## NCERT Solutions for class 7 Maths Chapter 1: Integers## Exercise 1.1
(a) Observe this number line and write the temperature of the places marked on it.
The negative numbers on a number line increases from right to left.
The negative numbers on a number line increases from right to left.
The positive numbers on a number line increases from left to right. The process of representing positive numbers is opposite to that of the negative numbers.
The positive numbers on a number line increases from left to right.
The positive numbers on a number line increases from left to right.
Hottest place: Bengaluru with the temperature of 22 °C Coldest place: Lahulspiti with the temperature of -8 °C Difference = Hottest - Coldest = 22 - (- 8) = 22 + 8 = 30 °C Thus, the difference between the hottest and coldest place is 30 °C.
Srinagar with the temperature of -2 °C Lahulspiti has the temperature of -8 °C Difference = -2 - (- 8) = -2 + 8 = 8 - 2 = 6 °C Thus, the difference between the Lahulspiti and Srinagar is 6 °C.
Temperature of Shimla: 5 °C Together = Temperature of Srinagar + Temperature of Shimla = -2 °C + 5 °C = 3 °C So,
The temperature of Srinagar (-2 °C) is less than the temperature of Srinagar and Shimla taken together (3 °C). So, The second condition is not true.
Total score = 25 + (-5) + (-10) + 15 + 10 Total score = 25 - 5 - 10 + 15 + 10 Total score = 35 Thus, the total marks scored by the Jack in five successive rounds were
The drop in the temperature means the decrease in the temperature by 2°C. Temperature in Srinagar on Tuesday = - 5°C - (2°C) = The rise in temperature means the increase in the temperature by 4°C. Temperature in Srinagar on Wednesday = Temperature in Tuesday + 4°C = - 7°C + (4°C) =
The height of submarine below the sea level = 1200 m The vertical distance between the plane and the submarine = height of plane above the seal level + height of submarine below the sea level = 5000 + 1200 = 6200 m Thus, the vertical distance between the plane and the submarine is 6200 m.
The money deposited by Mohan inhis bank account = ₹ 2,000 The money withdrawal by Mohan fromhis bank account = ₹ 1,642 Balance = Money deposited - Money withdrawal Balance =₹ 2,000 - ₹ 1,642 Balance =₹ 358
The four directions are represented in the given diagram: 20 km towards east from a point A to the point B can be represented as: 30 km towards west from point B along the same road can be represented as: The final position of Rita from the point A is: 30 km - 20 km = 10 km The direction in the west is represented by a negative integer. So, we will represent it as
i.
Sum = 5 + (-1) + (-4) = 5 - 1 - 4 = 0
Sum = -5 + (-2) + 7 = -5 - 2 + 7 = 0
Sum = 0 + 3 + (-3) = 0 + 3 - 3 = 0 The sum of all the three rows is equal.
Sum = 5 + (-5) + 0 = 5 - 5 + 0 = 0
Sum = -1 + (-2) + (3) = -1 - 2 + 3 = -3 + 3 = 0
Sum = -4 + 7 + (-3) = - 4 + 7 - 3 = 0 The sum of all the three columns is also equal to 0.
Sum = 5 + (-2) + (-3) = 5 - 2 - 3 = 0
Sum = -4 + (-2) + 0 = -4 - 2 + 0 = -6 The sum of two diagonals is not equal to 0. Hence, it is not a magic square. ii
Sum = 1 + (-10) + (0) = 1 - 10 + 0 = -9
Sum = -4 + (-3) + (-2) = -4 - 3 - 2 = -9
Sum = (-6) + (4) + (-7) = - 6 + 4 - 7 = -9 The sum of all the three rows is equal to -9.
Sum = 1 + (-4) + (-6) = 1 - 4 - 6 = -9
Sum = -10 + (-3) + 4 = -10 - 3 + 4 = -9
Sum = 0 + (-2) + (-7) = 0 - 2 - 7 = -9 The sum of all the three columns is also equal to -9.
Sum = 1 + (-3) + (-7) = 1 - 3 - 7 = -9
Sum = 0 + (-6) + (-3) = 0 - 6 - 3 = -9 The sum of two diagonals is also equal to -9. Hence, it is a magic square. Hence, only option (ii) is a magic square.
a - (- b) = 21 - (- 18) When two negative integers are added we get a positive integer. = 21 + 18 = 39
a + b = 21 + 18 = 39 Hence, verified.
a - (- b) = 118 - (- 125) When two negative integers are added we get a positive integer. = 118 + 125 = 243
a + b = 118 + 125 = 243 Hence, verified.
a - (- b) = 75 - (- 84) When two negative integers are added we get a positive integer. = 75 + 84 = 159
a + b = 75 + 84 = 159 Hence, verified.
a - (- b) = 28 - (- 11) When two negative integers are added we get a positive integer. = 28 + 11 = 39
a + b = 28 + 11 = 39 Hence, verified.
(- 8) + (- 4) = -8 - 4 = - 12
(-8) - (- 4) = -8 + 4 = - 4 LHS < RHS Hence, (- 8) + (- 4) < (-8) - (- 4)
(- 3) + 7 - (19) = -3 + 7 - 19 = -15
15 - 8 + (- 9) = 15 - 8 - 9 = -2 LHS < RHS Thus, (- 3) + 7 - (19) < 15 - 8 + (- 9)
23 - 41 + 11 = -7
23 - 41 - 11 = -29 LHS > RHS Thus, 23 - 41 + 11 > 23 - 41 - 11
39 + (- 24) - (15) = 39 - 24 - 15 When one negative and positive integer is added, the result is negative. = 39 - 39 = 0 When two negative integers are added we get a negative integer with the sum.
36 + (- 52) - (- 36) = 36 - 52 + 36 When one negative and positive integer is added, the result is negative. = 72 - 52 = 20 LHS < RHS Thus, 39 + (- 24) - (15)
(e) - 231 + 79 + 51 ___ -399 + 159 + 81
- 231 + 79 + 51 = -231 + 130 = -101
-399 + 159 + 81 = -399 + 240 = -159 LHS > RHS Thus, - 231 + 79 + 51
In a first jump, monkey jumps 3 steps down. In the second jump, monkey jumps 2 steps up. The process continues the same for further jumps. The monkey is currently at the first step. Here, we will represent the down steps by a positive integer and the up steps by a negative integer. In the Steps = 1 + 3 = 4 In the Steps = 4 + (-2) (2 steps down) = 4 - 2 = 2 In the Steps = 2 + 3 = 5 In the Steps = 5 + (-2) (2 steps down) = 5 - 2 = 3 In the Steps = 3 + 3 = 6 In the Steps = 6 + (-2) (2 steps down) = 6 - 2 = 4 In the Steps = 4 + 3 = 7 In the Steps = 7 + (-2) (2 steps down) = 7 - 2 = 5 In the Steps = 5 + 3 = 8 In the Steps = 8 + (-2) (2 steps down) = 8 - 2 = 6 In the Steps = 6 + 3 = 9 Thus, the monkey took 11 jumps to reach the level at 9
In a first jump, monkey jumps 4 steps up. In the second jump, monkey jumps 2 steps down. The process continues the same for further jumps. The monkey is currently at the ninth step. Here, we will represent the down steps by a positive integer and the up steps by a negative integer. In the Steps = 9 + (-4) (4 steps up) = 5 The monkey in the first jump is at the 5 In the Steps = 5 + (2) (2 steps down) = 7 The monkey in the second jump is at the 7 In the Steps = 7 + (-4) (4 steps up) = 3 The monkey in the third jump is at the 3 In the Steps = 3 + (2) (2 steps down) = 5 The monkey in the fourth jump is at the 5 In the Steps = 5 + (-4) (4 steps up) = 1 Now, the monkey has reached the step 1. Thus, it took 5 steps by monkey to reach back to the top step.
In (a) the sum (- 8) represents going down by eight steps. So, what will the sum 8 in (b) represent?
## Exercise 1.2
(a) sum is -7
- -3 + (-4) = -7
- 3 + (-10) = -7
- 2 + (-9) = - 7
- 1 + (-8) = -7
- (-2) + (-5) = -7
- (-1) + (-6) = -7
We can write any one pair of our choice.
- -3 - (7) = -10
- -4 - (6) = -10
- 2 - (12) = - 10
- 1 - (11) = -10
- (-5) - (5) = -10
- (-1) - (9) = -10
- 5 - 15 = -10
- -2 - 8 = - 10
We can write any one pair of our choice.
- -3 + 3 = 0
- -2 + 2 = 0
- -1 + 1 = 0
- -4 + 4 = 0
- -5 + 5 = 0
- -6 + 6 = 0
- -7 + 7 = 0
- -8 + 8 = 0
- -9 + 9 = 0
We can write any one pair of our choice.
- 12 - (4) = 8
- 16 - (8) = 8
- 10 - (2) = 8
- -2 - (-10) = 8
We can write any one pair of our choice.
- 3 + (-8) = -5
- 2 + (-7) = -5
- 1 + (-6) = -5
- 4 + (-9) = -5
- 5 + (-10) = -5
We can write any one pair of our choice.
- -1 - (2) = -3
= -40 + 10 + 0 = -30 Score of Team B = Score in first round + score in the second round + score in the third round = 10 + 0 + (-40) = -30 Thus, both the team scored the same. Yes, we can add integers in any order. The result will be the same n every case.
Thus, the other side should have the value as (-5) to make the statement true. (-5) + (- 8) = (- 8) +
The numbers on both sides are already equal. So, we will add the number 0 so that it does not affect the equation. -53 + 0 = -53
To make the result as 0, the opposite integer with the same value is added to the number. So, we will add (-17) on the left side of the equation to make the result as 0. 17 +
Thus, the other side should have the value as (-7) to make the statement true. [13 + (- 12)] +
Thus, the other side should have the value as (-3) to make the statement true. (- 4) + [15 + (-3)] = [- 4 + 15] + ## Exercise 1.3
Let's understand the multiplication of integers. Negative integer × Negative integer = Positive integer Negative integer × Positive integer = Negative integer Positive integer × Positive integer = Positive integer Positive integer × Negative integer = Negative integer
Positive integer × Negative integer = Negative integer
Negative integer × Positive integer = Negative integer
Negative integer × Negative integer = Positive integer
Any number multiplied with the 1 result in the same number. Negative integer × Negative integer = Positive integer
Any series of number multiplied with 0 is equal to 0. 0 does not have any negative value.
= 132 × (10) = 1320 Negative integer × Negative integer = Positive integer Positive integer × Positive integer = Positive integer Or Negative integer × Negative integer × Positive integer = Positive integer
= 9 × 18 = 162 Negative integer × Negative integer = Positive integer Positive integer × Positive integer = Positive integer Or Positive integer × Negative integer × Negative integer = Positive integer
= 90 × (- 4) = -360 Negative integer × Negative integer = Positive integer Positive integer × Negative integer = Negative integer Or Negative integer × Negative integer × Negative integer = Negative integer
= 2 × (-3) × 4 = (-6) × 4 = -24
= 18 × (-2) × (-1) = (-36) × (-1) Negative integer × Negative integer = Positive integer = 36 ## Note: We can multiply the integers in any order. The result will be the same in every case of multiplication.
18 × [7 + (-3)] The bracket value is always calculated first. = 18 × [7 - 3] = 18 × [4] = 72
[18 × 7] + [18 × (-3)] = [126] + [-54] = 126 - 54 = 72 LHS = RHS Hence, verified.
(-21) × [(- 4) + (- 6)] The bracket value is always calculated first.
Negative integer × Negative integer = Positive integer
[(-21) × (- 4)] + [(-21) × (- 6)] Negative integer × Negative integer = Positive integer = [84] + [126] = 210 LHS = RHS Hence, verified.
Negative integer × Positive integer = Negative integer Negative integer × Negative integer
A × (-1) = -22 A = 22 (-1) multiplied with any positive integer A, makes it a negative integer.
A × (-1) = 37 A = -37 (-1) multiplied with any negative integer A, makes it a positive integer. Negative integer × Negative integer = Positive integer
- (-1) × 5 = -5
- (-1) × 4 = -4
-4 = -5 + 1 - (-1) × 3 = -3
-3 = -4 + 1 - (-1) × 2 = -2
-2 = -3 + 1 - (-1) × 1= -1
-1 = -2 + 1 - (-1) × 0 = 0
0 = -1 + 1
The product of two negative integers is always positive. The product of one negative integer and one positive integer is negative. So,
Negative integer × Negative integer = Positive integer
a × (b + c) = a × b + a × c 26 × (- 48) + (- 48) × (-36) Here, a = -48 b = 26 c = -36 = -48 × (26 + (-36)) = -48 × (26 -36)) = -48 × (-10) Negative integer × Negative integer = Positive integer = 480
8 × 53 × (-125) (The integers can be multiplied in any order) = 8 × (-125) × 53 = -1000 × 53 = -53000
15 × (-25) × (- 4) × (-10) = (-25) × (- 4) × (-10) × 15 (The integers can be multiplied in any order) = 100 × (-10) × 15 = (-1000) × 15 = -15000
(- 41) × 102 = (- 41) × (100 + 2) = (- 41) × 100 + (- 41) × 2 = - 4100 + (- 82) Two negative integers are always added. = - 4100 - 82 = -4182
a × (b + c) = a × b + a × c 625 × (-35) + (- 625) × 65 The second 625 is a negative integer. So, we will convert it into positive first. 625 × (-35) + (625) × (-1) × 65 = 625 × (-35) + (625) × (-65) = 625 (-35 + (-65)) = 625 (-35 - 65) = 625 (-100) = -62500
a × (b + c) = a × b + a × c 7 × (50 - 2) = 7 × 50 - 7 × 2 = 350 - 14 = 336
a × (b + c) = a × b + a × c (-17) × (-29) = (-17) × (-30 + 1) = (-17) × (-30) + (-17) × 1 = 510 - 17 = 493
a × (b + c) = a × b + a × c (-57) × (-19) + 57 = (-57) × (-19) + 57 × 1 We have used 1 with the number because 1 multiplied with any number does not affect that number. = (-57) × (-19) + 57 × 1 The first 57 is a negative integer. So, we will convert it into positive first. = (57) × (-1) × (-19) + 57 × 1 = (57) × (19) + 57 × 1 ((-1) × (-19) = 19) = (57) × [(19) + 1] = 57 × 20 = 1140
The temperature decreases at the rate of 5°C every hour. The temperature decrease in 1 hour = 5°C The temperature decrease in 10 hours = 5°C × 10 = -50°C It is a decrease in the temperature. So, we will represent in using a negative integer. The temperature decrease in 10 hours = -50°C The room temperature 10 hours after the process begins = Current temperature +the temperature decrease in 10 hours The room temperature 10 hours after the process begins = 40°C + (-50°C) =
Marks awarded for every incorrect answer = (-2) Correct answers got by Mohan = 4 Marks awarded for 4 correct answers = 4 × 5 = 20 Incorrect answers got by Mohan = 6 Marks awarded for 6 incorrect answers = 6 × (-2) = -12 Total marks = Marks awarded for correct answers + Marks awarded for incorrect answers Total marks = 20 + (-12) = 20 - 12 = 8 Thus, Mohan scored total 8 marks.
Marks awarded for every incorrect answer = (-2) Correct answers got by Reshma = 5 Marks awarded for 5 correct answers = 5 × 5 = 25 Incorrect answers got by Reshma = 5 Marks awarded for 5 incorrect answers = 5 × (-2) = -10 Total marks = Marks awarded for correct answers + Marks awarded for incorrect answers Total marks = 25 + (-10) = 25 - 10 = 15 Thus, Reshma scored total 15 marks.
Marks awarded for every incorrect answer = (-2) Correct answers got by Heena = 2 Marks awarded for 2 correct answers = 2 × 5 = 10 Incorrect answers got by Heena = 5 Marks awarded for 5 incorrect answers = 5 × (-2) = -10 Total marks = Marks awarded for correct answers + Marks awarded for incorrect answers Total marks = 10 + (-10) = 10 - 10 = 0 Thus, Heena scored total 0 marks.
Loss earned by selling one bag of grey cement in a month= ₹ 5 Total bag of white cement sold = 3000 Price of 3000 bags of white cement sold = 3000 × price per bag = 3000 × 8 = 24000 Profit earned = ₹ 24000 Total bag of grey cement sold = 5000 Price of 5000 bags of grey cement sold = 5000 × price per bag = 5000 × 5 = 25000 Loss = 25000 The price of grey cements in more than the price of white cement sold in a month. It means that the loss is greater than the profit. Difference = 25000 - 24000 = 1000 Thus, The company had a loss of rupees 1000.
Loss earned by selling one bag of grey cement in a month= ₹ 5 Price of 6400 bags of grey cement sold =6400 × price per bag = 6400 × 5 = 32000 Loss = 32000 Profit earned by selling one bag of white cement in a month = ₹ 8 For no profit or no loss, the amount of loss should be equal to the amount of profit. Let the total bags of white cement be A. Price of A bags of white cement sold =A × price per bag = A × 8 A × 8 = Loss amount A × 8 = 32000 A = 32000/8 A = (8 × 4000)/8 A = 4000 Thus, 4000 number of white cement bags it must sell to have neither profit nor loss.
(-3) × A = 27 A = 27/ (-3) A = (9 × 3)/ (-3) Multiplication of two negative integers results into a positive integer. So, we can write the above expression as: A = (-9 × -3)/ (-3) (-9 × -3) = 27 Cancelling out the common terms, we get: A = (-9)
5 × A = -35 A = (-35) /5 A = (-7 × 5)/ (5) Multiplication of one negative integer and one positive integer results into a negative integer. So, we can write the above expression as: A = (-7 × 5)/ (5) Cancelling out the common terms, we get: A = (-7)
A × (- 8) = -56 A = (-56) / (- 8) Multiplication of one negative integer and one positive integer results into a negative integer. So, we can write the above expression as: A = (-8 × 7)/ (- 8) Cancelling out the common terms, we get: A = (7)
A × (- 12) = 132 A = (132) / (- 12) Multiplication of two negative integers results into a positive integer. So, we can write the above expression as: A = (-11 × -12)/ (- 12) Cancelling out the common terms, we get: A = (-11) ## Exercise 1.4
= (-30) / 10 Factorizing the numerator, = (-3 × 10) / 10 Multiplication of one negative integer and one positive integer results into a negative integer. Cancelling out the common terms, we get: = -3
Factorizing the numerator, = (-5 × -10) / -5 Multiplication of two negative integers results into a positive integer. Cancelling out the common terms, we get: = -10
Factorizing the numerator, = (-9 × 4) / (-9) Multiplication of one negative integer and one positive integer results into a negative integer. Cancelling out the common terms, we get: = 4
Factorizing the numerator, = (-1 × 49) / (49) Multiplication of one negative integer and one positive integer results into a negative integer. Cancelling out the common terms, we get: = -1
The values inside the bracket are always calculated first. = 13 / [(-1] 1 divided by any number gives the same number. Factorizing the numerator, = (-1 × -13) / -1 Multiplication of two negative integers results into a positive integer. Cancelling out the common terms, we get: = -13
The values inside the bracket are always calculated first. = (-31) / [-30 - 1] = (-31) / [-31] = 1
The values inside the bracket are always calculated first. = [(-36) / 12] ÷ 3 Factorising the numerator, = [(12 × - 3) / 12] ÷ 3 Cancelling out the common terms, we get: [- 3] ÷ 3 = [- 3] / 3 Factorising the numerator again, = [3 × - 1] / 3 = - 1
The values inside the bracket are always calculated first. = [- 6 + 5)] ÷ [(-2) + 1] = [- 1)] / [(-2) + 1] Now solving the other bracket, = [- 1)] / [-2 + 1] = [- 1)] / [- 1)] = 1
a ÷ (b + c) 12 ÷ (-4 + 2) 12 ÷ (-2) 12/-2 = -6
(a ÷ b) + (a ÷ c) (12 ÷ - 4) + (12 ÷ 2) = (-3) + (6) = 6 - 3 = 3 LHS is not equal to RHS Hence, verified.
Only multiplication is distributive. The division is not distributive. Hence, this property is only valid for multiplication.
a ÷ (b + c) (-10) ÷ (1 + 1) (-10) ÷ (2) (-10) /2 = -5
(a ÷ b) + (a ÷ c) ((-10) ÷ 1) + ((-10) ÷ 1) = (-10) + (-10) Any number divided by 1 gives back the same number. = -20 LHS is not equal to RHS Hence, verified.
(a) 369 ÷ _____ = 369
369 ÷ A = 369 369/A = 369 A = 1 The number on both the sides is the same. Any number divided by 1 gives back the same number. Thus, The blank space is 1.
(-75) ÷ _____ = -1 Let the blank space be A. (-75)/A = -1 A = 75
(-206) ÷ _____ = 1 Let the blank space be A. (-206)/A = 1 A =-206
-87 ÷ A = 87 -87 /A = 87 A = -1 The number on both the sides is the same. Any number divided by 1 gives back the same number. Thus, The blank space is -1.
_____ ÷ 1 = - 87 Let the blank space be A. A ÷ 1 = - 87 A/1 = - 87 A = - 87
_____ ÷ 48 = - 1 Let the blank space be A. A ÷ 48 = - 1 A/48 = - 1 A = - 48
20 ÷ A = -2 20/ A = -2 A = 20/-2 A = -10
A ÷ (4) = -3 A/4 = -3 A = 4 × (-3) A = -12
**(-6, 2)** -6 ÷ (2) = (-3)**(9, -3)** 9 ÷ (-3) = (-3)**(12, -4)** 12 ÷ (-4) = (-3)**(-3, 1)** -3 ÷ (1) = (-3)**(-15, 5)** -15 ÷ (5) = (-3)
The rate of decrease of temperature per hour = 2°C The temperature below zero becomes negative. Thus, the temperature 8°C below zero = -8°C Difference = 10°C - (-8°C) = 10°C + 8°C = 18°C The rate of decrease per degree Celsius = 18°C The rate of decrease per two degree Celsius = 18°C/2°C = 9 Thus, the time at which the temperature be 8°C below zero = 12 noon + 9 = 9 p.m. ( Midnight = 12:00 The temperature at 9.p.m = -8°C The rate of decrease of temperature per hour = 2°C The rate of decrease of temperature for 3 hours = 2°C × 3 = -6°C (It is represented as the negative integer because it shows the decrease in temperature) The temperature at midnight = -8°C + (-6°C) = -14°C
Marks given for every incorrect answer = - 2 Marks given for not attempting any question = 0 Total marks scored by Radhika = 20 Correct answers given by Radhika = 12 Marks for 12 correct answers = 12 × 3 = 36 Difference = Total marks scored - Total marks for correct answers Difference = 20 - 36 = - 16 Marks given for every incorrect answer = - 2 Incorrect questions = Difference/ Marks for 1 incorrect question = -16/ - 2 = 8 Thus, Radhika got
Marks given for every incorrect answer = - 2 Marks given for not attempting any question = 0 Total marks scored by Mohini= -5 Correct answers given by Mohini=7 Marks for 7 correct answers = 7 × 3 = 21 Difference = Total marks scored - Total marks for correct answers Difference = = -5 - 21 = -26 Marks given for every incorrect answer = - 2 Incorrect questions = Difference/ Marks for 1 incorrect question = -26/ - 2 = 13 Thus, Mohinigot
The ground level is considered at 0. The level measured above the ground is positive and the level measured below the ground is negative. The start level = 10 m The end level = - 350 m (350 m below the ground level) Total time taken = (Start level - End level)/ Rate per minute Total time taken = (10 - (- 350))/ 6 Total time taken = (10 + 350)/ 6 Total time taken = (360)/ 6 Total time taken = 60 minutes Or Total time taken = 1 hour (1 hour = 60 minutes). Thus, the elevator descends took Next TopicClass 7 Maths Chapter 2 |