NCERT solutions for class 7 maths chapter 1 Integers Class 6 CBSE NCERT maths introduced the chapter integers. The first chapter of class 7 starts with recollecting the concepts of integers studied in the previous class. The first exercise explained in the solutions of NCERT for class 7 maths chapter 1 integers is based on these recollected concepts. In NCERT class 7 maths chapter 1 integers, you will study more about the integers, their properties, and operations. There are a total of 4 exercises with 30 questions. All the questions including topic wise are explained in the CBSE NCERT solutions for class 7 maths chapter 1 integers. The NCERT solutions help students in their preparation of final examination. NCERT Class 7 maths chapter 1 integers is an important chapter for a math student. All the topics included in this chapter are extremely important for the students because these basic concepts of integers are also included in the higher classes. In this chapter, we will study important topics like properties of addition, subtraction, multiplication, and division of Integers, multiplication of a positive integer with a negative integer, multiplication of two or more than two negative integers and also learn closure under addition, subtraction and multiplication. The NCERT solutions for class 7 maths chapter 1 integers help students to understand how to apply the concepts in an applicationlevel problem.
1.1 Introduction
1.2 Recall
1.3 Properties of Addition and Subtraction of Integers
1.3.1 Closure under Addition
1.3.2 Closure under Subtraction
1.3.3 Commutative Property
1.3.4 Associative Property
1.3.5 Additive Identity
1.4 Multiplication of Integers
1.4.1 Multiplication of a Positive and a Negative Integers
1.4.2 Multiplication of two Negative Integers
1.4.3 Product of three or more Negative Integers
1.5 Properties of Multiplication of Integers
1.5.1 Closure under Multiplication
1.5.2 Commutativity of Multiplication
1.5.3 Multiplication by Zero
1.5.4 Multiplicative Identity
1.5.5 Associativity for Multiplication
1.5.6 Distributive Property
1.5.7 Making Multiplication Easier
1.6 Division of Integers
1.7 Properties of Division of Integers
1. A number line representing integers is given below
–3 and –2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O?
First, we complete the number line.
Now, the integers marked by:
B = 6
D = 4
H = 0
J = 2
M = 5
O = 7
The given number are: 7, –5, 4, 0 and – 4
Arranging them in ascending order (increasing order)
On a number line, as we move towards right, the number increases.
(a) 7, 3, – 1, – 5,
The pattern is :
(b) – 2, – 4, – 6, – 8,
The pattern is :
(c) 15, 10, 5, 0,
The pattern is :
(d) – 11, – 8, – 5, – 2,
The pattern is :
(a) a negative integer :
(b) zero :
(c) an integer smaller than both the integers.
(d) an integer smaller than only one of the integers.
(e) an integer greater than both the integers.
(a) a negative integer :
(b) zero :
(c) an integer smaller than both the integers.
(d) an integer smaller than only one of the integers.
(e) an integer greater than both the integers.
1. Find:
(i) 6 × (–19)
(ii) 12 × (–32)
(iii) 7 × (–22)
Multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer
(i)
(ii)
(iii)
We know, multiplication of a positive and negative integer is given by:
(a)
(b)
(c)
(d)
Write five more such examples.
We know, when multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer.
(a)
L.H.S =
R.H.S =
Therefore, L.H.S = R.H.S
(b)
L.H.S =
R.H.S =
Therefore, L.H.S = R.H.S
Five more examples:
We know,
Multiplication of two negative integers :
We know,
If the number of negative integers in a product is even, then the product is a positive integer and if the number of negative integers in a product is odd, then the product is a negative integer.
Hence, the product is positive whereas the product is negative
(ii) What will be the sign of the product if we multiply together:
(a) 8 negative integers and 3 positive integers?
(b) 5 negative integers and 4 positive integers?
(d) (–1), 2m times, m is a natural number?
We know,
If the number of negative integers in a product is even, then the product is a positive integer and if the number of negative integers in a product is odd, then the product is a negative integer
And, any number of positive integers will always give a positive integer.
So, the sign of the product will be decided by the number of negative integers.
(a) 8 negative integers and 3 positive integers: Even number of negative integers, hence product is positive.
(b) 5 negative integers and 4 positive integers: Odd number of negative integers, hence product is negative.
(c) (–1), twelve times: Even number of negative integers, hence product is positive.
(d) (–1), 2m times, m is a natural number: Even number of negative integers, hence product is positive.
NCERT solutions for class 7 maths chapter 1 integers topic 1.5.6
(i) Is 10 × [(6 + (–2)] = 10 × 6 + 10 × (–2)?
L.H.S =
R.H.S =
Therefore,L.H.S = R.H.S
Hence,
L.H.S =
R.H.S =
Therefore, L.H.S = R.H.S
Hence,
(ii) Is (–15) × [(–7) – (–1)] = (–15) × (–7) – (–15) × (–1)?
L.H.S =
R.H.S =
Therefore, L.H.S = R.H.S
Solutions for NCERT class 7 maths chapter 1 integers topic 1.5.7
Q. Find (– 49) × 18; (–25) × (–31); 70 × (–19) + (–1) × 70 using distributive property.
We know,
Distributive law: for any integers a, b and c,
Solutions for NCERT class 7 maths chapter 1 integers topic 1.6
Find:
(a) (–100) ÷ 5 (b) (–81) ÷ 9
When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient
(a)
(b)
(c)
(d)
When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient.
(a)
(b)
(c)
Find: (a) (–36) ÷ (– 4) (b) (–201) ÷ (–3) (c) (–325) ÷ (–13)
When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+).
(a)
(b)
(c)
Solutions of NCERT class 7 maths chapter 1 integers topic 1.7
Q. Is (i) 1 ÷ a = 1?
(ii) a ÷ (–1) = – a? for any integer a.
Take different values of a and check.
(i) is true only for .
(ii)
Taking,
Hence, this is true for every integer.
(a) Observe this number line and write the temperature of the places marked on it.
(b) What is the temperature difference between the hottest and the coldest places among the above?
(c) What is the temperature difference between Lahulspiti and Srinagar?
(a) The temperature of the places in degree Celsius (°C) marked on it are :
Places 
Temperature 
Lahaul Spiti 

Srinagar 

Shimla 

Ooty 

Bangalore 

(b) The hottest temperature is and coldest is
The temperature difference between them =
(c) The temperatures at Lahulspiti is and Srinagar is
The temperature difference between Lahulspiti and Srinagar is =
(d) The temperatures at Srinagar is and Shimla is
The temperature of Srinagar and Shimla took together =
which is less than the temperature of Shimla.
But, it is not less than the temperature at Srinagar.
Given, Jack’s scores in five successive rounds are
Therefore, his total score =
His total score at the end was
The temperature of Srinagar on Monday =
According to question,
The temperature of Tuesday =
The temperature of Srinagar on Tuesday =
Again, according to question
The temperature of Wednesday=
The temperature of Srinagar on Wednesday =
The plane is flying at the height of 5000 m above the sea level
The distance of the plane from sea level =
Also, the submarine floating 1200 m below the sea level
The distance of the submarine from sea level =
Distance between them =
Given,
The amount deposited in bank account =
Amount withdrawn from bank account =
Balance in account = Amount deposited in bank account + Amount withdrawn from the bank account
Therefore,balance in Mohan’s account after the withdrawal is
We will represent the distance travelled towards west with negative integer.
Distance travelled towards east =
Distance travelled towards west =
Distance travelled from A =
As the distance is negative, so final position of Rita from A is towards the west direction.
5  1  4 
5  2  7 
0  3  3 
Taking Rows
Taking Columns
Taking Diagonals
As the sum of one of the diagonals is not equal to 0, it is not a magic square.
1  10  0 
4  3  2 
6  4  7 
Answer:
Taking Rows
Taking Columns
Taking Diagonals
As the sum of all the rows, columns and diagonals are equal, this is a magic square.
(i)
L.H.S =
R.H.S =
L.H.S = R.H.S
Hence verified.
(ii)
L.H.S =
R.H.S =
L.H.S = R.H.S
Hence verified.
(iii)
L.H.S =
R.H.S =
L.H.S = R.H.S
Hence verified.
(iv)
L.H.S =
R.H.S =
L.H.S = R.H.S
Hence verified.
(Note: On a number line, the number on the right is greater than a number towards its left.)
(a) (– 8) + (– 4) (–8) – (– 4)
L.H.S =
R.H.S =
Clearly,
(b) (– 3) + 7 – (19) 15 – 8 + (– 9)
L.H.S =
R.H.S =
Clearly,
(c) 23 – 41 + 11 23 – 41 – 11
L.H.S =
R.H.S =
Clearly,
(d) 39 + (– 24) – (15) 36 + (– 52) – (– 36)
L.H.S =
R.H.S =
Clearly,
(e) – 231 + 79 + 51 –399 + 159 + 81
L.H.S =
R.H.S =
Clearly,
Let the number of steps moved down be represented by positive integers and the number of steps moved up by negative integers.
Given,
Monkey is at the step = 1
Now,
The step after:
1^{st} jump =
2^{nd} jump =
3^{rd} jump =
4^{th} jump =
5^{th} jump =
6^{th} jump =
7^{th} jump =
8^{th} jump =
9^{th} jump =
10^{th} jump =
11^{th} jump =
Therefore, after 11 jumps, the monkey reaches the water level which is at the ninth step.
Let the number of steps moved down be represented by positive integers and the number of steps moved up by negative integers.
Given,
Monkey is at the step = 9
Now,
The step after:
1^{st} jump =
2^{nd} jump =
3^{rd} jump =
4^{th} jump =
5^{th} jump =
Therefore, after 5 jumps, the monkey will reach back at the top.
If we represent the down steps by negative and up steps by positive then:
Moves in (i):
Also,
Moves in part (ii):
Therefore, the sum +8 represents going up by eight steps.
1. Write down a pair of integers whose:
(a) sum is –7 (b) difference is –10 (c) sum is 0
A pair of integers whose:
(a) sum is –7 :
(b) difference is –10 :
(c) sum is 0 :
(a) A pair of negative integers whose difference gives 8 :
(b) A negative integer and a positive integer whose sum is –5:
:
(c) A negative integer and a positive integer whose difference is –3:
Given,
Team A scored :
Team A's total =
Team B scored :
Team B's total =
Therefore, both the team scored same.
Yes, we can add integers in any order.
(i) (–5) + (– 8) = (– 8) + ()
By the commutative property
(ii) –53 + = –53
is the additive identity. The number added to gives the same number.
(iii) 17 + = 0
By additive identity property.
(iv) [13 + (– 12)] + () = 13 + [(–12) + (–7)]
By associative property
(v) (– 4) + [15 + (–3)] = [– 4 + 15] +
By associative property
1. Find each of the following products:
(a) 3 × (–1) (b) (–1) × 225
(c) (–21) × (–30) (d) (–316) × (–1)
(e) (–15) × 0 × (–18)
(a)
(b)
(c)
Product of two negative integers is a positive integer
(d)
(e)
(f) (–12) × (–11) × (10) (g) 9 × (–3) × (– 6)
(h) (–18) × (–5) × (– 4) (i) (–1) × (–2) × (–3) × 4
Product of even number of negative integers is an even integer.
(f)
(g)
(h)
(i)
(j)
18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]
L.H.S =
R.H.S =
L.H.S = R.H.S
Hence, verified.
2 (b). Verify the following:
[(–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)]
L.H.S =
R.H.S
Therefore, L.H.S = R.H.S
Hence verified.
(i) For any integer a, , i.e, the additive inverse of the given integer.
(ii) The integer whose product with gives the following are:
Given,
Therefore,
Using suitable properties of multiplication:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Given,
Initial temperature =
Also,
Change in temperature per hour =
Therefore,Change in temperature after 10 hours =
Hence,
Final temperature =
(i) Mohan gets four correct and six incorrect answers. What is his score?
Given,
Total questions =
Marks for correct answer =
Marks for incorrect answers =
Marks for unattempted question =
According to question,
Mohan gets four correct and six incorrect answers
Total marks =
Therefore, Mohan scored 8 marks.
(ii) Reshma gets five correct answers and five incorrect answers, what is her score?
Given,
Total questions =
Marks for correct answer =
Marks for incorrect answers =
Marks for unattempted question =
According to question,
Reshma gets five correct answers and five incorrect answers
Total marks =
Therefore, Reshma scored 15 marks.
Given,
Total questions =
Marks for correct answer =
Marks for incorrect answers =
Marks for unattempted question =
According to question,
Heena gets two correct and five incorrect answers.
Therefore, number of unattempted questions =
Total marks =
Therefore, Heena scored 0 marks.
Given,
Profit earned by selling 1 bag of white cement =
Profit earned by selling 3000 bags of white cement=
Loss on 1 bag of grey cement =
Loss of 5000 bags of grey cement =
Total earnings =
This means that the company incurred a loss of
Given,
Profit earned by selling 1 bag of white cement =
Loss on 1 bag of grey cement =
Loss on 6400 bags of grey cement =
To be neither in profit nor loss, the profit from white cement must be
Number of white cement bags to be sold =
Therefore the company has to sell 4000 white cement bags.
(a)Given,
The product is positive, therefore there should be even number of negative integers.
(b) Given,
The product is negative, therefore there should be odd number of negative integers.
(c)
The product is negative, therefore there should be odd number of negative integers.
(d)
The product is positive, therefore there should be even number of negative integers.
(h) [(–36) ÷ 12] ÷ 3 (i) [(– 6) + 5)] ÷ [(–2) + 1]
Points to keep in mind:
When we divide a negative integer by a positive integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient.
When we divide a positive integer by a negative integer, we first divide them as whole numbers and then put a minus sign (–) before the quotient.
When we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+).
(a)
(b)
(c)
(d)
(e)
(f )
(g)
(h)
(i)
(a) a = 12, b = – 4, c = 2
L.H.S =
R.H.S =
Therefore.
Hence verified.
(b) a = (–10), b = 1, c = 1
L.H.S =
R.H.S =
Therefore.
Hence verified.
(a) Given,
A number divided by 1 gives the number itself.
(b)
The product is negative, therefore there must be odd number of negative integers.
(c)
A number divided by itself gives 1
(d)
The product is positive, therefore there must be even number of negative integers.
(e)
A number divided by 1 gives the number itself.
(f)
The product is negative, therefore there must be odd number of negative integers.
(g)
The product is negative, therefore there must be odd number of negative integers.
(h)
The product is negative, therefore there must be odd number of negative integers.
Five pairs of integers such that are:
Given,
The temperature at 12 noon =
Final temperature =
The decrease in temperature =
Time taken for the temperature to decrease by =
Time taken for the temperature to decrease by =
Now,
Time until midnight =
The decrease in temperature in =
Therefore, the temperature at midnight =
Therefore, the temperature at midnight will be below zero.
Given,
Marks for every correct answer =
Marks for every wrong answer =
(i) According to question,
Marks obtained by Radhika =
Number of correct answers =
Marks obtained for correct answers =
Marks obtained for incorrect answers = Total marks – Marks obtained for correct answers
Number of incorrect answers =
Therefore, Radhika attempted 8 questions incorrectly.
(ii)
According to question,
Marks obtained by Mohini =
Number of correct answers =
Marks obtained for correct answers =
Marks obtained for incorrect answers = Total marks – Marks obtained for correct answers
Number of incorrect answers =
Therefore, Mohini attempted 13 questions incorrectly.
7. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach – 350 m.
Given,
Initial height =
Final depth =
Total distance the elevator has to descend = Final position  Initial position
=
Also,
Time the elevator takes to descend =
Time the elevator takes to descend =
Chapter No. 
Chapter Name 
Chapter 1 
NCERT Solutions for class 7 maths chapter 1 Integers 
Chapter 2 
CBSE NCERT solutions for class 7 maths chapter 2 Fractions and Decimals 
Chapter 3 

Chapter 4 
Solutions of NCERT for class 7 maths chapter 4 Simple Equations 
Chapter 5 
CBSE NCERT solutions for class 7 maths chapter 5 Lines and Angles 
Chapter 6 
NCERT solutions for class 7 maths chapter 6 The Triangle and its Properties 
Chapter 7 
Solutions of NCERT for class 7 maths chapter 7 Congruence of Triangles 
Chapter 8  NCERT solutions for class 7 maths chapter 8 comparing quantities 
Chapter 9 
CBSE NCERT solutions for class 7 maths chapter 9 Rational Numbers 
Chapter 10 
NCERT solutions for class 7 maths chapter 10 Practical Geometry 
Chapter 11 
Solutions of NCERT for class 7 maths chapter 11 Perimeter and Area 
Chapter 12 
CBSE NCERT solutions for class 7 maths chapter 12 Algebraic Expressions 
Chapter 13 
NCERT solutions for class 7 maths chapter 13 Exponents and Powers 
Chapter 14 
If a student is familiar with the CBSE NCERT solutions for class 7 maths chapter 1 integers, then good marks in the exam is not a difficult task. Also, the NCERT solutions for class 7 maths chapter 1 integers help in solving homework problems