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# NCERT Solutions Class 7 Maths Chapter 1 - Integers

Last updated date: 14th Aug 2024
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## NCERT Solution for Maths Class 7 Chapter 1 Integers - Free PDF Download

In NCERT Solutions for Class 7 Maths Chapter 1, Integers, students are introduced to the world of integers, including positive and negative whole numbers. Our NCERT solutions provide step-by-step explanations and solutions for all the exercises and problems in the chapter. Our solutions cover a wide range of topics, including understanding integers, addition and subtraction of integers, properties of integers, and more. Each question is explained in a detailed yet easy-to-understand manner, ensuring that you grasp the concepts effectively.

Table of Content
1. NCERT Solution for Maths Class 7 Chapter 1 Integers - Free PDF Download
2. Glance of NCERT Solutions for Class 7 Maths Chapter 1 Integers | Vedantu
3. Access Exercise Wise NCERT Solutions for Chapter 1 Maths Class 7
4. Exercises under NCERT Class 7 Maths Chapter 1 - Integers
4.1What is an Integer?
4.3Multiplication of Integers
4.4Division of Integers
5. Access NCERT Solutions for Maths Chapter 1 – Integers
5.1Exercise 1.1
5.2Exercise 1.2
5.33. Fill in the Blanks:
6. NCERT Solutions for Class 7 Chapter 1 Maths Integers – Free PDF Download
6.1How are Integers Applicable in Our Real Life?
6.2Properties of Integers
6.3Facts
6.4Example:
6.5Important Note
6.6Number Line
7. Overview of Deleted Syllabus for CBSE Class 7  Maths Chapter - Integers
8. Class 10 Maths Chapter 1: Exercise Breakdown
9. Other Related Links for CBSE Class 7 Maths Chapter 1
10. NCERT Solutions for Class 7 Maths - Chapter-wise List
FAQs

## Glance of NCERT Solutions for Class 7 Maths Chapter 1 Integers | Vedantu

• In this article, delve deeper into Integers, addition and subtraction of Integers along with multiplication and division of Integers.

• Learn about Properties of Integers, Properties of multiplication of Integers and Properties of Division of Integers and about Number Line and steps involved in drawing a number line.

• Types of Integers:

• Positive Integers: These are whole numbers greater than zero (1, 2, 3, and so on).

• Negative Integers: These are whole numbers less than zero (-1, -2, -3, and so on).

• Zero: Zero is a special case, considered neither positive nor negative.

• We can compare integers using greater than (>) and less than (<) signs. Just like on a number line, higher numbers are to the right and considered greater.

• A number line is a great tool to represent integers. Zero is placed in the middle, positive integers extend to the right, and negative integers extend to the left. The farther a number is from zero, the greater its value (positive or negative).

• This article contains chapter notes, formulas, exercise links and important questions for Chapter 1 - Integers.

• There are three exercises (15 fully solved questions) in Class 7th Maths Chapter 1 Integers

## Access Exercise Wise NCERT Solutions for Chapter 1 Maths Class 7

 S. No Current Syllabus Exercises of Class 7 Maths Chapter 1 1 NCERT Solutions of Class 7 Maths Integers Exercise 1.1 2 NCERT Solutions of Class 7 Maths Integers Exercise 1.2 3

## Exercises under NCERT Class 7 Maths Chapter 1 - Integers

Chapter 1 of the CBSE Class 7 syllabus includes three exercises on Integers. Each exercise contains problems that aim to help students understand the concept of Integers and their practical applications in solving various word problems.

• Exercise 1.1 -  This Exercise contains 4 problems, with multiple parts. These problems aim to introduce students to the fundamentals of Integers which includes addition between two negative integers, one positive and one negative integer, and two negative integers.

• Exercise 1.2 -  This Exercise contains 4 problems, with multiple parts. This exercise covers several problems that involve multiplication of any two integers, distributive property and associative property.

• Exercise 1.3 -  This is the final exercise that contains 7 problems with multiple parts. This exercise deals with division.

### What is an Integer?

The word ‘Integer’ is derived from the Latin word intact or whole. So, integers are always the whole number, which consists of positive or negative numbers or zero, which is simply called a combination of negative and whole numbers. Integers will never be fractional or decimal numbers.  Usually, the set of integers is denoted by Z.

Eg, Z = { -5, -2, 0, 3, 9}

### Addition and Subtraction of Integers

Learning the addition and subtraction of Integers is most important to make simple calculations in day-to-day life for performing simple calculations. Like calculating the pocket money spent by you in a day. Calculating average among integers.

### Multiplication of Integers

Multiplication is a simple one for many of you. But, while multiplying the integers it is important to note a sign of the numbers. This will mainly help while simplifying an equation.

### Multiplication of a Positive and Negative Integer

Multiplication of a positive and negative integer will always leave the answer as a negative integer.

### Multiplication of Two Negative Integers

Multiplication of two negative integers will always result in a positive integer. Should keep in the note of this topic while simplifying the quadratic equation.

### Properties of Multiplication of Integers

1. Multiplication of two positive integers leaves a positive result

2. Multiplication of two negative integers leaves a positive result

3. Multiplication of a positive integer and a negative integer leaves a negative result

4. Multiplication of zero with any number is zero.

### Properties of Division of Integers

1. Division of two positive integers leaves a positive result.

2. The division of two negative integers leaves a positive result.

3. The division of a positive integer and a negative integer leaves a negative result.

4. The division of zero by any number is zero.

5. The division of any number by zero is infinite.

## Access NCERT Solutions for Maths Chapter 1 – Integers

### Exercise 1.1

1. Write down a pair of integers whose:

a) sum is $-7$

Ans: The pair of integers whose sum is $-7$ is $\left( -4,-3 \right)$ i.e., $-4+\left( -3 \right)=-7$.

b) difference is $-10$

Ans: The pair of integers whose difference is $-10$ is $\left( -3,7 \right)$ i.e., $-3-7=-10$

c) sum is $0$

Ans: The pair of integers whose sum is $0$ is $\left( -30,30 \right)$ i.e., $-30+30=0$.

2. Find the integers in the below questions:

a) Write a pair of negative integers whose difference gives $8$.

Ans: The pair of negative integers whose difference is $8$ is $\left( -1,-9 \right)$ i.e., $-1-\left( -9 \right)=-1+9=8$

b) Write a negative integer and a positive integer whose sum is $-5$.

Ans: The pair of negative and positive integers whose sum is $-5$ is $\left( -9,4 \right)$ i.e., $\left( -9 \right)+4=-5$.

c) Write a negative integer and a positive integer whose difference is $-3$.

Ans: The pair of negative and positive integers whose difference is $-3$ is $\left( -1,2 \right)$ i.e., $\left( -1 \right)-2=-1-2=-3$.

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?

Ans: Given that the team $A$ scored $-40,10,0$. Therefore, total score of the team $A$ $=-40+10+0=-30$.

Given that the team $B$ scored $10,0,-40$.Therefore, total score of Team $B$ $=10+0+\left( -40 \right)=-30$.

Therefore, scores of both teams are same and we can add integers in any order due to commutative property.

4. Fill in the blanks to make the following statements true:

i) $\left( -5 \right)+\left( -8 \right)=\left( -8 \right)+\left( ...... \right)$

Ans: Using Commutative Property we can fill the blank as $-5$.

$\therefore$ $\left( -5 \right)+\left( -8 \right)=\left( -8 \right)+\left( -5 \right)$.

ii) $-53+\left( ...... \right)=-53$

Ans: Using Zero additive property we can fill the blank as $0$ .

$\therefore$ $-53+0=-53$

iii) $17+\left( ...... \right)=0$

Ans: Using Zero Additive identity we can fill the blank as $-17$ .

$\therefore$ $17+\left( -17 \right)=0$

iv) $\left[ 13+\left( -12 \right) \right]+\left( ...... \right)=13+\left[ \left( -12 \right)+\left( -7 \right) \right]$

Ans: Using Associative property we can fill the blank as $-7$.

$\therefore$ $\left[ 13+\left( -12 \right) \right]+\left( -7 \right)=13+\left[ \left( -12 \right)+\left( -7 \right) \right]$

v) $\left( -4 \right)+\left[ 15+\left( -3 \right) \right]=\left[ -4+15 \right]+\left( ...... \right)$

Ans: Using Associative property we can fill the blank as $-3$.

$\therefore$ $\left( -4 \right)+\left[ 15+\left( -3 \right) \right]=\left[ -4+15 \right]+\left( -3 \right)$

### Exercise 1.2

1. Find each of the following products:

a) $3\,\,\times \,\,\left( -1 \right)$

Ans: While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e.,

$3\times \left( -1 \right)=-3$

b) $\left( -1 \right)\,\,\times \,\,225$

Ans: While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e.,

$\left( -1 \right)\times 225=-225$

c) $\left( -21 \right)\,\,\times \,\,\left( -30 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e.,

$\left( -21 \right)\times \left( -30 \right)=630$

d) $\left( -316 \right)\,\,\times \,\,\left( -1 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e.,

$\left( -316 \right)\times \left( -1 \right)=316$

e) $\left( -15 \right)\,\,\times \,\,0\,\,\times \,\,\left( -30 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e.,

$\left( -15 \right)\times \,0\times \left( -18 \right)=0$

f) $\left( -12 \right)\,\,\times \,\,\left( -11 \right)\,\,\times \,\,\left( 10 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e.,

$\left[ \left( -12 \right)\times \left( -11 \right) \right]\times \left( 10 \right)=132\times 10=1320$

g) $9\,\,\times \,\,\left( -3 \right)\,\,\times \,\,\left( -6 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e.,

$9\times \left[ \left( -3 \right)\times \left( -6 \right) \right]=9\times 18=162$

h) $\left( -18 \right)\,\,\times \,\,\left( -5 \right)\,\,\times \,\,\left( -4 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product i.e., $\left[ \left( -18 \right)\times \left( -5 \right) \right]\times \left( -4 \right)=90\times \left( -4 \right)$   ….. (1)

While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e., from (1),

$\left[ \left( -18 \right)\times \left( -5 \right) \right]\times \left( -4 \right)=-360$

i) $\left( -1 \right)\,\,\times \,\,\left( -2 \right)\,\,\times \,\,\left( -3 \right)\,\,\times \,\,4$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product and while multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e.,

$\left[ \left( -1 \right)\times \left( -2 \right) \right]\times \left[ \left( -3 \right)\times 4 \right]=2\times \left( -12 \right)=-24$

j) $\left( -3 \right)\,\,\times \,\,\left( -6 \right)\,\,\times \,\,\left( 2 \right)\,\,\times \,\,\left( -1 \right)$

Ans: While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product and while multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product i.e.,

$\left[ \left( -3 \right)\times \left( -6 \right) \right]\times \left[ \left( 2 \right)\times \left( -1 \right) \right]=\left( 18 \right)\times \left( -2 \right)=-36$

2. Verify the following:

a) $18\,\,\times \,\,\left[ 7+\left( -3 \right) \right]=\left[ 18\times 7 \right]+\left[ 18\times \left( -3 \right) \right]$

Ans: Given expression, $18\times \left[ 7+\left( -3 \right) \right]=\left[ 18\times 7 \right]+\left[ 18\times \left( -3 \right) \right]$.

Simplifying the given expression by first solving the square brackets.

While multiplying a negative integer and a positive integer, multiply them as whole numbers and then put a minus sign $\left( - \right)$ before the product.

$\Rightarrow \,\,18\times \left[ 4 \right]=\left[ 126 \right]+\left[ -54 \right]$

$\Rightarrow \,\,72=72$

$\Rightarrow \,\,\text{L}\text{.H}\text{.S}\text{.}=\text{R}\text{.H}\text{.S}\text{.}$

Hence verified.

b) $\left( -21 \right)\times \left[ \left( -4 \right)+\left( -6 \right) \right]=\left[ \left( -21 \right)\times \left( -4 \right) \right]+\left[ \left( -21 \right)\times \left( -6 \right) \right]$

Ans: Given expression, $\left( -21 \right)\times \left[ \left( -4 \right)+\left( -6 \right) \right]=\left[ \left( -21 \right)\times \left( -4 \right) \right]+\left[ \left( -21 \right)\times \left( -6 \right) \right]$

Simplifying the given expression by first solving the square brackets.

While multiplying two negative integers, multiply them as whole numbers and then put a plus sign $\left( + \right)$ before the product

$\Rightarrow \,\,\left( -21 \right)\times \left( -10 \right)=84+126$

$\Rightarrow \,\,210=210$

$\Rightarrow \,\,\text{L}\text{.H}\text{.S}\text{.}=\text{R}\text{.H}\text{.S}\text{.}$

Hence verified.

3. Solve the following:

i)  For any integer $a$, what is $\left( -1 \right)\,\,\times \,\,a$ equal to?

Ans: $\left( -1 \right)\times a=-a,\,\text{ where }a\text{ is an integer}\text{.}$

ii) Determine the integer whose product with $\left( -1 \right)$ is:

a) $-22$

Ans: The integer whose product with $-1$ is $-22$ is $22$ i.e., $\left( -1 \right)\times \left( 22 \right)=-22$.

b) $37$

Ans: The integer whose product with $-1$ is $37$ is $-37$ i.e., $\left( -1 \right)\times \left( -37 \right)=37$.

c) $0$

Ans: The integer whose product with $-1$ is $0$ is $0$ i.e., $-1\times 0=0$.

4. Starting from $\left( -1 \right)\,\,\times \,\,5$ write various products showing some patterns to show $\left( -1 \right)\,\,\times \,\,\left( -1 \right)=1$.

Ans: Consider the product, $\left( -1 \right)\times 5=-5$

Also, $\left( -1 \right)\times 4=-4$, $\left( -1 \right)\times 3=-3$, $\left( -1 \right)\times 2=-2$, $\left( -1 \right)\times 1=-1$, etc.

Thus, we can observe that the product of one negative integer and one positive integer is negative integer.

Similarly, $\left( -1 \right)\times \left( -1 \right)=1$ i.e., the product of two negative integers is a positive integer.

Exercise 1.2

1. Evaluate each of the following:

a) $\left( -30 \right)\div 10$

Ans: While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, $\left( -30 \right)\div \text{10}=-\dfrac{30}{10}=-3$.

b) $50\div \left( -5 \right)$

Ans: While dividing a positive integer by a negative integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, $50\div \left( -5 \right)=-\dfrac{50}{5}=-10$.

c) $\left( -36 \right)\div \left( -9 \right)$

Ans: While dividing a positive integer by a positive integer, divide them as whole numbers and then put a plus sign $\left( + \right)$ before the quotient. Therefore, $\left( -36 \right)\div \left( -9 \right)=\dfrac{36}{9}=4$.

d) $\left( -49 \right)\div 49$

Ans: While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, $\left( -49 \right)\div 49=-\dfrac{49}{49}=-1$.

e) $13\div \left[ \left( -2 \right)+1 \right]$

Ans: Simplifying the given expression, $13\div \left[ \left( -\text{2} \right)+1 \right]=13\div \left( -1 \right)$ ….(1)

While dividing a positive integer by a negative integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore from (1), $13\div \left( -1 \right)=-13$.

f) $0\div \left( -12 \right)$

Ans: While dividing $0$ by any integer, the quotient is $0$. Therefore  $0\div \left( -12 \right)=0$.

g) $\left( -31 \right)\div \left[ \left( -30 \right)\div \left( -1 \right) \right]$

Ans: While dividing a positive integer by a positive integer, divide them as whole numbers and then put a plus sign $\left( + \right)$ before the quotient. Therefore, $\left( -30 \right)\div \left( -1 \right)=30$.  …..(1)

Hence from (1), $\left( -31 \right)\div \left[ \left( -30 \right)\div \left( -1 \right) \right]=\left( -31 \right)\div \left( 30 \right)$ ….. (2)

While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore from (2), $\left( -31 \right)\div \left[ \left( -30 \right)\div \left( -1 \right) \right]=-\dfrac{31}{30}$.

h) $\left[ \left( -36 \right)\div 12 \right]\div 3$

Ans: While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, $\left( -36 \right)\div 12=-\dfrac{36}{12}=-3$  …..(1)

While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore from (1), $\left[ \left( -36 \right)\div 12 \right]\div 3=\left( -3 \right)\div 3=-1$.

i) $\left[ \left( -6 \right)+5 \right]\div \left[ \left( -2 \right)+1 \right]$

Ans: Simplifying the given expression, $\left[ \left( -6 \right)+5 \right]\div \left[ \left( -2 \right)+1 \right]=\left( -1 \right)\div \left( -1 \right)$ ….(1)

While dividing a negative integer by a negative integer, divide them as whole numbers and then put a plus sign $\left( + \right)$ before the quotient. Therefore from (1), $\left[ \left( -6 \right)+5 \right]\div \left[ \left( -2 \right)+1 \right]=1$.

2. Verify that $a\div \left( b+c \right)\ne \left( a\div b \right)+\left( a\div c \right)$  for each of the following values of $a,b\text{ and }c.$

a) $a=12,b=-4,c=2$

Ans: Given, $a=12,b=-4,c=2$. We have to verify $a\div \left( b+c \right)\ne \left( a\div b \right)+\left( a\div c \right)$.

Substituting the given values of $a,b,c$ in LHS we get,

L.H.S.$=12\div \left( -4+2 \right)=12\div \left( -2 \right)$ ….. (1)

While dividing a positive integer by a negative integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore from (1), $12\div \left( -4+2 \right)=-6$.   …..(2)

Substituting the given values of $a,b,c$ in RHS we get,

R.H.S.$=\left[ 12\div \left( -4 \right) \right]\text{+}\left( 12\div 2 \right)\text{ }$ …..(3)

While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, from (3), $\left[ 12\div \left( -4 \right) \right]\text{+}\left( 12\div 2 \right)=\left( -3 \right)\text{+}6\text{ }$

$\Rightarrow RHS=3$ …..(4)

From (2) and (4), we can conclude that $\,\text{L}\text{.H}\text{.S}\text{.}\ne \text{R}\text{.H}\text{.S}\text{.}$ Hence verified.

b) $a=\left( -10 \right),b=1,c=1$

Ans: Given, $a=-10,b=1,c=1$. We have to verify $a\div \left( b+c \right)\ne \left( a\div b \right)+\left( a\div c \right)$.

Substituting the given values of $a,b,c$ in LHS we get,

L.H.S.$=-10\div \left( 1+1 \right)=-10\div \left( 2 \right)$ ….. (1)

While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore from (1), $-10\div \left( 1+1 \right)=-5$.   …..(2)

Substituting the given values of $a,b,c$ in RHS we get,

R.H.S.$=\left[ -10\div 1 \right]\text{+}\left( -10\div 1 \right)\text{ }$ …..(3)

While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, from (3), $\left[ -10\div 1 \right]\text{+}\left( -10\div 1 \right)\text{ }=\left( -10 \right)\text{+}\left( -10 \right)\text{ }$

$\Rightarrow RHS=-20$ …..(4)

From (2) and (4), we can conclude that $\,\text{L}\text{.H}\text{.S}\text{.}\ne \text{R}\text{.H}\text{.S}\text{.}$ Hence verified.

### 3. Fill in the Blanks:

a) $369\div \_\_\_\_\_=369$

Ans: While dividing any integer by $1$, the quotient is the original integer. Therefore, $369\div \underline{1}=369$.

b) $\left( -75 \right)\div \_\_\_\_\_=\left( -1 \right)$

Ans: While dividing any integer by itself, the quotient is $1$. Also, while dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, $\left( -75 \right)\div \underline{75}=\left( -1 \right)$

c) $\left( -206 \right)\div \_\_\_\_\_=1$

Ans: While dividing any integer by itself, the quotient is $1$. Also, while dividing a negative integer by a negative integer, divide them as whole numbers and then put a plus sign $\left( + \right)$ before the quotient. Therefore, $\left( -206 \right)\div \underline{\left( -206 \right)}=1$

d) $\left( -87 \right)\div \_\_\_\_\_=87$

Ans: While dividing any integer by $1$, the quotient is the original integer. Also, while dividing a negative integer by a negative integer, divide them as whole numbers and then put a plus sign $\left( + \right)$ before the quotient. Therefore, $\left( -87 \right)\div \underline{\left( -1 \right)}=87$.

e) $\_\_\_\_\_\div 1=-87$

Ans: While dividing any integer by $1$, the quotient is the original integer. Therefore, $\underline{\left( -87 \right)}\div 1=-87$.

f) $\_\_\_\_\_\div 48=-1$

Ans: While dividing any integer by $1$, the quotient is the original integer. Also, while dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, $\underline{\left( -48 \right)}\div 48=-1$

g) $20\div \_\_\_\_\_=-2$

Ans: While dividing a positive integer by a negative integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient. Therefore, $20\div \underline{\left( -10 \right)}=-2$

h) $\_\_\_\_\_\div \left( 4 \right)=-3$

Ans: While dividing a negative integer by a positive integer, divide them as whole numbers and then put a minus sign $\left( - \right)$ before the quotient.$\left( -12 \right)\div \left( 4 \right)=-3$

4. Write five pairs of integers $\left( a,b \right)$ such that $a\div b=-3$. One such pair is $\left( 6,-2 \right)$ because $6\div \left( -2 \right)=\left( -3 \right)$.

Ans: Five pair of integers $\left( a,b \right)$ such that $a\div b=-3$ are:

i) The pair of integers $\left( -9,3 \right)$ is such that $\left( -9 \right)\div 3=-3$.

ii) The pair of integers $\left( -15,5 \right)$ is such that $\left( -15 \right)\div 5=-3$.

iii) The pair of integers $\left( 12,-4 \right)$ is such that $12\div \left( -4 \right)=-3$.

iv) The pair of integers $\left( -3,1 \right)$ is such that $\left( -3 \right)\div 1=-3$.

v) The pair of integers $\left( 21,-7 \right)$ is such that $21\div \left( -7 \right)=-3$.

5. The temperature at noon was ${{10}^{\circ }}\text{C}$ above zero. If it decreases at the rate of ${{2}^{\circ }}\text{C}$ per hour until mid-night, at what time would the temperature be ${{8}^{\circ }}\text{C}$ below zero? What would be the temperature at mid-night?

Ans: Given that the temperature at $12$ noon is ${{10}^{\circ }}C$ above the zero i.e., $+{{10}^{\circ }}C$. Now, the temperature decreases ${{2}^{\circ }}C$ each hour until midnight. Therefore, following number line represents the temperature path:

Temperature at $12$ noon $={{10}^{\circ }}C$.

Temperature at $1PM$ $={{\left( 10-2 \right)}^{\circ }}C={{8}^{\circ }}C$.

Temperature at $2PM$ $={{\left( 8-2 \right)}^{\circ }}C={{6}^{\circ }}C$.

Temperature at $3PM$ $={{\left( 6-2 \right)}^{\circ }}C={{4}^{\circ }}C$.

Therefore, temperature at $nPM$ $={{\left( 10-2n \right)}^{\circ }}C$.

According to the question, $\left( 10-2n \right)=-8$ ….(1)

Solving (1) by rearranging terms we get,

$10+8=2n$

$\Rightarrow n=9$

Therefore, at $9\,\text{pm}$ the temperature would be ${{8}^{\circ }}\text{C}\,\,\text{below}\,\,{{0}^{\circ }}\text{C}$.

6. In a class test $\left( +3 \right)$ marks are given for every correct answer and $\left( -2 \right)$ 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: Given that Radhika scored $20$ marks and she has got $12$ correct answers.

Let the number of incorrect answers be $x$ then,

Marks given for one correct answer $=3$

Marks given for $12$ correct answers $=3\times 12=36$ ….. (1)

Marks given for one wrong answer $=-2$

Marks given for $x$ wrong answers $=-2x$ ….. (2)

Also, Radhika scored $20$ marks. Hence from (1) and (2),

$20=36+\left( -2x \right)$ ….. (3)

Solving (3) by rearranging terms we get,

$2x=16$

$\Rightarrow x=8$

Therefore, Radhika has attempted $8$ incorrect questions.

ii) Mohini scores $\left( -5 \right)$ marks in this test, though she has got $7$ correct Answers. How many questions has she attempted incorrectly?

Ans: Given that Mohini scored $-5$ marks and she has got $7$ correct answers.

Let the number of incorrect answers be $x$ then,

Marks given for one correct answer $=3$

Marks given for $7$ correct answers $=3\times 7=21$ ….. (1)

Marks given for one wrong answer $=-2$

Marks given for $x$ wrong answers $=-2x$ ….. (2)

Also, Mohini scored $-5$ marks. Hence from (1) and (2),

$-5=21+\left( -2x \right)$ ….. (3)

Solving (3) by rearranging terms we get,

$2x=26$

$\Rightarrow x=13$

Therefore, Mohini has attempted $13$ incorrect questions.

7. An elevator descends into a mine shaft at the rate of $6\text{ m/min}$ . If the descent starts from $10$ above the ground level, how long will it take to reach $\text{-350 m}$?

Ans: Given that the starting position of mine shaft is $10\,\,\text{m}$ above the ground.

And its destination is $350m$ below the ground.

Let us denote the distance above the ground by $+$ sign and the distance below the ground by $-$ sign. Therefore, starting from $10m$ it has to go $-350m$.

Total distance covered by mine shaft $=10\,\text{m}-\left( -350 \right)\text{m}=10+350=360\,\text{m}$…..(1)

Now, it is given that elevator takes $1\text{ }\min$ to cover the distance of $6m$ i.e.,

Time taken to cover a distance of $6\,\text{m}$$=1$ minute.    …..(2)

Hence from (2),

Time taken to cover a distance of $1\,\text{m}$ $=\dfrac{1}{6}$ minute.   …..(3)

Hence from (1) and (3), time taken to cover a distance of $360\,\text{m}$

$=\,\dfrac{1}{6}\times 360=60$ minutes $=1$ hour.

Therefore, in $1$ hour the mine shaft reaches $-350$ below the ground.

## NCERT Solutions for Class 7 Chapter 1 Maths Integers – Free PDF Download

Integers : The set of natural numbers like ……., -4, -3, -2, -1, 0, 1, 2, 3, 4 ….. are called integers.

### How are Integers Applicable in Our Real Life?

Integers are applied in many ways in our real-life besides math class. We can use integers for calculating the efficiency of positive and negative numbers in all fields.

In our day-to-day life, we come across many situations where integers are used:

i. Positive integers are used to determine profit, income, increase, rise, high, north, east, above depositing, climbing and so on.

ii. Negative integers are used to determine quantities like loss, expenditure, decrease, fall, low, south, west, below, withdrawing, sliding and so on.

Example: A point A is on a mountain which is 4680 m above sea level and a point B is in a mine which is 765 m below sea-level. What is the distance between A and B?

In this example, we can consider a point O at the sea level.

Then, height OA = + 4680m;

Height OB = - 765m.

Distance between A and B  = [OA] + [OB]

= {[ + 4680 ]  + [ -765]} m

= (4680 + 765) m = 5445 m

### Properties of Integers

The properties of integers include numbers for addition and multiplication through patterns. They also include the whole numbers as well. Integers involve expressing communicative and associative properties in a general form.

### Facts

• The counting numbers 1, 2, 3, 4, 5, …….. and so on are called Natural Numbers, whereas the set of natural numbers together with zero like  0, 1, 2, 3, 4, 5, ……. and so on are called whole numbers.

• On a number line, we represent the negative integers by the points to the left of zero and positive integers by the points to the right of zero.

• The integer 0 is an additive identity for integers by the points to the left of zero and positive integers by the points to the right of zero.

• The integer 0 is neither positive nor negative.

• The absolute value of an integer is its numerical value of the integer regardless of its sign. The absolute value of an integer a is denoted by | a |.

### Example:

Absolute value of 8 i.e. | 8 | = 8

Absolute value of –8 i.e. | -8 | = 8

• If ‘a’ and ‘b’ are integers, then (a + b), (a - b) and (a x b) are also integers. Integers are used for addition, subtraction and multiplication.

• If ‘a’ and ‘b’ are integers then a x b = b x a and a + b = b + a. Multiplication is a commutative property for integers.

• For any three integers a, b and c, we have:

1. Addition is associative property for integers. a + ( b + c ) = ( a + b ) + c = c + ( b + a )

2. Multiplication is associative property for integers. a x ( b x c ) = ( a x b ) x c = c x ( b x a)

• For any three integers a, b and c, we have:

1. a x (b + c) = (a x b) + ( a x c)

2. a x (b-c) = (a x b) – (a x c)

• 1 is the multiplicative identity for integers and 0 is the identity under addition.

Ex: a + 0 = a = 0 + a and a x 1= a =1 x a

• If the integers are of like signs, their product is positive.

• When we add two positive or negative integers with like signs, we add their numerical values and assign the sign of the numbers added with the sum.

Ex: 6 + 5 = 11

- 6 + ( - 5 ) = - 6  - 5 = -11

• If the integers have unlike signs, their product is negative.   a x –b = - ab

Ex:  5 x – 4 = -20

- 5 x 4 = -20

• When we add two integers with unlike signs then we take the difference of their numerical values and assign the sign of the integer with the greater numerical value.

Ex: 2 + ( - 8 ) = - 2

8 + ( - 2 ) = 2

• The product of an integer and 0 is always 0.

• If the dividend and divisor are integers of like signs, then the quotient is a positive integer.

• If the dividend and divisor are integers of unlike signs, then the quotient is a negative integer.

• Division by 0 is not possible. However, 0 divided by any integer (except 0) is equal to 0.

### Important Note

• 0 is neither positive nor negative.

• The + sign is not written before a positive number.

• ½  and 0.5 are not integers because they are not whole numbers.

• Negative numbers are usually placed in brackets to avoid confusion arising due to two signs in evaluations.

Ex: 4 + ( - 2 ) = -2

### Number Line

A number line represents natural numbers, whole numbers, positive integers and negative integers. The identities are marked at equal intervals on a line to determine numerical operations. Number lines are important because they represent numbers that are used in our daily life.

Steps for drawing a number line

1. Draw a straight line of any length.

2. Mark points at equal intervals on the drawn line to divide it into the required number.

3. Mark any one of the points, marked on the line in step 2, as 0.

4. Starting from 0 and on the right-hand side of the line, mark the positive numbers + 1, + 2, + 3, and so on. Similarly, starting from 0 on the left side of mark the negative integers -1, -2, -3, and so on.

5. The arrowheads on both the sides of the drawn line indicate that the numbers continue up to infinity.

## Overview of Deleted Syllabus for CBSE Class 7  Maths Chapter - Integers

 Chapter Dropped Topics Integers Introduction - Page Number (1–5) Recall - Page Number (14–15) 1.4.3 Product of three or more negative numbers - Page Number(18–22) 1.5.7 Making multiplication easier - Page Number(27)

## Class 10 Maths Chapter 1: Exercise Breakdown

 Exercise Number of Questions Exercise 1.1 Solutions 4 Questions & Solutions Exercise 1.2 Solutions 7 Questions & Solutions Exercise 1.3 Solutions 7 Questions and Solutions

## Conclusion

The NCERT Solutions for Class 7 Maths Chapter 1 Integers, provided by Vedantu, are essential for understanding the basics of integers. These solutions help students grasp concepts like addition, subtraction, multiplication, and division of integers, as well as their properties. It's important to focus on solving the examples and practice problems to reinforce these concepts. The step-by-step explanations make complex topics easier to understand, which is crucial for building a strong foundation in mathematics. For best results, regularly practice the exercises and review the key concepts highlighted in these solutions.

## Other Related Links for CBSE Class 7 Maths Chapter 1

 Sr.No Important Links for Chapter 1 Integers 1. Integers Important Questions 2. Integers Revision Notes 3. Integers Formulas 4. Integers RD Sharma Solutions 5. Integers NCERT Exemplar Solutions 6. Integers RS Aggarwal Solutions

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

Given below are the chapter-wise NCERT Solutions for Class 7 Maths. These solutions are provided by the Maths experts at Vedantu in a detailed manner. Go through these chapter-wise solutions to be thoroughly familiar with the concepts.

## FAQs on NCERT Solutions Class 7 Maths Chapter 1 - Integers

1. Write a Pair of Integers Whose Sum Gives

1. Negative Integer.

2. Zero

3. an Integer Smaller than Only one of the Integers.

4. an Integer Greater than Both the Integers.

5. an Integer Greater than Both the Integers.

1. (-20) and 8

sum: ( -20) + 8 = -2 ( -2 is a negative integer)

2. -15 and 15

sum: (-15) + 15 = 0

3. ( -8 ) and ( -2 )

sum: ( -8 ) + ( -2 ) = - 10 ( - 10 is smaller than - 8 and -2 )

4. 5 and - 7

sum: 5 + ( -7) = -2 ( -2 is smaller than 5)

5. 15 and 20

sum: 15 + 20 = 35 ( 35 is greater than 15 and 20)

2. Simplify the Following, Using Suitable Properties:

1. ( -1500 ) x ( -100 ) + ( -1500 ) x ( -50 )

1. (2500 x 845) – (579 x 845)

1. (-1500) x (-100) + (-1500) x (-50)

= 1500 ((-1) x (-100) + (-1) x (-50))

= 1500 (100 + 50)

= 1500(150)

=225000

2. (2500 x 845) – (579 x 845)

= 845 (2500 - 579)

= 845 (1921)

= 1623245

3. Choose the Correct Answers from the Given Choice and Fill in the Following Blanks.

1. -3, -2, -1 are __________________ Integers.  (positive/ negative)

2. The Greatest Negative Integer is __________________. (0/ -1)

3. Every Positive Integer is _________________ then Every Negative Integer. (Smaller/ Greater)

4. The Product of Two Integers is Always ______________________. (an Integer/ not an Integer)

1. Negative

2. - 1

3. Greater

4. An integer

4. What are the 4 rules of integers?

The four basic rules for integers are:

The sum of two positive integers is a positive integer. (Ex: 9 + 2 = 11)

The sum of two negative integers is a negative integer. (Ex: -3 + (-5) = -8)

The sum of a positive integer and a negative integer depends on their values. The integer with the larger absolute value determines the sign of the sum. (Ex: 5 + (-3) = 2, -2 + 4 = 2)

Subtraction:

Subtracting a positive integer is the same as adding a negative integer with the same absolute value. (Ex: 6 - 4 = 6 + (-4) = 2)

Subtracting a negative integer is the same as adding a positive integer with the same absolute value. (Ex: 10 - (-2) = 10 + (2) = 12)

Multiplication:

The product of two positive integers is a positive integer. (Ex: 4 x 6 = 24)

The product of two negative integers is a positive integer. (Ex: -4 x (-6) = 24)

The product of a positive integer and a negative integer is a negative integer. (Ex: 2 x (-8) = -16)

Division:

Dividing two positive integers results in a positive integer (assuming no remainder). (Ex: 10 ÷ 5 = 2)

Dividing two negative integers results in a positive integer (assuming no remainder). (Ex: -18 ÷ (-2) = 9)

Dividing a positive integer by a negative integer results in a negative integer (assuming no remainder). (Ex: 20 ÷ (-5) = -4)

Dividing a negative integer by a positive integer results in a negative integer (assuming no remainder). (Ex: -16 ÷ 4 = -4)

5. What are integers for 7th class?

Integers is the first chapter in the NCERT Class 7 Maths textbook. The set of natural numbers like -4, -3, -2, -1, 0, 1, 2, 3, 4, and so on are called integers. They have a lot of applications in real life. Both positive and negative integers play an important role in daily calculations and transactions. It is essential to master elementary topics like Integers to be able to do well in Maths in higher classes.

6. What I found challenging in chapter integers?

Understanding negative numbers: It might be difficult to understand negative numbers and how they differ from positive numbers.It can be difficult to see them as a number line or to understand their behavior when they are operating.

Applying Signs Correctly: It might be challenging to recall the rules for positive and negative signs, particularly in operations like subtraction and multiplication. Students might struggle to determine the resulting sign (+ or -) based on the signs of the numbers being operated on.

7. What is the fraction for 7th class?

Fractions and Decimals is the 2nd chapter in the NCERT CBSE Class 7 Maths textbook. An introduction to fractions has already been provided in earlier classes. The addition and subtraction of fractions have also already been discussed previously. The chapter Fractions in class 7 mostly deals with the multiplication and division of fractions. The concept behind reciprocal fractions and mixed fractions is taught in detail and questions are set on the same.

8. What are the formulas of integers?

You do not need any formulas for Integers as long as your concepts are clear. However, you can remember some tricks like the product of two integers of the same sign is always positive whereas the product of two integers of different signs is always negative. To add two integers of the same signs, you just need to add their absolute values while to add two integers of different signs, you need to subtract their absolute values.

9. Where can I get NCERT Solutions for Class 7th Maths Chapter 1?

The best place to get all your questions answered and find solutions for NCERT Class 7 Maths Chapter 1 is Vedantu. Now you need not worry if you get stuck on a sum or do not know the method of solving a complicated maths problem. All you need to do is click on Vedantu's NCERT Class 7 Maths Chapter 1 Solutions and get all your doubts solved instantly in a comprehensive and easy-to-understand  manner. You can download the solution free of cost in PDF format from the Vedantu website and from the Vedantu app.

10. What are the 3 types of integers?

Positive Integers: These are the counting numbers we all know, starting from 1 and going up forever (like 1, 2, 3, 4, 5...). They represent positive quantities.

Negative Integers: Imagine a number line where zero is the center. Negative integers live to the left of zero. They represent quantities less than zero, like -1, -2, -3, and so on.

Zero: Zero stands alone at the center of the number line. It's neither positive nor negative, but it's still considered an integer.

11. What is the full form of integers?

The word integer originated from the Latin word “Integer” which means whole or intact.