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**Exercise 1.2 **

**Q.1. ****Write down a pair of integers whose****(a) Sum is -7.****(b) Difference is -10. ****(c) Sum is 0.****Solution. ****(a)** One such pair whose sum is -7: -5 + (-2) = -7**(b)** One such pair whose difference is -10: -2 - (+8) = -10**(c)** One such pair whose sum is 0: -5 + 5 = 0**Q.2. ****(a) Write a pair of negative integers whose difference gives 8. ****(b) Write a negative integer and a positive integer whose sum is -5. ****(c) Write a negative integer and a positive integer whose difference is -3. ****Solution****. ****(a)** (-2) - (-10) = - 2 + 10 = 8**(b)** (-7) + 2 = -5**(c)** (-2) - 1 = - 2 - 1 = -3**Q.3. ****In a quiz, team A scored -40,10,0 and team B scores 10, 0, -40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order? ****Solution****. **Team A scored -40, 10, 0

⇒ Total score of Team A = -40 + 10 + 0 = - 30

Team B scored 10, 0, -40

⇒ Total score of Team B = 10 + 0+ (-40) - 10 + 0 - 40 = -30

Thus, the scores of both teams are same.

Yes, we can add integers in any order due to the commutative property.**Q.4. ****Fill in the blanks to make the following statements true: ****(i) (-5) + (8) = (-8) + (....)****(ii) -53 +... = -53****(iii) 17 +... = 0****(iv) [13 + (-12)] + (....) = 13 + [(-12) + (-7)]****(v) (-4) + [15 + (-3)] = [-4 + 15] + ..... ****Solution****.****(i)** (-5) + (-8) = (-8) + __(-5)__ [Commutative property]**(ii)** -53 + 0 = -53 [Zero additive property]**(iii)** 17 + (-17__)__ = 0 [Additive identity]**(iv)** [13 + (12)] + __(__-__7)__ = 13 + [(-12) + (-7)] [Associative property]**(v)** (-4) + [5 + (-3)] = [-4 + 15] + __(__-__3)__ [Associative property]

**Exercise 1.3**

**Q.1. ****Find the each of the following products****(a) 3 x (-1) ****(b) (-1) x 225 ****(c) (-21) x (-30) ****(d) (-316) x (-1) ****(e) (-15) x 0 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 ****(j) (-3) x (-6) x (2) x (-1) ****Solution. ****(a) **3 x (-1) = -3**(b)** (-1) x 225 = -225**(c)** (-21) x (-30) = 630**(d)** (-316) x (-1) = 316**(e)** (-15) x 0 x (-18) = 0**(f)** (-12) x (-11) x (10) = 132 x 10 = 1320**(g)** 9 x (-3) x (-6) = 9 x 18 = 162**(h)** (-18) x (-5) x (-4) = 90 x (-4) = -360**(i)** (-1) x (-2) x (-3) x 4 = (-6 x 4) = -24**(j)** (-3) x (-6) x (2) x (-1) = (-18) x (-2) = 36**Q.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)** 18 x [7 + (-3)] = [18 x 7] + [18 x (-3)]

⇒ 18 x 4 = 126 + (-54)

⇒ 72 = 72

⇒ L.H.S. = R.H.S.

Hence, Verified.

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

⇒ (-21) x (-10) = 84 + 126

⇒ 210 = 210

⇒ L.H.S. = R.H.S.

Hence, Verified.

**Q.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) **(-1) x a = -a, where a is an integer.**(ii) ****(a)** (-1) x (-22) = 22**(b) **(-1) x 37 = -37**(c) **(-1) x 0 = 0**Q.4. ****Starting from (-1) x 5, write various products showing some patterns to show (-1) x (-1) = 1****Solution****. **According to the pattern,

(−1)×(5) = −5

(−1)×(4) = −4

(−1)×(3) = −3

(−1)×(2) = −2

(−1)×(1) = −1

(−1)×(0) = 0

(−1)×(−1) = 1

Thus, we can conclude that this pattern shows the product of one negative integer and one positive integer is a negative integer whereas the product of two negative integers is a positive integer.**Q.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) **(-7) x (-29)**(h) **(-57) x (-19)+57

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

⇒ (-48) x [26 + (-36)] [Distributive property]

⇒ (-48) x (-10)

⇒ 480

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

⇒ 53 x [8 x (-125)] [Commutative property]

⇒ 53 x (-1000)

⇒ -53000

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

⇒ 15 x [(-25) x (-4) x (-10)] [Commutative property]

⇒ 15 x (-1000)

⇒ -15000

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

⇒ -41 x [100+2] [Distributive property]

⇒ [(-41) x 100]+[(-41) x 2]

⇒ -4100 + (-82)

⇒ -4182

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

⇒ 625 x [(-35) + (-65)] [Distributive property]

⇒ 625 x (-100)

⇒ -62500

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

⇒ 7 x 50 - 7 x 2 [Distributive property]

⇒ 350-14 = 336

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

⇒ (-17) x [(-30) + 1] [Distributive property]

⇒ (-17) x (30) + (- 17)x 1

⇒ 510 + (-17)

⇒ 493

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

⇒ (-57) x (-19) + 57 x 1

⇒ 57 x 19 + 57 x 1

⇒ 57 x (19+1) [Distributive property]

⇒ 57 x 20 = 1140**Q.6. ****A certain freezing process requires that room temperature be lowered from 40 ^{0}C at the rate of 5^{0}C every hour. What will be the room temperature 10 hours after the process begins?**

► Decreasing the temperature every hour = 5

► Room temperature after 10 hours = 40

= 40

= - 10

Thus, the room temperature after 10 hours is - 10

► He gets marks for six incorrect questions = 6 x (-2) = -12

► Therefore, total scores of Mohan = (4 x 5) + [6 x (-2)]

= 20- 12 = 8

Thus, Mohan gets 8 marks in a class test

**(ii) **Reshma gets marks for five correct questions = 5 x 5 = 25

► She gets marks for five incorrect questions = 5 x (-2)=-10

► Therefore, total score of Resham = 25 + (-10) = 15

► Thus, Reshma gets 15 marks in a class test

**(iii) **Heena gets marks for two correct questions = 2 x 5 = 10

► She gets marks for five incorrect questions = 5 x (-2) = -10

► Therefore, total score of Resham = 10 + (-10) = 0

Thus, Reshma gets 0 marks in a class test.**Q.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? ****(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****. ****Given: **Profit of 1 bag of white cement = Rs 8

► And Loss of 1 bag of grey cement = Rs 5**(a)** Profit on selling 3000 bags of white cement = 3000 x Rs 8 = Rs 24,000

► Loss of selling 5000 bags of grey cement = 5000 x Rs 5 = Rs 25,000

Since, Profit < Loss

Therefore, his total loss on selling the grey cement bags = Loss - Profit

= Rs 25,000 - Rs 24,000

= Rs 1,000

Thus, he has lost Rs 1,000 on selling the grey cement bags.

**(b)** Let the number of bags of white cement be x.

According to question,

Thus, he must sell 4000 white cement bags to have neither profit nor loss.**Q.9. ****Replace the blank with an integer to make it a true statement.****(a) **(-3) x _____= 27 ** ****(b)** 5 x _____ = -35**(c) ** _____ x (-8) = -56 ** ****(d) **_____ x (-12) = 132**Ans: ****(a) **(-3) x (-9) = 27**(b) **5 x (-7) = -35**(c) **7 x (-8) = -56**(d) **(-11) x (-12) = 132

**Exercise 1.4**** **

**Q.1. Evaluate each of the following****(a) **(-30) ÷ 10 **(b) **50 ÷ (-5)**(c) **(-36) ÷ (-9) ** ****(d) **(-49) ÷ 49**(e) **13 + [(-2) + 1] ** ****(f) **0 ÷ (-12)**(g) **(-31) ÷ [(-30) ÷(-1)] **(h) **[(-36) ÷ 12] ÷ 3**(i) **[(-6)+5] ÷ [(-2) + 1]**Ans.****Q.2. ****Verify that a + (b + c) ≠ (a + b) + (a ÷ c) for each of the following values of a, b and c. ****(a) a = 12, b = -4,c = 2 ****(b) a = (-10), b = 1 c = 1****Solution**. **(a) **__Given:__

Putting the given values in L.H.S,

Putting the given values in L.H.S.

Since, L.H.S. ≠ R.H.S. Hence verified.

Putting the given values in L.H.S, = 12 +(-4 +2)

**(b)** **Given:** a÷(b+c)≠(a÷b) + (a÷c)

a = -10,b=1,c=1

Putting the given values in L.H.S. = -10 ÷ (l + 1)

= -10 ÷ (2) = -5

Putting the given values in R.H.S. = [-10÷1]+ (-10÷ 1)

= -10 -10 = -20

Since, L.H.S. ≠ R.H.S.

Hence verified.**Q.3. ****Fill in the blanks****(a) **369 ÷ _____ = 369 **(b) **(-75) ÷ _____ = (-1)**(c) **(-206) ÷ _____ = 1 ** ****(d) **(-87) ÷ _____ = 87**(e) **_____ ÷1 = -87 **(f) **_____ ÷ 48 = -1**(g) **20 ÷ _____ = -2 ** ****(h) **_____ ÷ (4) = -3**Ans.****(a) **369 ÷ 1 = 369 **(b) **(-75) ÷ 75 = (-1)**(c) **(-206) ÷ (-206) = 1 ** ****(d) **(-87) ÷ (-1) = 87**(e) **__(__-87)÷ = -87 ** ****(f) **(-48) ÷ 48 = -1**(g) **20 ÷ (-10) = -2 ** ****(h) **(-12) ÷ (4) = -3**Q.4. ****Write five pairs of integers (a,b) such that a ÷ b = -3. One such pair is (6,-2) because 6 ÷ (-2) = (-3)****Solution**. **(i) **(-6) ÷ 2 = -3 ** ****(ii) **9 ÷ (-3) = -3**(iii) **12 ÷ (-4) = -3 ** ****(iv) **(-9) ÷ 3 = -3**(v) **(-15) ÷ 5 = -3**Q.5. ****The temperature at noon was 10 ^{0}C above zero. If it decreases at the rate of 2^{0}C per hour until midnight, at what time would the temperature be 8^{0}C below zero? What would be the temperature at mid-night? **

**Solution**.__The following number line is representing the temperature:__

Temperature in (^{0}C)The temperature decreases 2^{0}C = 1 hour

The temperature decreases 1^{0}C = 1/2 hour

The temperature decreases

Total time = 12 noon + 9 hours = 21 hours = 9 pm

Thus, at 9 pm the temperature would be 8^{0}C below 0^{0}C.**Q.6. ****In a class test (+3) marks are given for every correct answer and (-2) marks are given for every incorrect answer and no marks for not attempting any question. ****(i) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly? ****(ii) Mohini scores (-5) marks in this test, though she has got 7 correct answers.****How many questions has she attempted incorrectly? **

**Solution**. **(i)** Marks are given for one correct answer = 3

► Marks given for 12 correct answers = 3 x 12 = 36

Radhika scored 20 marks.

► Therefore, Marks obtained for incorrect answers = 20 - 36 = -16

Now, marks given for one incorrect answer = -2

Therefore, number of incorrect answers = (-16) ÷(-2) = 8

Thus, Radhika has attempted 8 incorrect questions.

**(ii)** Marks given for seven correct answers = 3 x 7 = 21

► Mohini scores = -5

► Marks obtained for incorrect answers = -5 -21 = -26

► Now, marks given for one incorrect answer = -2

► Therefore, number of incorrect answers = (-26) ÷ (-2) = 13

Thus, Mohini has attempted 13 incorrect questions.**Q.7. ****An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 above the ground level, how long will it take to reach -350 m? **** Solution. **The starting position of mine shaft is 10 m above the ground but it moves in the opposite direction so it travels the distance (-350) m below the ground.

► So total distance covered by mineshaft = 10 m - (-350) m = 10 + 350 = 360 m

► Now, time taken to cover a distance of 6 m by it = 1 minute

► So, time taken to cover a distance of 1 m by it = 1/6 minute

► Therefore, time taken to cover a distance of 360 m

= 60 minutes = 1 hour

[Since 60 minutes = 1 hour]

Thus, in one hour the mine shaft reaches -350m below the ground.

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