## Integers

EXERCISE 1.3
CBSE Class 7 Maths

Q1: Find each of the following products:
(a) 3 × (–1)
(b) (–1) × 225
(c) (–21) × (–30)
(d) (–316) × (–1)
(e) (–15) × 0 × (–18)
(f) (–12) × (–11) × (10)
(g) 9 × (–3) × (– 6)
(h) (–18) × (–5) × (– 4)
(i) (–1) × (–2) × (–3) × 4
(j) (–3) × (–6) × (–2) × (–1)

(a) 3 × (–1) = = –3

(b) (–1) × 225 = -225

(c) (–21) × (–30) = 630

(d) (–316) × (–1) = 316

(e) (–15) × 0 × (–18) = 0

(f) (–12) × (–11) × (10) = 132 × 10 = 1320

(g) 9 × (–3) × (– 6) = 9 × 18 = 162

(h) (–18) × (–5) × (– 4) = 90 × (–4) = –360

(i) (–1) × (–2) × (–3) × 4 = –6 × 4 = -24

(j) (–3) × (–6) × (–2) × (–1) = 18 × 2 = 36

Q2: Verify the following:
(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]
(b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]

(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]
⇒ 18 × 4 = 126 + (–54)
⇒ 72 = 72
LHS = RHS
Hence verified.

(b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]
⇒ (–21) × (–10) = 84 + 126
⇒ 210 = 210
LHS = RHS
Hence verified.

Q3: (i) For any integer a, what is (–1) × a equal to?
(ii) Determine the integer whose product with (–1) is
(a) –22
(b) 37
(c) 0

(i) -1 × a = -a
(ii)
(a) -1 × -22 = 22
(b) -1 ×  37 = -37
(b) -1 ×  0  = 0

Q4: Starting from (–1) × 5, write various products showing some pattern to show (–1) × (–1) = 1.

(-1) × 5 = -5
(-1) × 4 = -4
(-1) × 3 = -3
(-1) × 1 = -1
(-1) × 0 =  0
(-1) × (-1) = 1
(-1) × (-2) = 2

Q5: Find the product, using suitable properties:
(a) 26 × (– 48) + (– 48) × (–36)
(b) 8 × 53 × (–125)
(c) 15 × (–25) × (– 4) × (–10)
(d) (– 41) × 102
(e) 625 × (–35) + (– 625) × 65
(f) 7 × (50 – 2)
(g) (–17) × (–29)
(h) (–57) × (–19) + 57

(a) 26 × (– 48) + (– 48) × (–36)

= (-48) × [26 + (-36)] [Distributive property]
= (-48) × [26 - 36]
= (-48) × (-10)
= 480

(b) 8 × 53 × (–125)

= 53 × [8 × (–125)] [Commutative Property]
= 53 × -1000
= -53000

(c) 15 × (–25) × (– 4) × (–10)

=  15 × (–10) × (–25) × (– 4)  [Commutative Property]
=  [15 × (–10)] × [(–25) × (– 4)]
=  -150 × 100
= -15000

(d) (– 41) × 102
= (-41) × [100 + 2] [Distributive Property]
= (-41 × 100) + (-41 × 2)
= -4100 + (-82)
= -4100 - 82
= -4182

(e) 625 × (–35) + (– 625) × 65

= 625 × (–35) + 625 × (–65)
= 625 × [(–35) + (-65)] [Distributive Property]
= 625 × [-100]
= -62500

(f) 7 × (50 – 2)
= 7 × (50 – 2) [Distributive Property]
= (7 × 50) - (7 × 2)
= 350 - 14
= 336

(g) (–17) × (–29)

= (–17) × [(-30) + 1] [Distributive Property]
= (-17) × (-30) + (-17) × 1
= 510 + (-17)
= 510 - 17
= 493

(h) (–57) × (–19) + 57

= 57 × 19 + 57
= 57 × (19 + 1)
= 57 × 20
= 1140

Q6: A certain freezing process requires that room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

Answer: Present room temperature = 40°C
Decreasing in temperature per hour = 5°C
Room temperature after 10 hours = 40°C + 10 × (–5 o C )
= 40°C – 50°C
= –10°C

Thus, the room temperature after 10 hours is –10°C.

Q7: 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 questions not attempted.
(i) Mohan gets four correct and six incorrect answers. What is his score?
(ii) Reshma gets five correct answers 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?

Marks given for correct answer = +5
Marks deducted for incorrect = -2

(i) Mohan's correct answers = 4
Marks for correct answers = 4 × (+5) = 20
Marks for incorrect = 6 × (-2) = -12
Total marks for Mohan = 20 + (-12) = 20 - 12 = 8

(ii) Reshma's correct answers = 5
Marks for correct answers = 5 × (+5) = 25
Marks for incorrect = 5 × (-2) = -10
Total marks for Reshma = 25 + (-10) = 25 - 10 = 15

(iii) Heena's correct answers = 2
Marks for correct answers = 2 × (+5) = 10
Marks for incorrect = 5 × (-2) = -10
Total marks for Heena = 10 + (-10) = 10 - 10 = 0

Q8: 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 number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags.

Profit per white cement bag = ₹ +8
Loss per grey cement bag = ₹ -5

(a) Profit on 3000 white cement bags   = 3000 × ₹ +8 = ₹ +24,000
Loss on 5000 grey cement bags   = 5000 × ₹ -5 = ₹ -25,000
Total Profit or Loss = +24,000 + (-25,000)
= ₹ -1000
-ve value indicates it is a loss of ₹ 1000

(b) Let number of white cement bags = y
Profit =  y × 8
Number of grey bags = 6400
Loss on grey cement bags   = ₹-5 ×6,400
∵ Profit + Loss = 0
∴ y × 8 + (-5 × 6,400) = 0
y × 8 = 5 × 6,400
y = (5 × 6400)/8 = 5 × 800 = 4000 bags

Q9: Replace the blank with an integer to make it a true statement.
(a) (–3) × _____ = 27
(b) 5 × _____ = –35
(c) _____ × (– 8) = –56
(d) _____ × (–12) = 132

(a) (–3) × (-9) = 27
(b) 5 × (-7) = –35
(c) 7 × (– 8) = –56
(d) (-11) × (–12) = 132  