# NCERT Solutions for Class 7 Maths Chapter 1 Integers

**NCERT Solutions for Class 7 Maths Chapter 1 Integers ****- **The first chapter of class 7 starts with recollecting the concepts of integers studied in the previous class of NCERT. The first exercise explained in the NCERT class 7 maths chapter 1 solutions are based on these recollected concepts. In NCERT solutions for class 7 maths chapter 1, you will study more about the integers, their properties, and operations. There are a total of 4 exercises with 30 questions in this chapter. All the questions are explained in the CBSE NCERT solutions for class 7 maths chapter 1. The NCERT solutions are prepared by expert teachers to help students. NCERT Class 7 maths chapter 1 solutions are important for a student. All the topics included in NCERT Class 7 Maths chapter 1 solutions are extremely important for higher classes as well. In this NCERT Solutions for Class 7 Maths Chapter 1, we will study important topics of NCERT solutions for class 7 like properties of addition, subtraction, multiplication, and division of Integers. Here you will get NCERT Class 7 Maths chapter 1 solutions to all four exercises.

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**Topics of NCERT Grade 7 Maths Chapter 1 Integers **

- 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

**NCERT solutions for class 7 maths chapter 1 integers topic 1.2**

** 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?

** Answer: ** First, we complete the number line.

Now, the integers marked by:

B = -6

D = -4

H = 0

J = 2

M = 5

O = 7

** 2. ** Arrange 7, –5, 4, 0 and – 4 in ascending order and then mark them on a number line to check your answer.

** Answer: ** 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.

** Answer: **

(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 :

**NCERT soltions for**** class 7 maths chapter 1 integers ** ** topic 1.3.5 **

** Answer: **

(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.

** Answer: **

(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.

## ** NCERT solutions for class 7 maths chapter 1 integers ** ** topic 1.4.1 **

1. Find:

(i) 6 × (–19)

(ii) 12 × (–32)

(iii) 7 × (–22)

** Answer: **

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)

** 1. ** Find:

(a) 15 × (–16) (b) 21 × (–32)

(c) (– 42) × 12 (d) –55 × 15

** Answer: ** We know, multiplication of a positive and negative integer is given by:

(a)

(b)

(c)

(d)

** 2. ** Check if (a) 25 × (–21) = (–25) × 21 (b) (–23) × 20 = 23 × (–20)

Write five more such examples.

** Answer: ** 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:

** Q. ** Find: (–31) × (–100), (–25) × (–72), (–83) × (–28)

** Answer: ** We know,

Multiplication of two negative integers :

## ** ****NCERT soltions for** class 7 maths chapter 1 integers ** topic 1.4.3 **

**NCERT soltions for**class 7 maths chapter 1 integers

** Answer: ** 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?

** Answer: ** 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)?

** Answer: **

L.H.S =

R.H.S =

Therefore,L.H.S = R.H.S

Hence,

** (ii) ** Is (–15) × [(–7) + (–1)] = (–15) × (–7) + (–15) × (–1)?

** Answer: **

L.H.S =

R.H.S =

Therefore, L.H.S = R.H.S

Hence,

** (ii) ** Is (–15) × [(–7) – (–1)] = (–15) × (–7) – (–15) × (–1)?

** Answer: **

L.H.S =

R.H.S =

Therefore, L.H.S = R.H.S

## ** ****NCERT soltions for** class 7 maths chapter 1 integers ** topic 1.5.7 **

**NCERT soltions for**class 7 maths chapter 1 integers

** Q. ** Find (– 49) × 18; (–25) × (–31); 70 × (–19) + (–1) × 70 using distributive property.

** Answer: ** We know,

Distributive law: for any integers a, b and c,

## ** ****NCERT soltions for** class 7 maths chapter 1 integers ** topic 1.6 **

**NCERT soltions for**class 7 maths chapter 1 integers

Find:

(a) (–100) ÷ 5 (b) (–81) ÷ 9

** Answer: ** 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)

Find: (a) 125 ÷ (–25) (b) 80 ÷ (–5) (c) 64 ÷ (–16)

** Answer: ** 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)

** Answer: ** 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)

## ** ****NCERT soltions for** class 7 maths chapter 1 integers ** topic 1.7 **

**NCERT soltions for**class 7 maths chapter 1 integers

** Q. ** Is (i) 1 ÷ a = 1?

(ii) a ÷ (–1) = – a? for any integer a.

Take different values of a and check.

** Answer: ** (i) is true only for .

(ii)

Taking,

Hence, this is true for every integer.

## ** 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.

(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?

** Answer: **

(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.

** Answer: ** Given, Jack’s scores in five successive rounds are

Therefore, his total score =

His total score at the end was

** Answer: ** 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 =

** Answer: ** 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 =

** Answer: ** 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

** Answer: ** 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.

** **

**Answer: ** 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.

** Answer: ** Taking Rows-

Taking Columns-

Taking Diagonals-

As the sum of all the rows, columns and diagonals are equal, this is a magic square.

** 8. ** Verify for the following values of a and b.

** Answer: **

(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.

** Answer: ** (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,

** Answer: ** 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.

** Answer: ** 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.

** Answer: ** 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.

**NCERT solutions for class 7 maths chapter 1 integers exercise ** 1.2

** 1. ** Write down a pair of integers whose:

(a) sum is –7 (b) difference is –10 (c) sum is 0

** Answer: ** A pair of integers whose:

(a) sum is –7 :

(b) difference is –10 :

(c) sum is 0 :

** Answer: **

(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:

** Answer: ** 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.

** Answer: **

(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

## ** NCERT solutions for class 7 maths chapter 1 integers exercise 1.3 **

** 1. ** Find each of the following products:

(a) 3 × (–1) (b) (–1) × 225

(c) (–21) × (–30) (d) (–316) × (–1)

(e) (–15) × 0 × (–18)

** Answer: **

(a)

(b)

(c)

Product of two negative integers is a positive integer

(d)

(e)

** 1. ** Find each of the following products:

(f) (–12) × (–11) × (10) (g) 9 × (–3) × (– 6)

(h) (–18) × (–5) × (– 4) (i) (–1) × (–2) × (–3) × 4

** Answer: **

Product of even number of negative integers is an even integer.

(f)

(g)

(h)

(i)

(j)

** 2 (a). ** Verify the following:

18 × [7 + (–3)] = [18 × 7] + [18 × (–3)]

** Answer: **

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)]

** Answer: **

L.H.S =

R.H.S

Therefore, L.H.S = R.H.S

Hence verified.

** Answer: **

(i) For any integer a, , i.e, the additive inverse of the given integer.

(ii) The integer whose product with gives the following are:

** 4. ** Starting from (–1) × 5, write various products showing some pattern to show (–1) × (–1) = 1.

** Answer: **

Given,

Therefore,

** 5. ** Find the product, using suitable properties:

(a) 26 × (– 48) + (– 48) × (–36) (b) 8 × 53 × (–125)

(c) 15 × (–25) × (– 4) × (–10) (d) (– 41) × 102

(e) 625 × (–35) + (– 625) × 65 (f) 7 × (50 – 2)

(g) (–17) × (–29) (h) (–57) × (–19) + 57

** Answer: ** Using suitable properties of multiplication:

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

** 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?

** Answer: ** Given,

Initial temperature =

Also, Change in temperature per hour =

Therefore,Change in temperature after 10 hours =

Hence, Final temperature =

** 7 (i). ** 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?

** Answer: ** 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?

** Answer: ** 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.

** Answer: ** 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.

** Answer: ** 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

** Answer: ** 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.

** Answer: **

(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.

## **NCERT soltions for ****class 7 maths chapter 1 integers exercise 1.4 **

(h) [(–36) ÷ 12] ÷ 3 (i) [(– 6) + 5)] ÷ [(–2) + 1]

** Answer: ** 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)

** 2. ** Verify that 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

** Answer: **

(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.

** 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

** Answer: **

** (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.

** Answer: **

Five pairs of integers such that are:

** Answer: ** 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 mid-night will be below zero.

** Answer: ** 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.

** Answer: ** 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 =

### **NCERT Solutions for Class 7 Maths Chapter-wise **

**NCERT Solutions for Class 7 Subject wise **

**Important points of Class 7 Maths Chapter 1 Integers **

- Integers are closed for both addition and subtraction. That is, if a and b are any integers then a + b and a – b are also integers.
- The addition is commutative for the integers, such that for all integers 'a' and 'b'.
- The addition is associative for the integers, such that, for all the integers 'a', 'b' and 'c'.
- Integer 0 is the identity under addition. That is, for every integer 'a'.
- Integers are closed under multiplication. That is, is an integer for any two integers 'a' and 'b'.
- Multiplication is commutative for the integers. That is, for any integers 'a' and 'b.'
- The integer 1 is the identity under multiplication, such that, for any integer 'a'.
- Multiplication is associative for the integers, such that, for any three integers a, b and c.

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

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