2) = 40\)
\(40 \times (-5) = -200\)
\(-200 \times 7 = -1400\)
In simple words: Three negative signs result in a negative product.

Exam Tip: Group numbers conveniently, such as \((-2) \times (-5) = 10\), to simplify mental math.

Question 9. Use the sign \(>, <, =\)
i) \(29 + (-18) - 15 \square 36 - (-15) + 28\)
Answer: LHS = \(29 - 18 - 15 = 11 - 15 = -4\)
RHS = \(36 + 15 + 28 = 51 + 28 = 79\)
Since \(-4 < 79\), the correct sign is \(<\).
In simple words: Negative numbers are always smaller than positive numbers.

Exam Tip: Simplify both sides completely to single integers before comparing.

Question 9. ii) \(-241 + 76 + 86 \square -399 + 163 + 45\)
Answer: LHS = \(-241 + 162 = -79\)
RHS = \(-399 + 208 = -191\)
Since \(-79 > -191\), the correct sign is \(>\).
In simple words: A smaller negative number is greater in value than a larger negative number.

Exam Tip: Remember that closer to zero means greater among negative numbers.

Question 9. iii) \((-3) + 7 - (-18) \square 18 - 9 + (-6)\)
Answer: LHS = \(-3 + 7 + 18 = 4 + 18 = 22\)
RHS = \(18 - 9 - 6 = 9 - 6 = 3\)
Since \(22 > 3\), the correct sign is \(>\).
In simple words: Evaluate each side using standard addition and subtraction rules.

Exam Tip: Convert double negatives like \(-(-18)\) into \(+18\) first.

Question 9. iv) \((-8) + (-6) \square (-8) - (-6)\)
Answer: LHS = \(-8 - 6 = -14\)
RHS = \(-8 + 6 = -2\)
Since \(-14 < -2\), the correct sign is \(<\).
In simple words: Adding a negative decreases the value more than subtracting a negative.

Exam Tip: Pay close attention to operation signs inside parentheses versus between numbers.

Question 9. v) \((-18) + (18) \square (-31) + (31)\)
Answer: LHS = \(-18 + 18 = 0\)
RHS = \(-31 + 31 = 0\)
Since \(0 = 0\), the correct sign is \(=\).
In simple words: The sum of any integer and its additive inverse is zero on both sides.

Exam Tip: Opposite numbers always sum to zero.

Question 9. vi) \(86 - 45 + 23 \square -36 - (20) - (-8)\)
Answer: LHS = \(41 + 23 = 64\)
RHS = \(-36 - 20 + 8 = -56 + 8 = -48\)
Since \(64 > -48\), the correct sign is \(>\).
In simple words: Positive total is always greater than negative total.

Exam Tip: Work out operations from left to right for both expressions.

Question 10. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack's scores in five successive rounds were \(65, -10, -15, 20, 30\). What was his total score at the end?
Answer: Total score = \(65 + (-10) + (-15) + 20 + 30\)
\(= 65 - 10 - 15 + 20 + 30\)
\(= 55 - 15 + 20 + 30\)
\(= 40 + 20 + 30\)
\(= 60 + 30 = 90\)
Jack's total score was 90.
In simple words: Add all individual round scores together, taking negative scores as deductions.

Exam Tip: Group positive scores together and negative scores together to simplify addition.

Question 11. In a quiz, team A scored \(-50, 30, 0\) and team B scored \(60, 30, -40\) in three successive rounds. Which team scored more?
Answer: Total score of Team A = \(-50 + 30 + 0 = -20\)
Total score of Team B = \(60 + 30 + (-40) = 90 - 40 = 50\)
Since \(50 > -20\), Team B scored more.
In simple words: Team B has a higher cumulative total than Team A.

Exam Tip: Compare final sums rather than individual round scores.

Question 12. The temperature at 12 noon was \(10^\circ\text{C}\) above zero. If it decreases at the rate of \(2^\circ\text{C}\) per hour until midnight, at what time would the temperature be \(8^\circ\text{C}\) below zero? What would be the temperature at midnight?
Answer: Initial temperature at 12 noon = \(+10^\circ\text{C}\)
Target temperature = \(-8^\circ\text{C}\)
Total decrease needed = \(10 - (-8) = 10 + 8 = 18^\circ\text{C}\)
Rate of decrease = \(2^\circ\text{C}\) per hour
Time taken = \(18 \div 2 = 9\) hours
Time when temperature is \(-8^\circ\text{C}\) = 9 hours after 12 noon = 9:00 PM.
From 12 noon to midnight (12 AM) is 12 hours.
Total temperature drop in 12 hours = \(12 \times 2 = 24^\circ\text{C}\)
Midnight temperature = \(10^\circ\text{C} - 24^\circ\text{C} = -14^\circ\text{C}\).
In simple words: Calculate total temperature change over time and subtract it from the starting temperature.

Exam Tip: Use a number line concept where above zero is positive and below zero is negative.

Question 13. Replace the blank with an integer to make it a true statement.
a) \(\_\_\_\_\_\_\_\_ \times (-12) = -60\)
Answer: \(-60 \div (-12) = 5\)
The missing integer is \(5\).
In simple words: Divide the product by the given factor to find the missing multiplier.

Exam Tip: Check your answer by multiplying \(5 \times (-12)\) to ensure it equals \(-60\).

Question 13. b) \(5 \times \_\_\_\_\_\_\_\_ = -35\)
Answer: \(-35 \div 5 = -7\)
The missing integer is \(-7\).
In simple words: Divide \(-35\) by \(5\) to get the required negative factor.

Exam Tip: Positive multiplied by negative gives a negative product.

Question 13. c) \((-8) \times \_\_\_\_\_\_\_\_ = 72\)
Answer: \(72 \div (-8) = -9\)
The missing integer is \(-9\).
In simple words: Two negative factors produce a positive product.

Exam Tip: Ensure signs match the multiplication rule for integers.

Question 13. d) \(\_\_\_\_\_\_\_\_ \div (-3) = 9\)
Answer: \(9 \times (-3) = -27\)
The missing integer is \(-27\).
In simple words: Multiply the quotient by the divisor to find the dividend.

Exam Tip: In division, dividend = quotient \(\times\) divisor.

Question 13. e) \((-20) \div \_\_\_\_\_\_\_\_ = 5\)
Answer: \((-20) \div 5 = -4\)
The missing integer is \(-4\).
In simple words: Divide the dividend by the quotient to find the divisor.

Exam Tip: Verify by substituting \(-4\) back into the division expression.

I FILL IN THE BLANKS

1. ________________ is the additive identity and _______________ is the multiplicative identity of integers.

2. (-1) x even number of times = _______________

3. (-1) x odd number of times = _______________

4. 38 ÷ 0 = _______________

5. 0 ÷ 11 = _______________

6. 13 ÷ 1 + _______________

7. 55 + _______________ = 0

8. (-31) + _______________ = 0

9. (-55) + _______________ = -89

10. (-33) + _______________ = 79

11. 1000 + _______________ = - 1000

12. [ (-7) + 9] + (_______________) = 9 + [ -7 + 4]

13. (-6) + 2 = 2 (_______________)

14. 251 ÷ _______________ = 1

15. (-70) ÷ _______________ = 5