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## Chapter 1 Integers Exercise 1.3

Question 1: Find each of the following products:
(a) 3 × (–1)

= 3 × (-1)
= -3

(b) (–1) × 225

= (-1) × 225
= -225

(c) (–21) × (–30)

= (-21) × (-30)
= 630

(d) (–316) × (–1)

= (-316) × (-1)
= 316

(e) (–15) × 0 × (–18)

= (-15) × 0 × (–18)
= 0

(f) (–12) × (–11) × (10)

= (-12) × (-11) × (10)
= 132 × 10
= 1320

(g) 9 × (–3) × (– 6)

= 9 × (-3) × (-6)
= 9 × 18
= 162

(h) (–18) × (–5) × (– 4)

= (-18) × (-5) × (-4)
= 90 × -4
= -360

(i) (–1) × (–2) × (–3) × 4

= [(–1) × (-2)] × [(-3) × 4]
= 2 × (-12)
= -24

(j) (–3) × (–6) × (–2) × (–1)

= [(-3) × (-6)] × [(-2) × (-1)]
= 18 × 2
= 36

Question 2: Verify the following:
(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]

Left Hand Side = 18 × [7 + (-3)]
= 18 × [7 - 3]
= 18 × 4
= 72
Right Hand Side = [18 × 7] + [18 × (–3)]
= [126] + [-54]
= 126 - 54
= 72
Comparing LHS and RHS
72 = 72
LHS = RHS
Therefore, the given equation is verified.

(b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]

Left Hand Side = (-21) × [(-4) + (-6)]
= (-21) × [-4 - 6]
= (-21) × [-10]
= 210
Right Hand Side = [(-21) × (-4)] + [(-21) × (-6)]
= 84 + 126
= 210
Comparing LHS and RHS
210 = 210
LHS = RHS
Therefore, the given equation is verified.

Question 3:
(i) For any integer a, what is (–1) × a equal to?
(-1) × a = -a

(ii) Determine the integer whose product with (–1) is
(a) –22

= -22 × (-1)
= 22

(b) 37

= 37 × (-1)
= -37

(c) 0

= 0 × (-1)
= 0

Question 4: 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 × 2 = -2
= -1 × 1 = -1
= -1 × 0 = 0
= -1 × -1 = 1

Question 5: Find the product, using suitable properties:
(a) 26 × (– 48) + (– 48) × (–36)

= a × (b + c) = (a × b) + (a × c)
Let a be -48, b be 26 and c be -36
= 26 × (-48) + (-48) × (-36)
= -48 × (26 + (-36)
= -48 × (26 - 36)
= -48 × (-10)
= 480

(b) 8 × 53 × (–125)

= a × b = b × a
= 8 × [53 × (-125)]
= 8 × [(-125) × 53]
= [8 × (-125)] × 53
= [-1000] × 53
= -53000

(c) 15 × (–25) × (– 4) × (–10)

= a × b = b × a
= 15 × [(-25) × (-4)] × (-10)
= 15 × [100] × (-10)
= 15 × [-1000]
= -15000

(d) (– 41) × 102

= a × (b + c) = (a × b) + (a × c)
= (-41) × (100 + 2)
= (-41) × 100 + (-41) × 2
= -4100 - 82
= -4182

(e) 625 × (–35) + (– 625) × 65

= a × (b + c) = (a × b) + (a × c)
= 625 × [(-35) + (-65)]
= 625 × [-100]
= -62500

(f) 7 × (50 – 2)

= a × (b - c) = (a × b) - (a × c)
= (7 × 50) - (7 × 2)
= 350 - 14
= 336

(g) (–17) × (–29)

= a × (b + c) = (a × b) + (a × c)
= (-17) × [-30 + 1]
= [(-17) × (-30)] + [(-17) × 1]
= [510] + [-17]
= 493

(h) (–57) × (–19) + 57

= a × (b + c) = (a × b) + (a × c)
= (57 × 19) + (57 × 1)
= 57 [19 + 1]
= 57 × 20
= 1140

Question 6: 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?

Let the lowered temperature be negative.
Initial temperature = 40oC
Change in temperature per hour = -5oC
Change in temperature after 10 hours = (-5) × 10 = -50oC
The final room temperature after 10 hours of freezing process =
= 40oC + (-50oC)
= -10oC

Question 7: 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?

Marks awarded for 1 correct answer = 5
Total marks awarded for 4 correct answer = 4 × 5 = 20
Marks awarded for 1 wrong answer = -2
Total marks awarded for 6 wrong answer = 6 × -2 = -12
Total score obtained by Mohan =
= 20 + (-12)
= 20 – 12
= 8

(ii) Reshma gets five correct answers and five incorrect answers, what is her score?

Marks awarded for 1 correct answer = 5
Total marks awarded for 5 correct answer = 5 × 5 = 25
Marks awarded for 1 wrong answer = -2
Total marks awarded for 5 wrong answer = 5 × -2 = -10
Total score obtained by Reshma = 25 + (-10)
= 25 - 10
= 15

(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?

Marks awarded for 1 correct answer = 5
Total marks awarded for 2 correct answer = 2 × 5 = 10
Marks awarded for 1 wrong answer = -2
Total marks awarded for 5 wrong answer = 5 × -2 = -10
Marks awarded for questions not attempted is = 0
∴Total score obtained by Heena = 10 + (-10)
= 10 - 10
= 0

Question 8: A cement company earns a profit of ₹ 8 per bag of white cement sold and a loss of ₹ 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?

Let the profit be denoted by positive integer and loss denoted by negative integer.
Cement company earns a profit on selling 1 bag of white cement = ₹ 8 per bag
Cement company earns a profit on selling 3000 bags of white cement = 3000 × ₹ 8
= ₹ 24000
Loss on selling 1 bag of grey cement = -₹ 5 per bag
Loss on selling 5000 bags of grey cement = 5000 × -₹ 5
= -₹ 25000
Total loss or profit earned by the cement company = profit + loss
= 24000 + (-25000)
= -₹1000
Therefore, a loss of ₹ 1000 will be incurred by the company.

(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.

Let the profit be denoted by positive integer and loss denoted by negative integer.
Cement company earns a profit on selling 1 bag of white cement = ₹ 8 per bag
Let the number of white cement bags be x.
Cement company earns a profit on selling x bags of white cement = (x) × ₹ 8
= ₹ 8x
Loss on selling 1 bag of grey cement = -₹ 5 per bag
Loss on selling 6400 bags of grey cement = 6400 × -₹ 5
= -₹ 32000
= Profit + loss = 0
= 8x + (-32000) =0
= 8x = 32000
= x = 32000/8
= x = 4000
Therefore, the 4000 bags of white cement have neither profit nor loss.

Question 9: Replace the blank with an integer to make it a true statement.
(a) (–3) × _____ = 27

Let the missing integer be x.
= (-3) × (x) = 27
= x = - (27/3)
= x = -9
= (-3) × (-9) = 27

(b) 5 × _____ = –35

Let the missing integer be x.
= (5) × (x) = -35
= x = - (-35/5)
= x = -7
= (5) × (-7) = -35

(c) _____ × (– 8) = –56

Let the missing integer be x.
= (x) × (-8) = -56
= x = (-56/-8)
= x = 7
= (7) × (-8) = -56

(d) _____ × (–12) = 132