**Chapter 1 Integers Exercise 1.3**

Question 1.

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)

Solution:

(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) = 0 [∵ a × 0 = a]

(f) (-12) × (-11) × (10)

= (-) × (-) × 12 × 11 × 10 = 1320

(g) 9 × (-3) × (-6) = (-3) × (-6) × 9

= (—) × (-) × 3 × 6 × 9 = 162

(h) (-18) × (-5) × (-4)

= (-) × (-) × (-) × 18 × 5 × 4 = -360

(i) (-1) × (-2) × (-3) × 4

= (-) × (-) × (-) × 1 × 2 × 3 × 4 = -24

(j) (-3) × (-6) × (-2) × (-1)

= (-) × (-) × (-) × (-) × 3 × 6 × 2 × 1 = 36

Question 2.

Verify the following:

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

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

Solution:

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

LHS = 18 × [7 + (-3)] = 18 × 4 = 72

RHS = [18 × 7] + [18 × (-3)] = 126 + (-54)

= 126 – 54 = 72

LHS = RHS

Hence, verified.

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

LHS = (-21) × [(-4) + (-6)]

= (-21) × (-10)

= (-) × (-) × 21 × 10 = 210

RHS = [(-21) × (-4)] + [(-21) × (-6)]

= (84) + (126) = 84 + 126 = 210

LHS = RHS

Hence, verified.

Question 3.

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

(ii) Determine the integer whose product with (-1) is 0.

(a) -22

(b) 37

(c) 0

Solution:

(i) (-1) × a = -a

(ii) (-1) × 0 = 0 [∵ a × 0 = 0]

Hence (c) 0 is the required integer.

Question 4.

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

Solution:

(-1) × 5 = -5

(-1) × 4 = -4 = (-5) + 1

(-1) × 3 = -3 = (-4) + 1

(-1) × 2 = -2 = (-3) + 1

(-1) × (1) = -1 = (-2) + 1

(-1) × 0 = 0 – (-1) + 1

(-1) × (-1) = 1 = 0+1

Question 5.

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

Solution:

(a) 26 × (-48) + (-48) × (-36)

= -48 × [26 + (-36)] = -48 × [26 – 36] = -48 × -10 = 480 [Distributive property of multiplication over

addition]

(b) 8 × 53 × (-125) = 53 × [8 × (-125)]

[Associative property of multiplication] = 53 × (-1000) = -53000

(c) 15 × (-25) × (-4) × (-10)

= [(-25) × (-4)] × [15 × (-10)]

[Regrouping the terms] = 100 × (-150) = -15000

(d) (-41) × 102 = (-41) × [100 + 2]

= (-41) × 100 + (-41) × 2

[Distributive property of multiplication over addition] = -4100 – 82 = -4182

(e) 625 × (-35) + (-625) × 65

= 625 × [(-35) + (-65)]

[Distributive property of multiplication over addition]

= 625 × (-100) = -62500

(f) 7 × (50 – 2) = 7 × 48 = 336 or

7 × (50 – 2) = 7 × 50 -7 × 2 = 350 – 14 = 336 [Distributive property of multiplication over addition]

(g) (-17) × (-29) = (-17) × [30 + (-1)]

= (-17) × 30 + (-17) × (-1)

= -510 + 17 = -493

[Distributive property of multiplication over addition]

(h) (-57) × (-19) + 57 = 57 × 19 + 57

= 57 × 19 + 57 × 1 [Y (-) × (-) = (+)] [Distributive property of multiplication over addition]

= 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?

Solution:

Temperature of the room in the beginning = 40°C

Temperature after 1 hour

= 40°C – 1 × 5°C = 40°C – 5°C – 35°C

Similarly, temperature of the room after 10 hours

= 40°C – 10 × 5°C = 40°C – 50°C = -10°C

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 si× 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?

Solution:

(i) Marks awarded to Mohan = 4 × 5

=20 for correct answers Marks awarded to Mohan = 6 × (-2)

= -12 for incorrect answers.

∴ Total marks obtained by Mohan

= 20 + (-12) = 20 – 12 = 8

(ii) Marks awarded to Reshma for correct answers

= 5 × 5 = 25

Marks awarded to Reshma for incorrect answers

= 5 × (-2) = -10

∴ Total marks obtained by Reshma

= 25 + (-10) = 25 – 10 = 15

(iii) Marks awarded to Heena for correct answers

= 2 × 5 = 10

Marks awarded to Heena for incorrect answers

= 5 × (-2) = -10

Number of question not attempted by Heena

= 10 – (2 + 5) = 10 – 7 = 3

Marks awarded to Heena for non-attempted questions

=3×0=0

∴ Total marks obtained by Heena

= 10 + (-10) + 0 = 10-10+ 0 = 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 sold 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.

Solution:

(a) Profit on one white cement bag = ₹ 8

loss on one grey cement bag = ₹ – 5

Profit on 3,000 bags of white cement

= ₹ (8 × 3,000) = ₹ 24,000

Loss on 5,000 bags of grey cement

= ₹ (-5 × 5000) = – ₹ 25,000

Total loss = – ₹ 25,000 + ₹ 24,000

= – ₹ 1000 i.e. ₹ 1000

(b) Selling price of grey bags at a loss of ₹ 5

= ₹ (5 × 6,400) – ₹ 32,000

For no profit and no loss, the selling price of white bags = ₹ 32,000

Rate of selling price of white bags at a profit of ₹ 8 per bag.

∴ Number of white cement bags sold

=320008=4000

Hence, the required number of bags = 4,000

Ex 1.3 Class 7 Maths Question 9.

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

(a) (-3) × __ = 27

(b) 5 × __ = -35

(c) __ × (-8) = -56

(d) __ × (-12) = 132

Solution:

(a) (-3) × __ = 27 = (-3) × (-9) = 27 [∵ (-) × (-) = (+)]

(b) 5 × __ = -35 = 5 × (-7) = -35 [∵ (+) × (-) = (-)]

(c) __ × (-8) = -56 = 7 × (-8) = -56 [∵ (+) × (-) = (-)]

(d) __ × (-12) = 132 = (-11) × (-12) = 132 [∵ (-) × (-) = (+)]