The document NCERT Solutions(Part - 2) - Integers Class 7 Notes | EduRev is a part of the Class 7 Course Mathematics (Maths) Class 7.

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are the numbers that can be positive, negative or zero, but cannot be a fraction. These numbers are used to perform various arithmetic operations, like addition, subtraction, multiplication, and division. Integers

**Let's have a look at NCERT solutions of Integers from Exercise 1.2 to Exercise to 1.4**

**Exercise 1.2 **

**Q1. ****Write down a pair of integers whose**

**(a) Sum is -7.****Ans:**

= – 4 + (-3)

= – 4 – 3 … [∵ (+ × – = -)]

= – 7

**(b) Difference is -10.****Ans:**

= -25 – (-15)

= – 25 + 15 … [∵ (- × – = +)]

= -10

**(c) Sum is 0.****Ans:**

= 4 + (-4)

= 4 – 4

= 0**Q2. ****(a) Write a pair of negative integers whose difference gives 8.****Ans:**

= (-5) – (- 13)

= -5 + 13 … [∵ (- × – = +)]

= 8

**(b) Write a negative integer and a positive integer whose sum is -5.****Ans:**

= -25 + 20

= -5

**(c) Write a negative integer and a positive integer whose difference is -3. ****Ans: **= – 2 – (1)

= – 2 – 1

= – 3**Q3. ****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? ****Ans: **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.**Q4. ****Fill in the blanks to make the following statements true: **

**(i) (-5) + (8) = (-8) + (....)****Ans: **Let us assume the missing integer be x,

Then,

= (–5) + (– 8) = (– 8) + (x)

= – 5 – 8 = – 8 + x

= – 13 = – 8 + x

By sending – 8 from RHS to LHS it becomes 8,

= – 13 + 8 = x

= x = – 5

Now substitute the x value in the blank place,

(–5) + (– 8) = (– 8) + (- 5) … [This equation is in the form of Commutative law of Addition]

**(ii) -53 +... = -53****Ans: **Let us assume the missing integer be x,

Then,

= –53 + x = –53

By sending – 53 from LHS to RHS it becomes 53,

= x = -53 + 53 = x = 0

Now substitute the x value in the blank place,

= –53 + 0 = –53 … [This equation is in the form of Closure property of Addition]

**(iii) 17 +... = 0****Ans: **Let us assume the missing integer be x,

Then,

= 17 + x = 0

By sending 17 from LHS to RHS it becomes -17,

= x = 0 – 17

= x = – 17

Now substitute the x value in the blank place,

= 17 + (-17) = 0 … [This equation is in the form of Closure property of Addition]

= 17 – 17 = 0

**(iv) [13 + (-12)] + (....) = 13 + [(-12) + (-7)]****Ans: **Let us assume the missing integer be x,

Then,

= [13 + (– 12)] + (x) = 13 + [(–12) + (–7)]

= [13 – 12] + (x) = 13 + [–12 –7]

= [1] + (x) = 13 + [-19]

= 1 + (x) = 13 – 19

= 1 + (x) = -6

By sending 1 from LHS to RHS it becomes -1,

= x = -6 – 1 = x = -7

Now substitute the x value in the blank place,

= [13 + (– 12)] + (-7) = 13 + [(–12) + (–7)] … [This equation is in the form of Associative property of Addition]

**(v) (-4) + [15 + (-3)] = [-4 + 15] + .....****Ans: **Let us assume the missing integer be x,

Then,

= (– 4) + [15 + (–3)] = [– 4 + 15] + x

= (– 4) + [15 – 3)] = [– 4 + 15] + x

= (-4) + [12] = [11] + x

= 8 = 11 + x

By sending 11 from RHS to LHS it becomes -11,

= 8 – 11 = x

= x = -3

Now substitute the x value in the blank place,

= (– 4) + [15 + (–3)] = [– 4 + 15] + -3 … [This equation is in the form of Associative property of Addition]

**Exercise 1.3**

**Q1. ****Find the each of the following products**

**(a) 3 x (-1)****Ans: **By the rule of Multiplication of integers,

= 3 × (-1)

= -3 … [∵ (+ × – = -)]

**(b) (-1) x 225****Ans: **By the rule of Multiplication of integers,

= (-1) × 225

= -225 … [∵ (- × + = -)]

**(c) (-21) x (-30) ****Ans: **By the rule of Multiplication of integers,

= (-21) × (-30)

= 630 … [∵ (- × – = +)]

**(d) (-316) x (-1)****Ans: **By the rule of Multiplication of integers,

= (-316) × (-1)

= 316 … [∵ (- × – = +)]

**(e) (-15) x 0 x (-18) ****Ans: **By the rule of Multiplication of integers,

= (–15) × 0 × (–18) = 0

∵ Any integer is multiplied with zero and the answer is zero itself.

**(f) (-12) x (-11) x (10)****Ans: **By the rule of Multiplication of integers,

= (–12) × (-11) × (10)

First multiply the two numbers having same sign,

= 132 × 10 … [∵ (- × – = +)]

= 1320

**(g) 9 x (-3) x (-6)****Ans: **By the rule of Multiplication of integers,

= 9 × (-3) × (-6)

First multiply the two numbers having same sign,

= 9 × 18 … [∵ (- × – = +)]

= 162

**(h) (-18) x (-5) x (-4) ****Ans: **By the rule of Multiplication of integers,

= (-18) × (-5) × (-4)

First multiply the two numbers having same sign,

= 90 × -4 … [∵ (- × – = +)]

= – 360 … [∵ (+ × – = -)]

**(i) (-1) x (-2) x (-3) x 4****Ans: **By the rule of Multiplication of integers,

= [(–1) × (–2)] × [(–3) × 4]

= 2 × (-12) … [∵ (- × – = +), (- × + = -)]

= – 24

**(j) (-3) x (-6) x (2) x (-1)****Ans: **By the rule of Multiplication of integers,

= [(–3) × (–6)] × [(–2) × (–1)]

First multiply the two numbers having same sign,

= 18 × 2 … [∵ (- × – = +) = 36**Q2. ****Verify the following **

**(a) 18 x [7 + (-3)] = [18 x 7] + [18 x (-3)]****Ans: **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)] ****Ans:** (-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.

**Q3. ****(i) For any integer a, what is (-1) **x** a equal to?****Ans: **= (-1) × a = -a

Because, when we multiplied any integer a with -1, then we get additive inverse of that integer.

**(ii) Determine the integer whose product with (-1) is****(a) -22****Ans: **Now, multiply -22 with (-1), we get

= -22 × (-1)

= 22

Because, when we multiplied integer -22 with -1, then we get additive inverse of that integer.** ****(b) 37****Ans: **Now, multiply 37 with (-1), we get

= 37 × (-1)

= -37

Because, when we multiplied integer 37 with -1, then we get additive inverse of that integer.**(c) 0****Ans: **Now, multiply 0 with (-1), we get

= 0 × (-1)

= 0

Because, the product of negative integers and zero give zero only.** ****Q4. ****Starting from (-1) x 5, write various products showing some patterns to show (-1) x (-1) = 1****Ans: **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.**Q5. ****Find the product, using suitable properties **

**(a) ****26 x (-48) + (-48) x (-36)****Ans:**

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

⇒ (-48) x (-10)

⇒ 480

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

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

⇒ 53 x (-1000)

⇒ -53000

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

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

⇒ 15 x (-1000)

⇒ -15000

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

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

⇒ [(-41) x 100] + [(-41) x 2]

⇒ -4100 + (-82)

⇒ -4182

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

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

⇒ 625 x (-100)

⇒ -62500

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

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

⇒ 350 - 14 = 336

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

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

⇒ (-17) x (30) + (- 17) x 1

⇒ 510 + (-17)

⇒ 493

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

⇒ (-57) x (-19) + 57 x 1

⇒ 57 x 19 + 57 x 1

⇒ 57 x (19 + 1) [Distributive property]

⇒ 57 x 20 = 1140**Q6. ****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

**(i) Mohan gets four correct and six incorrect answers. What is his score?****Ans: **Mohan gets marks for four correct questions = 4 x 5 = 20

► 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 five correct answers and five incorrect answers, what is her score?****Ans: **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 two correct and five incorrect answers out of seven questions she attempts. What is her score?****Ans: **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?****Ans: ****Given: **Profit of 1 bag of white cement = Rs 8

And Loss of 1 bag of grey cement = Rs 5

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) 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. ****Ans: Given: **Profit of 1 bag of white cement = Rs 8

And Loss of 1 bag of grey cement = Rs 5

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.**Q9. ****Replace the blank with an integer to make it a true statement.**

**(a) ****(-3) x _____= 27Ans:** Let us assume the missing integer be x,

Then,

= (–3) × (x) = 27 = x = – (27/3)

= x = -9

Let us substitute the value of x in the place of blank,

= (–3) × (-9) = 27 … [∵ (- × – = +)]

**(b)** **5 x _____ = -35Ans:** Let us assume the missing integer be x,

Then,

= (5) × (x) = -35

= x = – (-35/5)

= x = -7

Let us substitute the value of x in the place of blank,

= (5) × (-7) = -35 … [∵ (+ × – = -)]

**(c) ** **_____ x (-8) = -56Ans:** Let us assume the missing integer be x,

Then,

= (x) × (-8) = -56

= x = (-56/-8)

= x = 7

Let us substitute the value of x in the place of blank,

= (7) × (-8) = -56 … [∵ (+ × – = -)]

**(d) ****_____ x (-12) = 132Ans:** Let us assume the missing integer be x,

Then,

= (x) × (-12) = 132

= x = – (132/12)

= x = – 11

Let us substitute the value of x in the place of blank,

= (–11) × (-12) = 132 … [∵ (- × – = +)]

**Exercise 1.4**** **

**Q1. Evaluate each of the following**

**(a)** **(-30) ÷ 10Ans:**

= (–30) ÷ 10

= – 3

When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.

**(b) ****50 ÷ (-5)Ans:**

= (50) ÷ (-5)

= – 10

When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.

**(c) ****(-36) ÷ (-9)Ans:** = (-36) ÷ (-9)

= 4

When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put positive sign (+) before the quotient.

**(d) ****(-49) ÷ 49Ans:** = (–49) ÷ 49

= – 1

When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.

**(e) ****13 + [(-2) + 1]Ans:**

= 13 ÷ [(–2) + 1]

= 13 ÷ (-1)

= – 13

When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.

**(f) ****0 ÷ (-12)Ans:**

= 0 ÷ (-12)

= 0

When we divide zero by a negative integer gives zero.

**(g) ****(-31) ÷ [(-30) ÷ (-1)]Ans:**

= (–31) ÷ [(–30) + (–1)]

= (-31) ÷ [-30 – 1]

= (-31) ÷ (-31)

= 1

When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put positive sign (+) before the quotient.

**(h) ****[(-36) ÷ 12] ÷ 3Ans:** First we have to solve the integers with in the bracket,

= [(–36) ÷ 12]

= (–36) ÷ 12

= – 3

Then, = (-3) ÷ 3 = -1

When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.

**(i)**** [(-6) + 5] ÷ [(-2) + 1]Ans: **The given question can be written as,

= [-1] ÷ [-1]

= 1

When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put positive sign (+) before the quotient.

**(a) a = 12, b = -4,c = 2 ****Ans:**

From the question,

a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)

Given, a = 12, b = – 4, c = 2

Now, consider

LHS = a ÷ (b + c)

= 12 ÷ (-4 + 2)

= 12 ÷ (-2) = -6

When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.

Then, consider

RHS = (a ÷ b) + (a ÷ c)

= (12 ÷ (-4)) + (12 ÷ 2)

= (-3) + (6) = 3

By comparing LHS and RHS = -6 ≠ 3

= LHS ≠ RHS

Hence, the given values are verified.

**(b)** **a = (-10), b = 1 c = 1**

From the question,

a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c)

Given, a = (-10), b = 1, c = 1

Now, consider

LHS = a ÷ (b + c) = (-10) ÷ (1 + 1)

= (-10) ÷ (2) = -5

When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put minus sign (-) before the quotient.

Then, consider

RHS = (a ÷ b) + (a ÷ c)

= ((-10) ÷ (1)) + ((-10) ÷ 1)

= (-10) + (-10)

= -10 – 10 = -20

By comparing LHS and RHS

= -5 ≠ -20

= LHS ≠ RHS

Hence, the given values are verified.**Q.3. ****Fill in the blanks**

**(a) ****369 ÷ _____ = 369Ans:** Let us assume the missing integer be x,

Then,

= 369 ÷ x = 369

= x = (369/369)

= x = 1

Now, put the valve of x in the blank.

= 369 ÷ 1 = 369

**(b) ****(-75) ÷ _____ = (-1)Ans: **Let us assume the missing integer be x,

Then,

= (-75) ÷ x = -1

= x = (-75/-1)

= x = 75

Now, put the valve of x in the blank.

= (-75) ÷ 75 = -1

**(c) ****(-206) ÷ _____ = 1****Ans:** Let us assume the missing integer be x,

Then,

= (-206) ÷ x = 1

= x = (-206/1)

= x = -206

Now, put the valve of x in the blank.

= (-206) ÷ (-206) = 1 ** **

**(d) ****(-87) ÷ _____ = 87Ans:** Let us assume the missing integer be x,

Then,

= (-87) ÷ x = 87

= x = (-87)/87

= x = -1

Now, put the valve of x in the blank.

= (-87) ÷ (-1) = 87

**(e) **_____ ÷1 = -87

Ans: Let us assume the missing integer be x,

Then, = (x) ÷ 1 = -87

= x = (-87) × 1

= x = -87

Now, put the valve of x in the blank.

= (-87) ÷ 1 = -87

**(f) **** _____ ÷ 48 = -1Ans:** Let us assume the missing integer be x,

Then,

= (x) ÷ 48 = -1

= x = (-1) × 48

= x = -48

Now, put the valve of x in the blank.

= (-48) ÷ 48 = -1

**(g) **20 ÷ _____ = -2**Ans: **Let us assume the missing integer be x,

Then,

= 20 ÷ x = -2

= x = (20) / (-2)

= x = -10

Now, put the valve of x in the blank.

= (20) ÷ (-10) = -2

**(h) ****_____ ÷ (4) = -3Ans: **Let us assume the missing integer be x,

Then,

= (x) ÷ 4 = -3

= x = (-3) × 4

= x = -12

Now, put the valve of x in the blank.

= (-12) ÷ 4 = -3

**Ans:**__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.**Q6. ****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?****Ans: **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) Mohini scores (-5) marks in this test, though she has got 7 correct answers.****How many questions has she attempted incorrectly? ****Ans:** 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.**Q7. ****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? ****Ans: **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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