**Integers**

**Question 1.****Find each of the following products :(a)** 3 x (-1)

**(b)**(-1) x 225

**(-21) x (-30)**

(c)

(c)

**(d)**(-316)

**x (-1)**

**(e)**(-15) x O x (-18)

**(f)**(-12) x (-11) x (10)

**(g)**9 x (-3) x (-6)

**(h)**(-18) x (-5) x (-4)

**(i)**(-1) x (-2) x (-3) x 4 Sol. (a) 3 x (-1) = – (3 x 1) = -3

**(j)**(-3) x (-6) x (-2) x (-1)

**Solution.****(a)** 3 x (-1) = – (3 x 1) = -3**(b)** (-1) x 225 = – (1 x 225) = -225**(c)** (-21) x (-30) = 21 x 30 =630**(d)** (-316) x (-1) = 316 x 1 = 316**(e)** (-15) x 0 x (-18) = [(-15) x 0] x (-18) = 0 x (-18) = 0**(f)** (-12) x (-11) x (10) = [(-12) x (-11)] x (10)

= (132) x (10) =1320**(g)** 9 x (-3) x (-6) = [9 x (-3)] x (-6) = (-27) x (-6) = 162**(h)** (-18) x (-5) x (-4) = [(-18) x (-5)] x (-4)

= 90 x (-4) – -360**(i)** (-1) x (-2) x (-3) x 4 = [(-1) x (-2)] x [(-3) x 4]

= (2)x (-12) = -24**(j)** (-3) x (-6) x (-2) x (-1) = [(-3) x (-6)] x [(-2) x (-1)] = (18) x (2) = 36

**Question 2.****Verify the following:(a)** 18 x [7 + (-3)] = [18 x 7] + [18 x (-3)]

**(b)**(-21) x [(-4) + (-6)] = [(-21) x (-4)] + [(-21) x (-6)]

**Solution.****(a)** We have,

18 x [7 + (-3)] = 18 x 4 = 72

and, [18 x 7] + [18 x (-3)] = 126 – 54 =72

18 x [7 + (-3)] = [18 x 7] + [18 x (-3)]**(b)** We have,

(-21) x [(-4) + (-6)] = (-21) x (-4 -6)

= (-21)(-10) = 210 and, [(-21) x (-4) + [(-21) x (-6)]

= 84+126 =210

∴ (-21) x [(-4) + (-6)] = [(-21) x (-4)] + [(-21) x (-6)]

**Question 3.****(i)** For any integer a, what is (-1) x a equal to?**(ii)** Determine the integer whose product with (-1) is**(a)** -22**(b)** 37**(c)** 0

**Solution.****(i)** For any integer a, (-1) x a = -a.**(ii) **We know that the product of any integer and (-1) is the additive inverse of integer.

The integer whose product with (-1) is**(a)** additive inverse of -22, t. e., 22.**(b)** additive inverse of 37, i.e., -37.**(c)** additive inverse of 0, i.e., 0.

**Question 4.Starting from (-1) x 5, write various products showing some pattern to show (-1) x (-1) = 1.**

**Solution.**(-1) x 5 = -5

(-1) x 4 = -4 = [-5 – (-1)] = -5 +1

(-1) x 3 = -3 = [-4 – (-1)] = -4 +1

(-1) x 2 = -2 = [-3 – (-1)] = -3 +1

(-1) x 1 = -1 = [-2 – (-1)] = -2 +1

(-1) x 0 = 0 = [-1 – (-1)] = -1 +1

(-1) x (-1) =[0 – (-1)] = 0 + 1 = 1

**Question 5.Find the product, using suitable properties :(a)** 26 x (-48) + (-48) x (-36)

**(b)**8 x 53 x (-125)

**(c)**15 x (-25) x (-4) x (-10)

**(d)**(-41) x 102

**(e)**625 x (-35) +(-625) x 65

**(f)**7 x (50-2)

**(g)**(-17) x (-29)

**(h)**(-57) x (-19) + 57

**Solution.****(a)** We have, 26 x (-48) + (-48) x (-36)

= (-48) x 26 + (-48) x (-36)

= (-48) x [26 + (-36)]

= (-48) x (26 – 36)

=(-48) x (-10)= 480

**(b)** We have,

8 x 53 x (-125) = [8 x (-125)] x 53

= (-1000) x 53 = -53000

**(c)** We have,

15 x (-25) x (-4) x (-10)

=15 x [(-25) x (-4)] x (-10)

= 15 x (100) x (-10)

= (15 x 100) x (-10)

= 1500 x (-10) = -15000

**(d)** We have,

(-41) x 102 = (-41) x (100 +2)

= (-41) x 100 + (-41) x 2 = -4100 – 82 = -4182

**(e)** We have, 625 x (-35) + (-625) x 65

= 625 x (-35) + (625) x (-65)

= 625 x [(-35)+ (-65)]

= 625 x (-100) = -62500

**(f)** 7 x (50-2) = 7 x 50 – 7 x 2

= 350 -14 =336

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

= (-17) x (-30) + (-17) x 1 = 510 – 17 = 493

**(h)** (-57) x (-19)+ 57 =57 x 19 + 57 x 1

= 57 x (19 +1)

= 57 x 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.**

Initial room temperature = 40 X

Temperature lowered every hour = (-5) °C

Temperature lowered in 10 hours = (-5) x 10 °C = -50 °C

∴ Room temperature after 10 hours = 40 X – 50 X = -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 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?

**Solution.****(i)** Marks awarded for one correct answer = 5

Marks scored for 4 correct answer = 5 x 4 = 20

Marks awarded for one incorrect answer = (-2)

Marks scored for 6 incorrect answer = (-2) x 6 = -12

Hence, Mohan’s score = 20 – 12 = 8 marks.

**(ii)** Reshma’s score for 5 correct answers = 5 x 5 = 25 marks

Reshma’s score for 5 incorrect answers = (-2) x 5 = -10 marks

Hence, Reshma’s score = 25-10 =15 marks

**(iii)** Heena’s score for 2 correct and 5 incorrect answers

= (5 x 2) + {(-2) x 5}

= 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?

**(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.**

Profit on sale of 1 bag of white cement = ₹ 8

Loss on sale of 1 bag of grey cement = – ₹ 5**(a)** Profit on sale of 3000 bags of white cement

= ₹ (3000 x 8)

= ₹ 24,000

Loss on sale of 5000 bags of grey cement = ₹ (5000 x -5)

= – ₹ 25,000

Difference between the two = ₹ 24,000 – ₹ 25,000 = – ₹ 1,000

Hence, there is a loss of ₹ 1000.

**(b)** Loss on the sale of 6400 bags of grey cement = ₹ (6400 x 5) = ₹ 32,000

In order to have neither profit nor loss, the profit on the sale of white cement should be ? 32,000.

Number of white cement bags sold

Hence, 4000 bags of white cement should be sold to have neither profit nor loss.

**Question 9.**

Replace the blank with an integer to make it a true statement.**(a)** (-3) x = 27**(b)** 5 x = -35**(c)** 7 x (-8) = -56**(d)** (-11) x (-12) = 132

**Solution.****(a)** (-3) x (-9) = 27**(b)** 5 x (-7) = (-35)**(c)** 7 x (-8) = (-56)**(d)** (-11) x (-12) =132