## Chapter 1 Integers Exercise 1.3

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

= 3 × (-1)

= -3

**(b) (–1) × 225**

Answer:

Answer:

= (-1) × 225

= -225

**(c) (–21) × (–30)**

Answer:

Answer:

= (-21) × (-30)

= 630

**(d) (–316) × (–1)**

Answer:

Answer:

= (-316) × (-1)

= 316

**(e) (–15) × 0 × (–18)**

Answer:

Answer:

= (-15) × 0 × (–18)

= 0

**(f) (–12) × (–11) × (10)**

Answer:

Answer:

= (-12) × (-11) × (10)

= 132 × 10

= 1320

**(g) 9 × (–3) × (– 6)**

Answer:

Answer:

= 9 × (-3) × (-6)

= 9 × 18

= 162

**(h) (–18) × (–5) × (– 4)**

Answer:

Answer:

= (-18) × (-5) × (-4)

= 90 × -4

= -360

**(i) (–1) × (–2) × (–3) × 4**

Answer:

Answer:

= [(–1) × (-2)] × [(-3) × 4]

= 2 × (-12)

= -24

**(j) (–3) × (–6) × (–2) × (–1)**

Answer:

Answer:

= [(-3) × (-6)] × [(-2) × (-1)]

= 18 × 2

= 36

**Question 2: Verify the following:**

(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]

Answer:

(a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]

Answer:

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)]**

Answer:

Answer:

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?

Answer:(-1) × a = -a

(i) For any integer a, what is (–1) × a equal to?

Answer:

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

(a) –22

Answer:

(a) –22

Answer:

= -22 × (-1)

= 22

**(b) 37**

Answer:

Answer:

= 37 × (-1)

= -37

**(c) 0**

Answer:

Answer:

= 0 × (-1)

= 0

**Question 4: Starting from (–1) × 5, write various products showing some pattern to show**

(–1) × (–1) = 1.

Answer:

(–1) × (–1) = 1.

Answer:

= -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)

Answer:

(a) 26 × (– 48) + (– 48) × (–36)

Answer:

= 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)**

Answer:

Answer:

= a × b = b × a

= 8 × [53 × (-125)]

= 8 × [(-125) × 53]

= [8 × (-125)] × 53

= [-1000] × 53

= -53000

**(c) 15 × (–25) × (– 4) × (–10)**

Answer:

Answer:

= a × b = b × a

= 15 × [(-25) × (-4)] × (-10)

= 15 × [100] × (-10)

= 15 × [-1000]

= -15000

**(d) (– 41) × 102**

Answer:

Answer:

= a × (b + c) = (a × b) + (a × c)

= (-41) × (100 + 2)

= (-41) × 100 + (-41) × 2

= -4100 - 82

= -4182

**(e) 625 × (–35) + (– 625) × 65**

Answer:

Answer:

= a × (b + c) = (a × b) + (a × c)

= 625 × [(-35) + (-65)]

= 625 × [-100]

= -62500

**(f) 7 × (50 – 2)**

Answer:

Answer:

= a × (b - c) = (a × b) - (a × c)

= (7 × 50) - (7 × 2)

= 350 - 14

= 336

**(g) (–17) × (–29)**

Answer:

Answer:

= a × (b + c) = (a × b) + (a × c)

= (-17) × [-30 + 1]

= [(-17) × (-30)] + [(-17) × 1]

= [510] + [-17]

= 493

**(h) (–57) × (–19) + 57**

Answer:

Answer:

= 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?**

Answer:

Answer:

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?

Answer:

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

Answer:

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?**

Answer:

Answer:

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?**

Answer:

Answer:

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?

Answer:

(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?

Answer:

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.**

Answer:

Answer:

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

Answer:

(a) (–3) × _____ = 27

Answer:

Let the missing integer be x.

= (-3) × (x) = 27

= x = - (27/3)

= x = -9

= (-3) × (-9) = 27

**(b) 5 × _____ = –35**

Answer:

Answer:

Let the missing integer be x.

= (5) × (x) = -35

= x = - (-35/5)

= x = -7

= (5) × (-7) = -35

**(c) _____ × (– 8) = –56**

Answer:

Answer:

Let the missing integer be x.

= (x) × (-8) = -56

= x = (-56/-8)

= x = 7

= (7) × (-8) = -56

**(d) _____ × (–12) = 132**

Answer:

Answer:

Let the missing integer be x,

= (x) × (-12) = 132

= x = - (132/12)

= x = - 11

= (-11) × (-12) = 132

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