NCERT Solutions for Class 7 Maths Chapter 1: Integers - Exercise 1.3

Exercise 1.3

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 a negative integer.

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

5. Find the product, using suitable properties:

a. $26\,\,\times \,\,\left( -48 \right)+\left( -48 \right)\,\,\times \,\,\left( -36 \right)$

Ans: $26\times \left( -48 \right)+\left( -48 \right)\times \left( -36 \right)$ …..(1)

Taking $-48$ common from (1) using distributive property we get,

$\Rightarrow \,\,\left( -48 \right)\times \left[ 26+\left( -36 \right) \right]$

$\Rightarrow \,\,\left( -48 \right)\times \left( -10 \right)$

$\Rightarrow \,\,480$

b. $8\,\,\times \,\,53\,\,\times \,\,\left( -125 \right)$

Ans: $8\times 53\times \left( -125 \right)$ …..(1)

Using Commutative property on (1) we get,

$\Rightarrow \,\,53\times \left[ 8\times \left( -125 \right) \right]$

\[\Rightarrow \,\,53\times \left( -1000 \right)\]

$\Rightarrow \,\,-53000$

c. $15\,\,\times \,\,\left( -25 \right)\times \left( -4 \right)\,\,\times \,\,\left( -10 \right)$

Ans: $15\times \left( -25 \right)\times \left( -4 \right)\times \left( -10 \right)$ 

$\Rightarrow \,\,15\times \left[ \left( -25 \right)\times \left( -4 \right)\times \left( -10 \right) \right]$   …..(1)

Using Commutative property on (1) we get,

\[\Rightarrow \,\,15\times \left( -4 \right)\times \left[ -25\times -10 \right]\]

\[\Rightarrow \,\,15\times \left( -4 \right)\times 250\] 

$\Rightarrow \,\,-15000$

d. $\left( -41 \right)\,\,\times \,\,\left( 102 \right)$

Ans: $\left( -41 \right)\times \left( 102 \right)$

$\Rightarrow \,\,-41\times \left[ 100+2 \right]$    …..(1)          

Using distributive property on (1) we get,

$\Rightarrow \,\,\left[ \left( -41 \right)\times 100 \right]+\left[ \left( -41 \right)\times 2 \right]$

$\Rightarrow \,\,-4100+\left( -82 \right)$

$\Rightarrow \,\,-4182$

e. $625\,\,\times \,\,\left( -35 \right)+\left( -625 \right)\,\,\times \,\,65$

Ans: $625\times \left( -35 \right)+\left( -625 \right)\times 65$ …..(1)

Taking $625$ common from (1) using distributive property we get,

$\Rightarrow \,\,625\times \left[ \left( -35 \right)+\left( -65 \right) \right]$

$\Rightarrow \,\,625\times \left( -100 \right)$

$\Rightarrow \,\,-62500$

f. $7\,\,\times \,\,\left( 50-2 \right)$

Ans: $7\times \left( 50-2 \right)$ ….. (1)

Using distributive property on (1) we get,

$\Rightarrow \,\,7\times 50-7\times 2$  

$\Rightarrow \,\,350-14=336$

g. $\left( -17 \right)\,\,\times \,\,\left( -29 \right)$

Ans: $\left( -17 \right)\times \left( -29 \right)$ 

$\Rightarrow \,\,\left( -17 \right)\times \left[ \left( -30 \right)+1 \right]$  …..(1)

Using distributive property on (1) we get,

$\Rightarrow \,\,\left( -17 \right)\times \left( -30 \right)+\left( -17 \right)\times 1$

$\Rightarrow \,\,510+\left( -17 \right)$       

$\Rightarrow \,\,493$

h. $\left( -57 \right)\,\,\times \,\,\left( -19 \right)+57$

Ans: $\left( -57 \right)\times \left( -19 \right)+57$ 

$\Rightarrow \,\,\left( -57 \right)\times \left( -19 \right)+57\times 1$

$\Rightarrow \,\,57\times 19+57\times 1$        …..(1)

Taking $57$ common from (1) using distributive property we get,

$\Rightarrow \,\,57\times \left( 19+1 \right)$ 

$\Rightarrow \,\,57\times 20=1140$

6. A certain freezing process requires that room temperature be lowered from ${{40}^{\circ }}\text{C}$ at the rate of ${{5}^{\circ }}\text{C}$ every hour. What will be the room temperature $10$ hours after the process begins?

Ans: It is given that the present temperature of the room is ${{40}^{\circ }}\text{C}$. Also, the temperature is decreasing every hour by ${{5}^{\circ }}\text{C}$.

Therefore, the temperature after $1$ hour is ${{40}^{\circ }}-{{5}^{\circ }}={{35}^{\circ }}C$.

Temperature after $2$ hours is ${{40}^{\circ }}-2\times \left( {{5}^{\circ }} \right)={{30}^{\circ }}C$. 

Temperature after $3$ hours is ${{40}^{\circ }}-3\times \left( {{5}^{\circ }} \right)={{25}^{\circ }}C$. 

Similarly, temperature after $n$ hours is ${{\left[ 40-n\left( 5 \right) \right]}^{\circ }}C$.    ….. (1)

Form (1), room temperature after 10 hours \[={{\left[ 40-10\times \left( 5 \right) \right]}^{\circ }}C\]

$\Rightarrow {{\left[ 40-50 \right]}^{\circ }}\text{C}=-{{10}^{\circ }}\text{C}$

Therefore, the room temperature $10$ hours after the process begins is $-{{10}^{\circ }}\text{C}$.

7. In a class test containing $10$ questions, $5$ marks are awarded for every correct answer and $\left( -2 \right)$ 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? 

Ans: Given that out of $10$ questions, Mohan gets $4$ correct and $6$ incorrect answers. 

Marks for $1$ correct answer $=5$ 

Marks for $4$ correct answers $=4\times 5=20$  …..(1)

Marks for $1$ wrong answer $=-2$ 

Marks for $6$ wrong answers $=6\times \left( -2 \right)=-12$  …..(2)

Therefore from (1) and (2), total scores of Mohan$=\left( 20 \right)+\left( -12 \right)=8$.

ii. Reshma gets five correct answers and five incorrect answers, what is her score? 

Ans: Given that out of $10$ questions, Reshma gets $5$ correct and $5$ incorrect answers. 

Marks for $1$ correct answer $=5$ 

Marks for $5$ correct answers $=5\times 5=25$  …..(1)

Marks for $1$ wrong answer $=-2$ 

Marks for $5$ wrong answers $=5\times \left( -2 \right)=-10$  …..(2)

Therefore from (1) and (2), total scores of Resham $=\left( 25 \right)+\left( -10 \right)=15$.

ii. Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?

Ans: Given that out of $10$ questions, Heena gets $2$ correct, $5$ incorrect and $3$ un-attempted.

Marks for $1$ correct answer $=5$ 

Marks for $2$ correct answers $=2\times 5=10$  …..(1)

Marks for $1$ wrong answer $=-2$ 

Marks for $5$ wrong answers $=5\times \left( -2 \right)=-10$  …..(2)

Marks for $1$ un-attempt question $=0$ 

Marks for $3$ un-attempt questions $=0\times 3=0$  …..(3)

Therefore from (1), (2) and (3), total scores of Heena $=\left( 10 \right)+\left( -10 \right)+0=0$.

8. A cement company earns a profit of ₹$8$ per bag of white cement sold and a loss of ₹$5$ per bag of grey cement sold.

a. The company sells \[3,000\] bags of white cement and \[5,000\] bags of grey cement in a month. What is its profit or loss?

Ans: Given that profit of 1 bag of white cement $=$ ₹$8$

Therefore, Profit on selling $3000$ bags of white cement

\[=3000\times 8=\] ₹$24,000$ ….. (1) 

And loss of 1 bag of grey cement $=$ ₹$5$.

Therefore, Loss on selling $5000$ bags of grey cement

\[=5000\times 5=\] ₹$25,000$ …..(2)

Let us denote profit by  $+$ and loss by $-$. Then from (1) and (2), 

Total selling $=24000+\left( -25000 \right)=-1000$.

Therefore, his total loss on selling cement bags is ₹$1,000$.

b. What is the number of white cement bags it must sell to have neither profit nor loss. If the number of grey bags sold is \[6,400\] bags.

Ans: Let the number of bags of white cement bags sold be $x$.

Given that profit of 1 bag of white cement $=$ ₹$8$

Therefore, Profit on selling $x$ bags of white cement ₹\[8x\] ….. (1) 

And loss of 1 bag of grey cement $=$ ₹$5$.

Therefore, Loss on selling $6400$ bags of grey cement 

\[=6400\times 5=\] ₹$32,000$ …..(2)

To have neither profit nor loss, from (1) and (2) we get,

$8x=32000$

$\Rightarrow \,\,x=\dfrac{32000}{8}\text{ }$ 

$\therefore x=4000$ 

Thus, he must sell $4000$ white cement bags to have neither profit nor loss.

9. Replace the blank with an integer to make it a true statement:

a. $\left( -3 \right)\,\,\times \,\,\_\_\_\_\_=27$

Ans: Let the number in the blank be $x$, then 

\[\left( -3 \right)\times x=27\] 

$\Rightarrow x=\dfrac{27}{-3}$ 

$\Rightarrow x=-9$ 

\[\therefore \left( -3 \right)\times \underline{\left( -9 \right)}=27\]

b. $5\,\,\times \,\_\_\_\_\_=-35$

Ans: Let the number in the blank be $x$, then

\[5\times x=-35\text{ }\] 

$\Rightarrow x=\dfrac{-35}{5}$ 

$\Rightarrow x=-7$ 

\[\therefore 5\times \underline{\left( -7 \right)}=-35\text{ }\]

c. $\_\_\_\_\_\times \left( -8 \right)=-56$

Ans: Let the number in the blank be $x$, then

$x\times \left( -8 \right)=-56$ 

$\Rightarrow x=\dfrac{-56}{-8}$ 

$\Rightarrow x=7$ 

$\therefore \underline{7}\times \left( -8 \right)=-56$

d. $\_\_\_\_\_\times \left( -12 \right)=132$

Ans: Let the number in the blank be $x$, then

\[x\times \left( -12 \right)=132\] 

\[\Rightarrow x=\dfrac{132}{-12}\] 

$\Rightarrow x=-11$ 

\[\therefore \underline{\left( -11 \right)}\times \left( -12 \right)=132\]

Chapter 1 Integers

The Maths chapter 1 INTEGERS covers the following topics:

1. Introduction

2. Properties of addition and subtraction

3. Division of integers

4. Properties of Multiplication of two negative integers

5. Properties of Multiplication of integers

6. Division of Integers

Introduction

In chapter 1 class 7 maths, this chapter will focus on making the students learn about Integers as a whole category. Students will learn that the integers form a bigger collection of numbers which contains both whole numbers and negative numbers. Integers can be simply explained as the numbers that can be written without fractional components.

Let's Recall

Before going deep into the chapter, let us first recall what we learned in the previous chapters. 

We learned to represent integers on a number line in ascending as well as descending order. Through this number line, we learned how to do addition and subtraction of integers and the properties of addition and subtraction using the concept of the integral number line.

Properties of Addition and Subtraction of Integers

Closure Property Under Addition

According to this property, the sum of two whole numbers is again a whole number. For example, a+b=c, where the result obtained c is a whole number.

Closure Property Under Subtraction

According to this property, the difference between two whole numbers is not always a whole number but the difference between two integers will always be an integer. For instance, a-b=c, where the result c is an integer.

Commutative Property

According to this property, the sum of any two whole numbers or integers, taken in any order, remains the same. For any two integers a and b, it can be represented mathematically as a+b=b+a.

This property is not applicable in the case of subtraction.

Associative Property

According to this property, the sum of the numbers remains the same regardless of how the parenthesis is placed or the way in which the groups are formed. For any three integers a,b and c, it can be represented mathematically as (a+b)+c = a+(b+c).

This property is also applicable in the case of multiplication.

Additive Identity

Additive Identity is defined as the element which when added to any number, say X, gives the same number X. In such cases, zero is the additive identity because when added to any number, the number remains unchanged. It can be represented mathematically as a+0=0+a.

Multiplication of Integers

Multiplication of a Positive and a Negative Integer

While multiplying one positive and one negative integer, both the integers are multiplied and a negative sign is put before their product. It can be represented mathematically as a*(-b)= (-a)*b= -(a*b).

Multiplication of two Negative Integers

In mathematics, a negative number is always a real number that is less than zero. Negative numbers come towards the left in a number line; therefore the product of two negative numbers always turns out as a positive number. This procedure is quite similar to the rule for addition and subtraction.

Product of Three or More Negative Integers

During the multiplication of negative integers, the sign depends on the number of negative integers that are being multiplied. If the negative integers are multiplied even number of times then the product is positive, and if the negative integers are multiplied an odd number of times then the product is negative.

Properties of Multiplication of Integers

Multiplication of integers means the product of two or more integers. multiplication of integers is basically repeated addition that is a*n= a+a+a+a……an up to n times.

The properties of multiplication of integers are:

1. Closure property 

2. Commutative property

3. Multiplication by zero

4. Multiplicative identity

5. Associative property

6. Distributive property
   
   

Closure Under Multiplication

According to this property if any two integers x and y are multiplied then their resultant x*y is also an integer. Therefore the integers are closed under multiplication. a*b is an integer for every integer a and b.

According to this property altering the order of operands or the integers does not affect the result of multiplication x*y= y*x for every integer x and y.

Multiplication by Zero

According to this property, multiplying any integer by zero, the result is always zero, that is, if x and y are two integers then x*0=0 and y*0=0

Multiplicative Identity

According to this property, multiplying any inter by 1, the result obtained will be the integer itself, that is, if x and y are two integers then, x*1=1*x=x 

Therefore 1 is the multiplicative identity of integers.

Associativity of Multiplication

According to this property, the result of the product of three or more integers is irrespective of the grouping of these integers, that is, if x, y, and z are three integers then x*(y*z)=(x*y)*z.

Distributive property

According to this property of multiplication of integers, if x, y, and z are the three integers then 

X*(y+z)=(x*y)+(x*z)

Making Multiplication Easier

According to this, the multiplication of big numbers and negative integers can be done easily by distributive and commutative properties.

Division of Integers

Arithmetic Operation is a branch of Mathematics that involves addition, subtraction, division, and multiplication on all types of real numbers including integers. Now let's come to the point what is actually Division of Integers.

Properties of Division of Integers

If the multiplication is the totaling of numbers, the division is the distribution of numbers. Dividing integers is the opposite operation of multiplication. But the rules for the division of integers are the same as multiplication. But it is not necessary that the quotient will always be an integer. For better explanation there are three rules:

RULE 1: The quotient of two positive integers will always be positive.

RULE 2: The quotient of two negative integers will always be positive.

RULE 3: The quotient of a positive integer and a negative integer will always be negative.

Topics Covered in Exercise 1.3

Based on your understanding of the properties of integers, you will have to answer 9 questions that are mentioned in exercise 1.3. Students have to carefully read the questions and solve them by applying the properties of integers. To solve these questions easily, you can refer to the examples.

Secret Tips to Understand the Chapter

Here are a few tips which will help you in understanding the concepts of the chapter in a better way:

• Practice as much as you can. Always remember that practice makes a man perfect. 

• Take notes and prepare charts and number lines wherever necessary.

• Alongwith the NCERT Solutions, practice our sample papers and previous year question papers as well. This will help you in excelling the chapter.

• Take proper breaks between your studies. Many of the students do this mistake by not taking proper breaks. But taking rest in between helps you focus more.

NCERT Solutions for Class 7 Maths

• Chapter 1 - Integers

• Chapter 2 - Fractions and Decimals

• Chapter 3 - Data Handling

• Chapter 4 - Simple Equations

• Chapter 5 - Lines and Angles

• Chapter 6 - The Triangle and Its Properties

• Chapter 7 - Congruence of Triangles

• Chapter 8 - Comparing Quantities

• Chapter 9 - Rational Numbers

• Chapter 10 - Practical Geometry

• Chapter 11 - Perimeter and Area

• Chapter 12 - Algebraic Expressions

• Chapter 13 - Exponents and Powers

• Chapter 14 - Symmetry

• Chapter 15 - Visualising Solid Shapes

NCERT Solution Class 7 Maths of Chapter 1 All Exercises

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