**Integers**

**Exercise 1.2**

*Questions with (*) sign are open-ended questions.*

## Exercise 1.3

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 X 10

= 1320

(g) 9 × (–3) × (– 6)

__9 × (–3)__ × (– 6) = -27 X (-6)

= 162

(h) (–18) × (–5) × (– 4)

__(–18) × (–5)__ × (– 4) = 90 X (-4)

= -360

(i) (–1) × (–2) × (–3) × 4

(–1) × (–2) × (–3) × 4 = __2 X (-3)__ X 4

= -6 X 4

= -24

(j) (–3) × (–6) × (–2) × (–1)

__(–3) × (–6)__ × (–2) × (–1) = 18 X (-2) X (-1)

= -36 X (-1)

= 36

2). Verify the following:

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

L. H. S. = 18 × [7 __+ (–3)__]

= 18 X __(7 – 3)__

= 18 X 4

= 72

R.H. S. = [__18 × 7__] + [18 × (–3)]

= 126 + __[18 × (–3)]__

= 126 + (-54)

= 126 – 54

= 72

L. H. S. = R. H. S.

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

L. H. S. = (–21) × [(– 4) __+ (– 6)__]

= (–21) × [– 4 – 6]

= (–21) × (-10)

= 210

R. H. S. = __[(–21) × (– 4)__] + [(–21) × (– 6)]

= 84 + __[(–21) × (– 6)]__

= 84 + 126

= 210

L. H. S. = R. H. S.

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 (b) 37 (c) 0

(a) –22

The required integer whose product with (-1) is -22 = 22

(b) 37

The required integer whose product with (-1) is 37 = -37

(c) 0

The required integer whose product with (-1) is 0 = 0

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

-1 X 5 = -5

-1 X 4 = -4

-1 X 3 = -3

-1 X 2 = -2

-1 X 1= -1

-1 X 0 = 0

-1 X (-1) = 1

Addtional Examples For Practice

5). Find the product, using suitable properties:

(a) 26 × (– 48) + (– 48) × (–36)

26 × __(– 48)__ + __(– 48) __× (–36)

= (-48) X [ 26 __+ (-36)__]

= (-48) X ( 26 -36)

= (-48) X ( -10)

= 480

Ans: 26 × (– 48) + (– 48) × (–36) = 480

(b) 8 X 53 X (–125)

8 X 53 X (–125)

= __8 X (-125)__ X 53

= -1000 X 53

= -53000

Ans: 8 X 53 X (–125) = -53000

(c) 15 X (–25) X (– 4) X (–10)

15 X __(–25) X (– 4)__ X (–10)

= __15 X 100__ X (-10)

= 1500 X (-10)

= -15000

Ans: 15 X (–25) X (– 4) X (–10) = -15000

(d) (– 41) × 102

(– 41) × 102 = (-41) X (100 + 2)

= (-41) X 100 + (-41) X 2

= -4100 + (-82)

= -4100 -82

= -4182

Ans: (– 41) × 102 = -4182

(e) 625 × (–35) + (– 625) × 65

625 × (–35) + (– 625) × 65

= 625 × (–35) – 625 × 65

= 625 [-35 – 65]

= 625 (-100)

= -62500

Ans: 625 × (–35) + (– 625) × 65 = -62500

(f) 7 × (50 – 2)

7 × (50 – 2) = 7 X 50 – 7 X 2

= 350 – 14

= 336

Ans: 7 × (50 – 2) = 336

(g) (–17) × (–29)

(–17) × (–29) = -17 X (1 – 30)

= -17 X 1 + 17 X 30

= -17 + 510

= 493

Ans: (–17) × (–29) = 493

(h) (–57) × (–19) + 57

(–57) × (–19) + 57 = 57 X 19 + 57

= 57 X (19 + 1)

= 57 X 20

= 1140

Ans: (–57) × (–19) + 57 = 1140

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?

The initial temperature of the room = 40^{0}C

Rate of lowering temperature = -5^{0} C per hour

Decrease in temperature after 10 hours = 10 X (-5)

= -50

Room temperature after 10 hours = 40 + (-50)

= 40 – 50

= -10

Room temperature after 10 hours = -10^{0}C

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 every correct answer = 5

Marks awarded for every incorrect answer = -2

Score of Mohan = 4 X 5 + 6 X (-2)

= 20 – 12

= 8

Score of Mohan = 8 marks

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

Marks awarded for every correct answer = 5

Marks awarded for every incorrect answer = -2

Score of Reshma = 5 X 5 + 5 X (-2)

= 25 – 10

= 15

Score of Reshma = 15 marks

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

Marks awarded for every correct answer = 5

Marks awarded for every incorrect answer = -2

Score of Heena = 2 X 5 + 5 X (-2)

= 10 – 10

= 0

Score of Heena = 0 marks

8). 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?

Profit of 1 bag of white cement = Rs 8

Loss on 1 bag of grey cement = Rs 5

Profit or Loss made by the company is = 3000 X 8 + 5000 X (-5)

= 24000 – 25000

= -1000

Negative mark indicates loss.

Loss made by the company = Rs 1000

(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.

Loss made by the company by selling grey cement = 6400 X 5

= 32000

Number of white cement bags to sell to make neither profit nor loss

= 32000 ÷ 8

= 4000.

9). Replace the blank with an integer to make it a true statement.

(a) (–3) × _____ = 27

(–3) × __9__ = 27

(b) 5 × _____ = –35

5 × __(-7)__ = –35

(c) _____ × (– 8) = –56

__7__ × (– 8) = –56

(d) _____ × (–12) = 132

__11__ × (–12) = 132

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