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Download Class 11 Geography NCERT Solutions | 11th Class Geography NCERT Solutions PDF

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If you are looking for help on Class 11 Geography Concepts you have come the right way. Refer to the Class 11 Geography NCERT Solutions and take your preparation to next level. Clarify your queries during preparation and assess your preparation level by practicing from the NCERT Solutions on a regular basis. Access the quick links available below for 11th Std Chapterwise Geography NCERT Solutions and download them for free of cost.

Chapterwise 11th Class Geography NCERT Solutions PDF

The NCERT Solutions for Class 11 Geography are prepared adhering to the latest Exam Pattern and CBSE Guidelines. You can use them for quick revision and get a good hold of the concepts. 11th Std NCERT Solutions on geography provides a deeper insight into the concepts and helps you score well in the board exams. Get to know the Chapterwise Class 11th Geography NCERT Solutions through the direct links available.

NCERT Class 11 Fundamentals of Physical Geography Solutions

NCERT Solutions for Class 11 Geography: Fundamentals of Physical Geography

NCERT Solutions for Class 11 Geography

NCERT Class 11 India Physical Environment Solutions

NCERT Solutions for Class 11 Geography: India Physical Environment

Practical Work in Geography Class 11 Solutions

NCERT Solutions for Class 11 Geography: Practical Work in Geography

Importance of Class 11 NCERT Solutions for Geography

Here are some of the advantages of referring to NCERT Solutions for Class 11 Geography. They are in the following fashion

  • Improve your subject knowledge and problem-solving skills by practicing from NCERT Solutions.
  • You can learn properly the concepts without any hassle by making use of these Solutions.
  • Step by Step Solutions provided helps you have a deeper insight into concepts.
  • Access the quick links and download the detailed solutions for free of cost to ace up your preparation.
  • All the Solutions are given in simple and easy to understand language.

FAQs on NCERT Solutions for 11th Grade Geography

1. Where do I find NCERT Solutions for Class 11th Geography?

You can find NCERT Solutions for Class 11th Geography in detail on our page.

2. How to download 11th Class Geography NCERT Solutions PDF?

You just need to click on the quick links available to download them for 11th Class Geography NCERT Solutions.

3. Where do I get Chapterwise Class 11 NCERT Solutions on Geography?

You can get Chapterwise Class 11 NCERT Solutions on Geography on our page.

Summary

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The post Download Class 11 Geography NCERT Solutions | 11th Class Geography NCERT Solutions PDF appeared first on Learn CBSE.


Mixed Numbers Calculator

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Use the free online tool ie, Mixed Number Calculator to make your mixed fractions or numbers calculations easy & quick. Simply enter the input whole number or fractions in the input field of the calculator and then click on the Calculate button to get the result in a mixed fraction or mixed number within seconds.

Mixed Numbers Calculator: Do you feel calculating the mixed numbers is tough? Not anymore with our handy Mixed numbers Calculator. A mixed number (also called mixed fraction) is a combination of a whole number and a proper fraction. By using this Online Mixed Numbers Calculator, you can effortlessly find all solutions for addition, subtraction, multiplication, and division of mixed numbers. Make use of this free mixed fractions calculator or mixed numbers calculator & make your calculations easy & learn the detailed concept behind it.

How to Add, Subtract, Multiply & Divide Mixed Numbers Easily?

Finding adding mixed numbers, subtracting mixed numbers, multiplying mixed numbers, dividing mixed numbers can be quite easy and simple by following the steps mentioned here below:

  • First and foremost, you have to take the given input whole numbers or proper fractions.
  • Now, check the question for what to find like addition, subtraction, multiplication, or division.
  • Before you start the calculating process, Convert the mixed numbers to improper fractions
  • Later, use the respective algebraic fractions formula to calculate the addition or subtraction or product or division mixed numbers.
  • Finally, reduce the fractions and simply into mixed fractions or numbers if possible.

Let’s see the closer look at how to solve the addition of mixed numbers from the given solved example and understand the concept behind it with a detailed explanation.

Example

Question: Add 2 1/4 and 1 2/6 and calculate the mixed numbers?

Solution:

Given inputs of mixed numbers are 2 1/4 and 1 2/6

At step-1, convert them to improper fractions

2 1/4 + 1 2/6 = 9/4 + 8/6

At step-2, Use the algebraic formula for the addition of fractions:

a/b + c/d = (axd)+(bxc) / bxd

9/4 + 8/6 = (9×6)+(4×8) / 4×6

= 54 + 32 / 24 = 86/24

At step-3, simply the fraction to mixed number if possible,

86/24 = 43/12 = 3 7/12.

Thus, 86/24 simplified to a mixed number or mixed fraction is 3 7/12.

FAQs on Mixed Numbers or Mixed Fractions Calculator

1. What is meant by mixed numbers?

In maths, A mixed number is a number consisting of a whole number and a proper fraction.


2. How to Simplify Mixed Fractions using a Calculator?

Simply enter the input numbers or fractions in the input field of the calculator and click the calculate button to get the mixed numbers or mixed fractions easily.


3. How do I add a whole number and a mixed fraction by hand?

To add a whole number and a mixed fraction manually, all you need to do is simply follow the above steps on how to find the addition of mixed numbers and make your calculation at a faster pace.


4. Where can I find the free online Mixed Numbers Calculator?

LearnCBSE.in gives the free Online Calculator for Mixed numbers or Mixed fractions and you can make use of this tool freely to calculate the addition, subtraction, multiplication, and division of mixed numbers.

The post Mixed Numbers Calculator appeared first on Learn CBSE.

Rational Numbers | Definition, Types, Properties, Standard Form of Rational Numbers

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In Maths, Rational Numbers sound similar to Fractions and they are expressed in the form of p/q where q is not equal to zero. Any fraction that has non zero denominators is called a Rational Number. Thus, we can say 0 also a rational number as we can express it in the form of 0/1, 0/2 0/3, etc. However, 1/0, 2/0 aren’t rational numbers as they give infinite values.

Continue reading further modules to learn completely about Rational Numbers. Get to know about Types of Rational Numbers, Difference Between Rational and Irrational Numbers, Solved Examples, and learn how to Identify Rational Numbers, etc. In order to represent Rational Numbers on a Number Line firstly change them into decimal values.

Definition of Rational Number

Rational Number in Mathematics is defined as any number that can be represented in the form of p/q where q ≠ 0. On the other hand, we can also say that any fraction fits into the category of Rational Numbers if bot p, q are integers and the denominator is not equal to zero.

How to Identify Rational Numbers?

You need to check the following conditions to know whether a number is rational or not. They are as follows

  • It should be represented in the form of p/q, where q ≠ 0.
  • Ratio p/q can be further simplified and expressed in the form of a decimal value.

The set of Rational Numerals include positive, negative numbers, and zero. It can be expressed as a Fraction.

Examples of Rational Numbers

p q p/q Rational
20 4 20/4 =5 Rational
2 2000 2/2000 = 0.001 Rational
100 10 100/10 = 10 Rational

Types of Rational Numbers

You can better understand the concept of sets by having a glance at the below diagram.

Rational Numbers

  • Real numbers (R) include All the rational numbers (Q).
  • Real numbers include the Integers (Z).
  • Integers involve Natural Numbers(N).
  • Every whole number is a rational number as every whole number can be expressed in terms of a fraction.

Standard Form of Rational Numbers

A Rational Number is said to be in its standard form if the common factors between divisor and dividend is only one and therefore the divisor is positive.

For Example, 12/24 is a rational number. It can be simplified further into 1/2. As the Common Factors between divisor and dividend is one the rational number 1/2 is said to be in its standard form.

Positive and Negative Rational Numbers

Positive Rational Numbers Negative Rational Numbers
If both the numerator and denominator are of the same signs. If numerator and denominator are of opposite signs.
All are greater than 0 All are less than 0
Example: 12/7, 9/10, and 3/4 are positive rational numbers Example: -2/13, 7/-11, and -1/4 are negative rational numbers

Arithmetic Operations on Rational Numbers

Let us discuss how to perform basic operations i.e. Arithmetic Operations on Rational Numbers. Consider p/q, s/t as two rational numbers.

Addition: Whenever we add two rational numbers p/q, s/t we need to make the denominator the same. Thus, we get (pt+qs)/qt.

Ex: 1/3+3/4 = (4+3)/12 = 7/12

Subtraction: When it comes to subtraction between rational numbers p/q, s/t we need to make the denominator the same and then subtract.

Ex: 1/2-4/3 = (3-8)/6 = -5/6

Multiplication: While Multiplying Rational Numbers p/q, s/t simply multiply the numerators and the denominators of the rational numbers respectively. On multiplying p/q with s/t then we get (p*s)/(q*t)

Ex: 1/3*4/2=4/6

Division: Division of p/q & s/t is represented as (p/q)÷(s/t) = pt/qs

Ex: 1/4÷4/3 =1*3/4*4 = 3/16

Properties of Rational Numbers

  • If we add a zero to a Rational Number you will get the Rational Number Itself.
  • Addition, Subtraction, Multiplication of a Rational Number yields in a Rational Number.
  • Rational Number remains the same on multiplying or dividing both the numerator and denominator with the same factor.

There are few other properties of rational numbers and they are given as under

  • Closure Property
  • Commutative Property
  • Associative Property
  • Distributive Property
  • Identity Property
  • Inverse Property

Representation of Rational Numbers on a Number Line

Number Line is a straight line diagram on which each and every point corresponds to a real number. As Rational Numbers are Real Numbers they have a specific location on the number line.

Rational Numbers Vs Irrational Numbers

There is a difference between Rational Numbers and Irrational Numbers. Fractions with non zero denominators are called Rational Numbers. All the numbers that are not Rational are Called Irrational Numbers. Rational Numbers can be Positive, Negative, or Zero. To specify a negative Rational Number negative sign is placed in front of the numerator.

When it comes to Irrational Numbers you can’t write them as simple fractions but can represent them with a decimal. You will endless non-repeating digits after the decimal point.

Pi (π) = 3.142857…

√2 = 1.414213…

Solved Examples

Example 1.

Identify whether Mixed Fraction 1 3/4 is a Rational Number or Not?

Solution: The Simplest Form of Mixed Number 1 3/4 is 7/4

Numerator = 7 which is an integer

Denominator = 4 which is an integer and not equal to 0.

Thus, 7/4 is a Rational Number.

Example 2.

Determine whether the given numbers are rational or irrational?

(a) 1.45 (b) 0.001 (c) 0.15 (d) 0.9 (d) √3

Solution:

Given Numbers are in Decimal Format and to find out whether they are rational or not we need to change them into fraction format i.e. p/q. If the denominator is non zero then the number is rational or else irrational.

Decimal Number Fraction Rational Number
1.45 29/20 Yes
0.001 1/1000 Yes
0.15 3/20 Yes
0.9 9/10 Yes
√ 3 1.732… No

FAQs on Rational Numbers

1. How to Identify a Rational Number?

If the Number is expressed in the form of p/q where p, q are integers and q is non zero then it called a Rational Number.

2. Is 5 a Rational Number?

Yes, 5 is a Rational Number as it can be expressed in the form of 5/1.

3. What do we get on adding zero to a Rational Number?

On Adding Zero to a Rational Number, you will get the Same Rational Number.

4. What is the difference between Rational and Irrational Numbers?

Rational Numbers are terminating decimals whereas Irrational Numbers are Non-Terminating Decimals.

 

The post Rational Numbers | Definition, Types, Properties, Standard Form of Rational Numbers appeared first on Learn CBSE.

Rational Numbers Class 8 Extra Questions Maths Chapter 1

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If you are looking for Extra questions for class 8 maths Rational Numbers, You have reached the correct page. You can also use these extra questions like Rational numbers class 8 worksheets with answers.

Rational Numbers Class 8 Extra Questions Maths Chapter 1

Extra Questions for Class 8 Maths Chapter 1 Rational Numbers

Rational Numbers Class 8 Extra Questions Very Short Answer Type

Question 1.
Pick up the rational numbers from the following numbers.
\(\frac { 6 }{ 7 }\), \(\frac { -1 }{ 2 }\), 0, \(\frac { 1 }{ 0 }\), \(\frac { 100 }{ 0 }\)
Solution:
Since rational numbers are in the form of \(\frac { a }{ b }\) where b ≠ 0.
Only \(\frac { 6 }{ 7 }\), \(\frac { -1 }{ 2 }\) and 0 are the rational numbers.

Question 2.
Find the reciprocal of the following rational numbers:
(a) \(\frac { -3 }{ 4 }\)
(b) 0
(c) \(\frac { 6 }{ 11 }\)
(d) \(\frac { 5 }{ -9 }\)
Solution:
(a) Reciprocal of \(\frac { -3 }{ 4 }\) is \(\frac { -4 }{ 3 }\)
(b) Reciprocal of 0, i.e. \(\frac { 1 }{ 0 }\) is not defined.
(c) Reciprocal of \(\frac { 6 }{ 11 }\) is \(\frac { 11 }{ 6 }\)
(d) Reciprocal of \(\frac { 5 }{ -9 }\) = \(\frac { -9 }{ 5 }\)

Question 3.
Write two such rational numbers whose multiplicative inverse is same as they are.
Solution:
Reciprocal of 1 = \(\frac { 1 }{ 1 }\) = 1
Reciprocal of -1 = \(\frac { 1 }{ -1 }\) = -1
Hence, the required rational numbers are -1 and 1.

Question 4.
What properties, the following expressions show?
(i) \(\frac { 2 }{ 3 } +\frac { 4 }{ 5 } =\frac { 4 }{ 5 } +\frac { 2 }{ 3 }\)
(ii) \(\frac { 1 }{ 3 } \times \frac { 2 }{ 3 } =\frac { 2 }{ 3 } \times \frac { 1 }{ 3 }\)
Solution:
(i) \(\frac { 2 }{ 3 } +\frac { 4 }{ 5 } =\frac { 4 }{ 5 } +\frac { 2 }{ 3 }\) shows the commutative property of addition of rational numbers.
(ii) \(\frac { 1 }{ 3 } \times \frac { 2 }{ 3 } =\frac { 2 }{ 3 } \times \frac { 1 }{ 3 }\) shows the commutative property of multiplication of rational numbers.

Question 5.
What is the multiplicative identity of rational numbers?
Solution:
1 is the multiplicating identity of rational numbers.

Question 6.
What is the additive identity of rational numbers?
Solution:
0 is the additive identity of rational numbers.

Question 7.
If a = \(\frac { 1 }{ 2 }\), b = \(\frac { 3 }{ 4 }\), verify the following:
(i) a × b = b × a
(ii) a + b = b + a
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q7
Extra Questions for Class 8 Maths Rational Numbers Q7.1

Question 8.
Multiply \(\frac { 5 }{ 8 }\) by the reciprocal of \(\frac { -3 }{ 8 }\)
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q8

Question 9.
Find a rational number between \(\frac { 1 }{ 2 }\) and \(\frac { 1 }{ 3 }\).
Solution:
Rational number between
Extra Questions for Class 8 Maths Rational Numbers Q9

Question 10.
Write the additive inverse of the following:
(a) \(\frac { -6 }{ 7 }\)
(b) \(\frac { 101 }{ 213 }\)
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q10

Question 11.
Write any 5 rational numbers between \(\frac { -5 }{ 6 }\) and \(\frac { 7 }{ 8 }\). (NCERT Exemplar)
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q11

Question 12.
Identify the rational number which is different from the other three : \(\frac { 2 }{ 3 }\), \(\frac { -4 }{ 5 }\), \(\frac { 1 }{ 2 }\), \(\frac { 1 }{ 3 }\). Explain your reasoning.
Solution:
\(\frac { -4 }{ 5 }\) is the rational number which is different from the other three, as it lies on the left side of zero while others lie on the right side of zero on the number line.

Rational Numbers Class 8 Extra Questions Short Answer Type

Question 13.
Calculate the following:
Extra Questions for Class 8 Maths Rational Numbers Q13
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q13.1
Extra Questions for Class 8 Maths Rational Numbers Q13.2

Question 14.
Represent the following rational numbers on number lines.
(a) \(\frac { -2 }{ 3 }\)
(b) \(\frac { 3 }{ 4 }\)
(c) \(\frac { 3 }{ 2 }\)
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q14

Question 15.
Find 7 rational numbers between \(\frac { 1 }{ 3 }\) and \(\frac { 1 }{ 2 }\).
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q15

Question 16.
Show that:
Extra Questions for Class 8 Maths Rational Numbers Q16
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q16.1

Question 17.
If x = \(\frac { 1 }{ 2 }\), y = \(\frac { -2 }{ 3 }\) and z = \(\frac { 1 }{ 4 }\), verify that x × (y × z) = (x × y) × z.
Solution:
We have x = \(\frac { 1 }{ 2 }\), y = \(\frac { -2 }{ 3 }\) and z = \(\frac { 1 }{ 4 }\)
LHS = x × (y × z)
Extra Questions for Class 8 Maths Rational Numbers Q17

Question 18.
If the cost of 4\(\frac { 1 }{ 2 }\) litres of milk is ₹89\(\frac { 1 }{ 2 }\), find the cost of 1 litre of milk.
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q18

Question 19.
The product of two rational numbers is \(\frac { 15 }{ 56 }\). If one of the numbers is \(\frac { -5 }{ 48 }\), find the other.
Solution:
Product of two rational numbers = \(\frac { 15 }{ 56 }\)
One number = \(\frac { -5 }{ 48 }\)
Other number = Product ÷ First number
Extra Questions for Class 8 Maths Rational Numbers Q19
Hence, the other number = \(\frac { -18 }{ 7 }\)

Question 20.
Let O, P and Z represent the numbers 0, 3 and -5 respectively on the number line. Points Q, R and S are between O and P such that OQ = QR = RS = SP. (NCERT Exemplar)
What are the rational numbers represented by the points Q, R and S. Next choose a point T between Z and 0 so that ZT = TO. Which rational number does T represent?
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q20
As OQ = QR = RS = SP and OQ + QR + RS + SP = OP
therefore Q, R and S divide OP into four equal parts.
Extra Questions for Class 8 Maths Rational Numbers Q20.1

Question 21.
Let a, b, c be the three rational numbers where a = \(\frac { 2 }{ 3 }\), b = \(\frac { 4 }{ 5 }\) and c = \(\frac { -5 }{ 6 }\) (NCERT Exemplar)
Verify:
(i) a + (b + c) = (a + b) + c (Associative property of addition)
(ii) a × (b × c) – (a × b) × c (Associative property of multiplication)
Solution:
Extra Questions for Class 8 Maths Rational Numbers Q21
Extra Questions for Class 8 Maths Rational Numbers Q21.1

Rational Numbers Class 8 Extra Questions Higher Order Thinking Skills (HOTS)

Question 22.
Rajni had a certain amount of money in her purse. She spent ₹ 10\(\frac { 1 }{ 4 }\) in the school canteen, bought a gift worth ₹ 25\(\frac { 3 }{ 4 }\) and gave ₹ 16\(\frac { 1 }{ 2 }\) to her friend. How much she have to begin with?
Solution:
Amount given to school canteen = ₹ 10\(\frac { 1 }{ 4 }\)
Amount given to buy gift = ₹ 25\(\frac { 3 }{ 4 }\)
Amount given to her friend = ₹ 16\(\frac { 1 }{ 2 }\)
To begin with Rajni had
Extra Questions for Class 8 Maths Rational Numbers Q22

Question 23.
One-third of a group of people are men. If the number of women is 200 more than the men, find the total number of people.
Solution:
Number of men in the group = \(\frac { 1 }{ 3 }\) of the group
Number of women = 1 – \(\frac { 1 }{ 3 }\) = \(\frac { 2 }{ 3 }\)
Difference between the number of men and women = \(\frac { 2 }{ 3 }\) – \(\frac { 1 }{ 3 }\) = \(\frac { 1 }{ 3 }\)
If difference is \(\frac { 1 }{ 3 }\), then total number of people = 1
If difference is 200, then total number of people
= 200 ÷ \(\frac { 1 }{ 3 }\)
= 200 × 3 = 600
Hence, the total number of people = 600

Question 24.
Fill in the blanks:
(a) Numbers of rational numbers between two rational numbers is ……….
Extra Questions for Class 8 Maths Rational Numbers Q24
Solution:
(a) Countless
(b) \(\frac { 6 }{ 11 }\)
(c) \(\frac { -3 }{ 2 }\)
(d) \(\frac { 3 }{ 5 }\)
(e) Commutative
(f) associative
(g) equivalent
(h) \(\frac { 3 }{ 11 }\)

Extra Questions for Class 8 Maths Rational Numbers 01
Maths Extra Questions for Class 8 Rationa Numbers
NCERT Solutions for Class 8 Maths Rational Numbers Extra Questions
Rational Numbers Extra Questions for Class 8 Maths 1
Extra Questions for Class 8 Maths Rational Numbers 05
Extra Questions for Class 8 Maths Rational Numbers 06
Extra Questions for Class 8 Maths Rational Numbers 07
Extra Questions for Class 8 Maths Rational Numbers 08
Extra Questions for Class 8 Maths Rational Numbers 09
Extra Questions for Class 8 Maths Rational Numbers 10
Maths Extra Questions for Class 8 Rationa Numbers 2
Maths Extra Questions for Class 8 Rationa Numbers 3
Maths Extra Questions for Class 8 Rationa Numbers 4
Extra Questions for Class 8 Maths Rational Numbers 09
Extra Questions for Class 8 Maths Rational Numbers 10
Maths Extra Questions for Class 8 Rationa Numbers 5
Maths Extra Questions for Class 8 Rationa Numbers 6
Maths Extra Questions for Class 8 Rationa Numbers 7
Extra Questions for Class 8 Maths Rational Numbers Q 19

Extra Questions for Class 8 Maths

NCERT Solutions for Class 8 Maths

The post Rational Numbers Class 8 Extra Questions Maths Chapter 1 appeared first on Learn CBSE.

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

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NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Exercise 1.1

Ex 1.1 Class 8 Maths Question 1.
Using appropriate properties find:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q1
Solution:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q1.1

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q1.2

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q1.3

Ex 1.1 Class 8 Maths Question 2.
Write the additive inverse of each of the following:
(i) \(\frac { 2 }{ 8 }\)
(ii) \(\frac { -5 }{ 9 }\)
(iii) \(\frac { -6 }{ -5 }\)
(iv) \(\frac { 2 }{ -9 }\)
(v) \(\frac { 19 }{ -6 }\)
Solution:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q2

Ex 1.1 Class 8 Maths Question 3.
Verify that -(-x) = x for
(i) x = \(\frac { 11 }{ 5 }\)
(ii) x = \(\frac { -13 }{ 17 }\)
Solution:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q3

Ex 1.1 Class 8 Maths Question 4.
Find the multiplicative inverse of the following:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q4
Solution:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q4.1

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q4.2

Ex 1.1 Class 8 Maths Question 5.
Name the property under multiplication used in each of the following:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q5

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q5.1
Solution:
(i) Commutative property of multiplication
(ii) Commutative property of multiplication
(iii) Multiplicative inverse property

Ex 1.1 Class 8 Maths Question 6.
Multiply \(\frac { 6 }{ 13 }\) by the reciprocal of \(\frac { -7 }{ 16 }\).
Solution:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q6

Ex 1.1 Class 8 Maths Question 7.
Tell what property allows you to compute
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q7
Solution:
Since a × (b × c) = (a × b) × c shows the associative property of multiplications.
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q7.1

Ex 1.1 Class 8 Maths Question 8.
Is \(\frac { 8 }{ 9 }\) the multiplicative inverse of -1\(\frac { 1 }{ 8 }\)? Why or Why not?
Solution:
Here -1\(\frac { 1 }{ 8 }\) = \(\frac { -9 }{ 8 }\).
Since multiplicative inverse of \(\frac { 8 }{ 9 }\) is \(\frac { 9 }{ 8 }\) but not \(\frac { -9 }{ 8 }\)
\(\frac { 8 }{ 9 }\) is not the multiplicative inverse of -1\(\frac { 1 }{ 8 }\)

Ex 1.1 Class 8 Maths Question 9.
If 0.3 the multiplicative inverse of 3\(\frac { 1 }{ 3 }\)? Why or why not?
Solution:
NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 Q9
Multiplicative inverse of 0.3 or \(\frac { 3 }{ 10 }\) is \(\frac { 10 }{ 3 }\).
Thus, 0.3 is the multiplicative inverse of 3\(\frac { 1 }{ 3 }\).

Ex 1.1 Class 8 Maths Question 10.
Write:
(i) The rational number that does not have a reciprocal.
(ii) The rational numbers that are equal to their reciprocals.
(iii) The rational number that is equal to its negative.
Solution:
(i) 0 is the rational number which does not have its reciprocal
[∵ \(\frac { 1 }{ 0 }\) is not defined]
(ii) Reciprocal of 1 = \(\frac { 1 }{ 1 }\) = 1
Reciprocal of -1 = \(\frac { 1 }{ -1 }\) = -1
Thus, 1 and -1 are the required rational numbers.
(iii) 0 is the rational number which is equal to its negative.

Ex 1.1 Class 8 Maths Question 11.
Fill in the blanks.
(i) Zero has ……….. reciprocal.
(ii) The numbers ……….. and ……….. are their own reciprocals.
(iii) The reciprocal of -5 is ………
(iv) Reciprocal of \(\frac { 1 }{ x }\), where x ≠ 0 is ……….
(v) The product of two rational numbers is always a …………
(vi) The reciprocal of a positive rational number is ……….
Solution:
(i) no
(ii) -1 and 1
(iii) \(\frac { -1 }{ 5 }\)
(iv) x
(v) rational number
(vi) positive

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-1

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-1.1

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-2

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-2.1

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-2.2

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-2.3

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-3

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-3.1

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-4

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-4.1

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-4.2

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-4.3

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-5

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-6

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-7

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-8

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-9

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers Ex 1.1 q-10

More CBSE Class 8 Study Material

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NCERT Exemplar Class 8 Maths Chapter 1 Rational Numbers

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NCERT Exemplar Class 8 Maths Chapter 1 Rational Numbers are part of NCERT Exemplar Class 8 Maths. Here we have given NCERT Exemplar Class 8 Maths Chapter 1 Rational Numbers.

NCERT  Exemplar Class 8 Maths Chapter 1 Rational Numbers

Multiple Choice Questions
Question . 1 A number which can be expressed as \(\frac { p }{ q }\) , where p and q are integers and \( q\neq 0\) is
(a) natural number (b) whole number
(c) integer (d) rational number
Solution. (d) A number which can be expressed as \(\frac { p }{ q }\), where p and q are integers and \( q\neq 0\) is a rational number.

Question . 2 A number of the form \(\frac { p }{ q }\) is said to be a rational number, if
(a) p, q are integers (b) p, q are integers and \( q\neq 0\)
(c) p, q are integers and \( p\neq 0\) (d) p, q are integers and \( p\neq 0\), also \( q\neq 0\)
Solution. (b) A number of the form \(\frac { p }{ q }\) is said to be a rational number, if p and q are integers and

Question . 3 The numerical expression \( \frac { 3 }{ 8 } +\frac { (-5) }{ 7 } =\frac { -19 }{ 56 }\) shows that
(a)rational numbers are closed under addition
(b) rational numbers are not closed under addition
(c) rational numbers are closed under multiplication
(d) addition of rational numbers is not commutative
Solution. (b) We have \( \frac { 3 }{ 8 } +\frac { (-5) }{ 7 } =\frac { -19 }{ 56 }\)
Show that rational numbers are closed under addition.
[\(\frac { 3 }{ 8 }\) and\( \frac { -5 }{ 7 }\) are rational numbers and their addition is \(\frac { -19 }{ 56 }\) which is also a rational number]
Note The sum of any two rational numbers is always a rational number.

Question . 4 Which of the following is not true?
(a) rational numbers are closed under addition
(b) rational numbers are closed under subtraction
(c) rational numbers are closed under multiplication
(d) rational numbers are closed under division
Solution. (d) Rational numbers are not closed under division.
As, 1 and 0 are the rational numbers but \( \frac { 1 }{ 0 }\) is not defined.

Question . 5 \(-\frac { 3 }{ 8 } +\frac { 1 }{ 7 } =\frac { 1 }{ 7 } +[\frac { -3 }{ 8 } ]\) is an example to show that
(a) addition of rational numbers is commutative
(b) rational numbers are closed under addition
(c) addition of rational numbers is associative
(d) rational numbers are distributive under addition
Solution.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-1
Clearly, a + b = b + a
So, addition is communication for rational numbers

Question . 6 Which of the following expressions shows that rational numbers are associative under multiplication.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-2
Solution.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-3
So, a x (b  x c) = (a x b) x c
Hence, the given expression shows that rational numbers are associative under multiplication.

Question . 7 Zero (0) is
(a) the identity for addition of rational numbers
(b) the identity for subtraction of rational numbers
(c) the identity for multiplication of rational numbers
(d) the identity for  division of rational numbers
Solution . (a) Zero (0) is the identity for addition of rational numbers.
That means,
If a is a rational number.
Then, a+0=0+a = a
Note Zero (0) is also the additive identity for integers and whole number as well.

Question . 8 One (1) is
(a) the identity for addition of rational numbers
(b) the identity for subtraction of rational numbers
(c) the identity for multiplication of rational numbers
(d) the identity for division of rational numbers
Solution . (c) One (1) is the identity for multiplication of rational numbers.
That means,
If a is  a rational number.
Then, a-1 = 1-a = a
Note One (1) is the multiplication identity for integers and whole number also.

Question . 9 The additive inverse  of \(\frac { -7 }{ 19 }\) is
ncert-exemplar-problems-class-8-mathematics-rational-numbers-4
Solution . (b) We know that, if a and b are the additive inverse of each other, then a + b = 0
Suppose, x is the additive inverse of \(\frac { -7 }{ 19 }\)
ncert-exemplar-problems-class-8-mathematics-rational-numbers-5

Question . 10 Multiplicative inverse of a negative rational number is
(a) a positive rational number (b) a negative rational number
(c) 0 (d) 1
Solution. (b) We know that, the product of two rational numbers is 1, taken they are multiplication inverse of each other, e.g.
Suppose, p is negative rational number, i.e.
\(\frac { 1 }{ p }\) is the  multiplicative inverse of-p, then, -p x \(\frac { 1 }{ -p }\) = 1
Hence, multiplicative inverse of a negative rational number is a negative rational number.

Question. 11 If x + 0 = 0 + x = x, which is rational number, then 0 is called
(a) identity for addition of rational numbers
(b) additive inverse of  x
(c) multiplicative inverse of x
(d) reciprocal of x
Solution . (a) We know that, the sum of any rational number and zero (0) is the rational number itself.
Now, x + 0 = 0+ x= x, which is a rational number, then 0 is called identity for addition of rational numbers.

Question . 12 To get the product 1,  we should multiply \( \frac { 8 }{ 21 }\) by
ncert-exemplar-problems-class-8-mathematics-rational-numbers-6
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-7

Question . 13 – (-x) is same as
(a)-x (b)x (c)\(\frac { 1 }{ x }\) (d)\(\frac { -1 }{ x }\)
Solution . (b) -(-x) = x
Negative of negative rational number is equal to positive rational number.

Question . 14 The multiplicative inverse of \( -1\frac { 1 }{ 7 }\) is
ncert-exemplar-problems-class-8-mathematics-rational-numbers-8
Solution . (d) We know that, if the product of two rational numbers is 1, then they are multiplicative inverse of each other.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-9

Question . 15 If x is any rational number, then x + 0 is equal to
(a)x (b) 0 (c)-x (d) Not defined
Solution . (a) If x is any rational number, then x + 0 = x [0 is the additive identity]

Question . 16  The reciprocal of 1 is ;
(a) 1 (b) -1 (c) 0 (d) Not defined
Solution . (a) The reciprocal of 1 is the number itself.

Question . 17  The reciprocal of -1 is
(a) 1 (b) -1 (c) 0 (d) Not defined
Solution . (b) The reciprocal of -1 is the number itself.

Question . 18 The reciprocal of 0 is
(a) 1 (b) -1 (c) 0 (d) Not defined
Solution . (d) The reciprocal of 0 is not defined.

Question . 19 The reciprocal of any rational number \(\frac { p }{ q }\) , where p and q are integers and \( q\neq 0\) is
(a)\(\frac { p }{ q }\) (b)1 (c)0 (d)\(\frac { q }{ p }\)
Solution . (d) The reciprocal of any rational number \(\frac { p }{ q }\), where p and q are integers and \( q\neq 0\) is \(\frac { q }{ p }\)

Question . 20 If y is the reciprocal of rational number x, then the reciprocal of y will be
(a)x (b) y (c) \(\frac { x }{ y }\) (d) \(\frac { y }{ x }\)
Solution . (a) If y be the reciprocal of rational number x, i.e. y = \(\frac { 1 }{ x }\) or x = \(\frac { 1 }{ y }\).
Hence, the reciprocal of y will be x.

Question .21
ncert-exemplar-problems-class-8-mathematics-rational-numbers-1
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-2

Question . 22 Which of the following is an example of distributive property of multiplication over addition for rational numbers.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-3
Solution . We know that, the distributive property of multiplication over addition for rational numbers can be expressed as a x (b + c) = ab + ac, where a, b and c are rational numbers.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-4
is the example of distributive property of multiplication over addition for rational numbers.

Question . 23 Between two given rational numbers, we can find
(a) one and only one rational number
(b) only two rational numbers
(c) only ten rational numbers
(d) infinitely many rational numbers
Solution . (d) We can find infinite many rational numbers between two given rational numbers.

Question .24
ncert-exemplar-problems-class-8-mathematics-rational-numbers-5
(a) Between x and y
(b) Less than x and y both
(c) Greater than x and y both
(d) Less than x but greater than y
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-6

Question . 25 Which of the following statements is always true?
ncert-exemplar-problems-class-8-mathematics-rational-numbers-7
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-8

Fill in the Blanks
In questions 26 to 47, fill in the blanks to make the statements true.
Question . 26 The equivalent of \( \frac { 5 }{ 7 } \) whose numerator is 45, is —.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-9

Question . 27 The equivalent rational number of \( \frac { 7 }{ 9} \) , whose denominator is 45 is——————.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-10

Question . 28 Between the numbers \(\frac {15 }{ 20} \) and \(\frac { 35 }{ 40} \), the greater number is———————-.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-11

Question . 29 The reciprocal of a positive rational number is—————.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-12

Question . 30 The reciprocal of a negative rational number is——————–.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-13

Question. 31 Zero has————reciprocal.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-14

Question. 32 The numbers ————–and————–are their own reciprocal.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-15

Question . 33 If y is the reciprocal of x, then the reciprocal of \({ y }^{ 2 }\) in terms of x will be—————-.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-16

Question . 34
ncert-exemplar-problems-class-8-mathematics-rational-numbers-17
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-18

Question . 35
ncert-exemplar-problems-class-8-mathematics-rational-numbers-19
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-20

Question . 36 The negative of 1 is—————-.
Solution . -1 The negative of 1 is -1.

Question . 37
ncert-exemplar-problems-class-8-mathematics-rational-numbers-21
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-22

Question . 38 \(\frac { -5 }{ 7 }\) is———————than -3.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-23

Question . 39 There are rational numbers between any two rational numbers.
Solution . Infinite
There are infinite rational numbers between any two rational numbers.

Question . 40 The rational numbers \(\frac { 1 }{ 3 }\) and \(\frac { -1 }{ 3 }\) are on the sides of zero on the number line.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-24

Question . 41 The negative of a negative rational number is always a—————-rational
number.
Solution. positive
Let x be a positive rational number.
Then, – x be a negative rational number.
Now, negative of a negative rational number = – (- x)= x =positive rational number.

Question . 42 Rational numbers can be added or multiplied in any————-.
Solution . order
Rational numbers can be added or multiplied in any order and this concept is known as commutative property.

Question . 43 The reciprocal of \(\frac { -5 }{ 7 }\) is——————.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-25

Question . 44 The multiplicative inverse of \(\frac { 4 }{ 3 }\) is———–.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-26

Question . 45 The rational number 10.11 in the form \(\frac { p }{ q }\) is ——–.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-27

Question .46
ncert-exemplar-problems-class-8-mathematics-rational-numbers-1
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-2

Question . 47 The two rational numbers lying between -2 and -5 with denominator as 1 are———–and————.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-3

True/False
In questions 48 to 99, state whether the given statements are True or False.
Question . 48 If \(\frac { x }{ y }\)is a rational number, then y is always a whole number.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-4

Question . 49 If \(\frac { p }{ q }\) is a rational number, then p Cannot be equal to zero.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-5

Question . 50 If \(\frac { r }{ s }\) is a rational number, then s cannot be equal to zero.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-6

Question . 51 \(\frac { 5 }{ 6 }\) lies between \(\frac { 2 }{ 3 }\) and 1.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-7

Question . 52 \(\frac { 5 }{ 10 }\) lies between \(\frac { 1 }{ 2 }\) and 1.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-8

Question . 53 \(\frac { 5 }{ 10 }\) lies between -3 and 4.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-9

Question . 54 \(\frac { 9 }{ 6 }\) lies between 1 and 2.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-10

Question . 55 If \(a\neq 0\) the multiplicative inverse of \(\frac { a }{ b }\) is \(\frac { b }{ a }\) .
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-11

Question . 56 The multiplicative inverse of \(\frac { -3 }{ 5 }\) is \(\frac { 5 }{ 3 }\) .
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-12

Question . 57 The additive inverse of \(\frac { 1 }{ 2 }\) is -2.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-13

Question . 58
ncert-exemplar-problems-class-8-mathematics-rational-numbers-14
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-15

Question . 59 For every rational number x, x + 1 = x.
Solution . False
For every rational number , x + 0 = x

Question . 60
ncert-exemplar-problems-class-8-mathematics-rational-numbers-16
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-17

Question . 61 The reciprocal of a non-zero rational number \(\frac { q }{ p }\) is the rational number \(\frac { q }{ p }\).
Solution .False
The reciprocal of a non-zero rational number \(\frac { q }{ p }\) . is the rational number\(\frac { p }{ q }\)

Question . 62 If x + y = 0, then -y is known as the negative of x, where x and y are rational numbers.
Solution . False
If x and y are rational numbers and x+ y = 0.
Then, y is known as the negative of x.

Question . 63 The negative of the negative of any rational number is the number itself.
Solution . True
Let x be a positive rational number. Then, -x be a negative rational number.
Now, negative of negative rational number = -(-x)= x = Positive rational number

Question . 64 The negative of 0 does not exist.
Solution . True
Since, zero is neither a positive integer nor a negative integer.

Question . 65 The negative of 1 is 1 itself.
Solution . False
The negative of 1 is -1.

Question . 66 For all rational numbers x and y,x-y = y- x
Solution . False
For all rational numbers x and y,
x-y = -(y-x)

Question . 67 For all rational numbers x and y, x x y = y x x.
Solution . True
For all rational numbers x and y,
x x y= y x x

Question . 68 For every rational number x, x x 0 = x.
Solution . False
For every rational number x,
x x 0 = 0

Question . 69 For every rational numbers x, y and z, x + (y x z) = (x + y) x (x + z)
Solution . False
For all rational numbers a, b and c.
a(b + c) = ab+ ac

Question . 70 For all rational numbers a, b and c,a (b + c) = ab + bc.
Solution . False
As, addition is not distributive over multiplication.

Question . 71 1 is the only number which is its own reciprocal.
Solution . False
Reciprocal of 1 is 1 and reciprocal of -1 is -1.

Question . 72 -1 is not the reciprocal of any rational number.
Solution . False
-1 is the reciprocal of -1.

Question . 73 For any rational number x, x + (-1) = – x.
Solution . False
For every rational number x,
x x (-1) = – x

Question . 74 For rational numbers x and y, if x < y, then x – y is a positive rational number.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-18

Question . 75 If x and y are negative rational numbers, then so is x + y.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-19

Question . 76 Between any two rational numbers there are exactly ten rational . numbers.
Solution . False
There are infinite rational numbers between any two rational numbers.

Question . 77 Rational numbers are closed under addition and multiplication but not under subtraction.
Solution . False
Rational numbers are closed under addition, subtraction and multiplication.

Question . 78 Subtraction of rational number is commutative.
Solution . False
Subtraction of rational numbers is not commutative, i.e. \(a-b\neq b-a\)
where, a and b are rational numbers.

Question . 79 \(-\frac { 3 }{ 4 }\) is smaller than -2 .
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-20

Question . 80 0 is a rational number.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-21

Question . 81 All positive rational numbers lie between 0 and 1000.
Solution . False
Infinite positive rational numbers lie on the right side of 0 on the number line.

Question. 82 The population of India in 2004-05 is a rational number.
Solution. True
The population of India in 2004-05 is a rational number.

Question. 83 There are countless rational numbers between \(-\frac { 5 }{ 6 }\) and \(-\frac { 8 }{ 9 }\).
Solution.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-22

Question. 84
ncert-exemplar-problems-class-8-mathematics-rational-numbers-23
Solution.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-24

Question. 85 The rational number \(-\frac { 57 }{ 23 }\) lies to the left of zero on the number line.
Solution. False
Since,\(-\frac { 57 }{ 23 }\) is a positive rational number.
So, it lies on the right of zero on the number line.

Question .86 The rational number \(-\frac { 7 }{ -4 }\) lies to the right of zero on the number line.
Solution . False
Since, \(-\frac { 7 }{ -4 }\) is a negative rational number.
So, it lies on the left of zero on the number line.

Question .87 The rational number \(-\frac { -8 }{ -3 }\) lies neither to the right nor to the left of zero on the number line.
Solution . False
\(-\frac { -8 }{ -3 }\) = \(-\frac { 8 }{ 3 }\) is a positive rational number.
Hence, it lies on the right of zero on the number line.

Question . 88 The rational numbers \(-\frac { 1 }{ 2 }\) and -1 are on the opposite sides of zero on the number tine.
Solution . True
Since, positive rational number and negative rational number are on the opposite sides of zero on the number line.’
Hence, \(-\frac { 1 }{ 2 }\) and -1 are on the opposite sides of zero on the number line.

Question . 89 Every fraction is a rational number.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-1

Question . 90 Every integer is a rational number.
Solution . True
Every integer is a rational number whose denominator remain 1.

Question . 91 The rational numbers can be represented on the number line.
Solution . True
The rational numbers can be represented on the number line.

Question . 92 The negative of a negative rational number is a positive rational number.
Solution . True
Let be a positive rational number.
Then, – x be the negative rational number.
Hence, negative of negative rational number = – (- x)= x = Positive rational number

Question . 93 If x and y are two rational numbers such that x > y, then x – y is always a positive rational number.
Solution . True
If x and y are two rational numbers such that x > y.
Then, there are three possible cases, i.e.
Case I x and y both are positive. ‘
Case II x is positive and y is negative.
Case III x and y both are negative.
In all three cases, x – y is always a positive rational number.

Question . 94 0 is the smallest rational number.
Solution . False
As the smallest rational number does not exist.

Question .95 Every whole number is an integer.
Solution .True
W (whole numbers) = {0,1,2, 3 }
Z (integers) = {…- 3, – 2, -1, 0,1,2, 3,…}
Every whole number is an integer, but every integer is not a whole number.

Question .96 Every whole number is a rational number.
Solution .True
Every whole number can be written in the form of \(-\frac { p }{ q }\), where p, q are integers and \(q\neq 0\).
Hence, every whole number is a rational number.

Question . 97 0 is whole number but it is not a rational number.
Solution . False
0 is a whole number and also a rational number.

Question . 98 The rational numbers \(-\frac { 1 }{ 2 }\)and \(-\frac -{ 5 }{ 2}\) are on the opposite sides of zero on the number line.
Solution . True
Positive rational number and negative rational number remain on opposite sides of zero on the number line.

Question .99 Rational numbers can be added (or multiplied) in any order
ncert-exemplar-problems-class-8-mathematics-rational-numbers-2
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-3

Question . 100 Solve the following, select the rational numbers from the list which are also the integers.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-4
Solution . From the given rational numbers, the numbers whose denominator is 1 and the numbers whose numerator is the multiple of denominator are the integers.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-5

Question . 101 Select those which can be written as a rational number with denominator 4 in their lowest form
ncert-exemplar-problems-class-8-mathematics-rational-numbers-6
Solution . From the given rational numbers, the number with denominator 4 in their lowest form is \(-\frac { 5 }{ -4 }\)

Question . 102 Using suitable rearrangement and find the sum
ncert-exemplar-problems-class-8-mathematics-rational-numbers-7
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-8

Question . 103 Verify – (-x) = x for
ncert-exemplar-problems-class-8-mathematics-rational-numbers-9
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-10

Question . 104 Give one example each to show that the rational numbers are closed under addition, subtraction and multiplication. Are rational numbers closed under division? Give two examples in support of your answer.
Solution . We know that, rational numbers are closed under addition, subtraction and multiplication. We can understand this from the following examples.
Rational numbers are closed under addition
ncert-exemplar-problems-class-8-mathematics-rational-numbers-11
ncert-exemplar-problems-class-8-mathematics-rational-numbers-12
But rational are not closed under division. If zero is excluded from the collection of rational numbers, then we can say that rational numbers are closed under division.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-13

Question . 105 Verify the property x + y = y + x of rational numbers by taking
ncert-exemplar-problems-class-8-mathematics-rational-numbers-14
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-15
ncert-exemplar-problems-class-8-mathematics-rational-numbers-16

Question . 106 Simplify each of the following by using suitable property. Also, name the property.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-17
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-18

Question . 107
ncert-exemplar-problems-class-8-mathematics-rational-numbers-19
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-20

Question . 108 Verify the property x x y = y x x of rational numbers by using
ncert-exemplar-problems-class-8-mathematics-rational-numbers-21
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-22
ncert-exemplar-problems-class-8-mathematics-rational-numbers-23

Question . 109 Verify the property x x (y x z)=i.(x x y) x z of rational numbers by using
ncert-exemplar-problems-class-8-mathematics-rational-numbers-24
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-25
ncert-exemplar-problems-class-8-mathematics-rational-numbers-26

Question . 110 Verify the property x x (y + z) = x x y + x x z of rational numbers by taking
ncert-exemplar-problems-class-8-mathematics-rational-numbers-27
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-28
ncert-exemplar-problems-class-8-mathematics-rational-numbers-29

Question . 111 Use the distributivity of multiplication of rational numbers over addition to simplify
ncert-exemplar-problems-class-8-mathematics-rational-numbers-1
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-2

Question. 112 Simplify
ncert-exemplar-problems-class-8-mathematics-rational-numbers-3
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-4

Question. 113 Identify the rational number that does not belong with the other three. Explain your reasoning
ncert-exemplar-problems-class-8-mathematics-rational-numbers-5
Solution . does not belong with the other three. Since,\( \frac { -7 }{ 3 }\) as it is smaller than -1 whereas rest of the numbers are greater than -1.

Question. 114 The cost of \( \frac { 19 }{ 4 }\) m of wire is Rs \( \frac { 171 }{ 2 }\) Find the cost of one metre of the wire.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-6

Question. 115 A train travels \( \frac { 1445 }{ 2 }\) km in \( \frac { 17 }{ 2 }\) h. Find the speed of the train in km/h.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-7

Question. 116 If 16 shirts of equal size can be made out of 24m of cloth, how much cloth is needed for making one shirt?
Solution . If 16 shirts are to be made by cloth of 24 m
Then, 1 shirt is to be made by cloth of = \( \frac { 24 }{ 16 }\) m = \( \frac { 3 }{ 2 }\) m = 1.5 m
Hence, 1.5 m cloth is needed for making one shirt.

Question. 117 \( \frac { 7 }{ 11 }\) of all the money in Hamid’s bank account is Rs 77000. How much money does Hamid have in his bank account?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-8

Question. 118 A 117\( \frac { 1 }{ 3 }\) m long rope is cut into equal pieces measuring 7\( \frac { 1 }{ 3 }\) m each. How many such small pieces are these?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-9

Question. 119 \( \frac { 1 }{ 6 }\) of the class students are above average, \( \frac { 1 }{ 4 }\) are average and rest are below average. If there are 48 students in all, how many students are below average in the class?
Solution . Number of above average students = \( \frac { 1 }{ 6 }\) of the class students
Number of average students = \( \frac { 1 }{ 4 }\)of the class students
ncert-exemplar-problems-class-8-mathematics-rational-numbers-10

Question. 120 \( \frac { 2 }{ 5 }\)of total number of students of a school come by car while \( \frac { 1 }{ 4 }\) of students come by bus to school. All the other students walk to school of which \( \frac { 1 }{ 3 }\)walk on their own and the rest are escorted by their parents. If 224 students come to school walking on their own, how many students study in that school?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-11

Question. 121 Huma, Hubna and Seema received a total of Rs 2016 as monthly allowance from their mother such that Seema gets \( \frac { 1 }{ 2 }\) of what Hubna gets and Huma gets 1\( \frac { 2 }{ 3 }\) times Seema’s share. How much money do the three sisters get individually?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-1
ncert-exemplar-problems-class-8-mathematics-rational-numbers-2

Question. 122 A mother and her two daughters got a room constructed for Rs 62000. The elder daughter contributes \( \frac { 3 }{ 8 }\)of her mother’s contribution while the younger daughter contributes \( \frac { 1 }{ 2 }\)of her mother’s share. How much do the three contribute individually?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-3
ncert-exemplar-problems-class-8-mathematics-rational-numbers-4

Question. 123 Tell which property allows you to compare
ncert-exemplar-problems-class-8-mathematics-rational-numbers-5
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-6

Question. 124 Name the property used in each of the following:
ncert-exemplar-problems-class-8-mathematics-rational-numbers-7
Solution . (i) Commutative property over multiplication
(ii) Distributive property over addition
(iii) Associative property over addition
(iv) Existence of additive identity
(v) Existence of multiplicative identity

Question. 125 Find the multiplicative inverse of(i)-1\( \frac { 1 }{ 8 }\) (ii)3 \( \frac { 1 }{ 3 }\)
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-8

Question. 126 Arrange the numbers is \( \frac { 1 }{ 4 }\) , \( \frac { 13 }{ 16 }\) , \( \frac { 5 }{ 8 }\)in the descending order.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-9

Question. 127 The product of two rational numbers is \( \frac { -14 }{ 27 }\) If one of the numbers be \( \frac { 7 }{ 9 }\) find the other.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-10

Question. 128 By what numbers should we multiply \( \frac { -15 }{ 20 }\) so that the product may be \( \frac { -5 }{ 7 }\) ?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-11

Question. 129 By what number should we multiply \( \frac { -8 }{ 13 }\) so that the product may be 24?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-12

Question. 130 The product of two rational numbers is -7. If one of the number is -5, find the other?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-13

Question. 131 Can you find a rational number whose multiplicative inverse is -1?
Solution . No, we cannot find a rational number whose multiplicative inverse is -1.

Question. 132 Find five rational numbers between 0 and 1.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-14

Question. 133 Find the two rational numbers whose absolute value is \( \frac { 1 }{ 5 }\) .
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-15

Question. 134 From a rope 40 m long, pieces of equal size are cut. If the length of one piece is \( \frac { 10 }{ 3 }\) m, find the number of such pieces.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-16

Question. 135 5 \( \frac { 1 }{ 2 }\) m long rope is cut into 12 equal pieces. What is the length of each piece?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-17

Question. 136 Write the following rational numbers in the descending order.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-18
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-19

Question. 137 Find
ncert-exemplar-problems-class-8-mathematics-rational-numbers-20
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-21

Question. 138 On a winter day the temperature at a place in Himachal Pradesh was -16°C. Convert it in degree Fahrenheit (°F) by using the formula
ncert-exemplar-problems-class-8-mathematics-rational-numbers-22
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-23

Question. 139 Find the sum of additive inverse and multiplicative inverse of 7.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-24

Question. 140 Find the product of additive inverse and multiplicative inverse of \( -\frac { 1 }{ 3 }\).
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-25

Question. 141 The diagram shows the wingspans of different species of birds. Use the diagram to answer the question given below
ncert-exemplar-problems-class-8-mathematics-rational-numbers-48
(a) How much longer is the wingspan of an Albatross than the wingspan of a Sea gull?
(b) How much longer is the wingspan of a Golden eagle than the wingspan of a Blue jay?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-26
ncert-exemplar-problems-class-8-mathematics-rational-numbers-27

Question. 142 Shalini has to cut out circles of diameter 1 \( -\frac { 1 }{ 4 }\) cm from an aluminium strip of dimensions 8\( -\frac { 3 }{ 4 }\) cm by 1 \( -\frac { 1 }{ 4 }\) cm. How many full circles can Shalini cut? Also, calculate the wastage of the aluminium strip.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-28
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-29
ncert-exemplar-problems-class-8-mathematics-rational-numbers-30

Question. 143 One fruit salad recipe requires \( -\frac { 1 }{ 2 }\) cup of sugar. Another recipe for the same fruit salad requires 2 tablespoons of sugar. If 1 tablespoon is 1 equivalent to \( -\frac { 1 }{ 16 }\) cup, how much more sugar does the first recipe require?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-31

Question. 144 Four friends had a competition to see how far could they hop on one foot. The table given shows the distance covered by each.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-32
(a) How farther did Soni hop than Nancy?
(b) What is the total distance covered by Seema and Megha?
(c) Who walked farther, Nancy or Megha?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-33

Question. 145 The table given below shows the distances, in kilo metres, between four villages of a state. To find the distance between two villages, locate the square, where the row for one village and the column for the other village intersect.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-34
(a) Compare the distance between Himgaon and Rawalpur to Sonapur and Ramgarh?
(b) If you drove from Himgaon to Sonapur and then from Sonapur to Rawalpur, how far would you drive?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-35
ncert-exemplar-problems-class-8-mathematics-rational-numbers-36

Question. 146 The table shows the portion of some common materials that are recycled.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-37
(a) Is the rational number expressing the amount of paper recycled more than \( -\frac { 1 }{ 2 }\) or less than \( -\frac { 1 }{ 2 }\) ?
(b) Which items have a Recycled amount less than \( -\frac { 1 }{ 2 }\) ?
(c) Is the quantity of aluminium fans recycled more (or less) than half of the quantity of aluminium cans?
(d) Arrange the rate of recycling the materials from the greatest to the smallest.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-38
ncert-exemplar-problems-class-8-mathematics-rational-numbers-39

Question. 147 The overall width in cm of several wide-screen televisions are 97.28 cm,98\( -\frac { 4 }{ 9 }\) cm, 98\( -\frac { 1 }{ 25 }\) cm and 97.94 cm. Express these numbers as rational numbers in the form \( -\frac { p }{ q }\) and arrange the widths in ascending order.
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-40
ncert-exemplar-problems-class-8-mathematics-rational-numbers-41

Question. 148 Roller coaster at an amusement park is \( -\frac { 2 }{ 3 }\) m high. If a new roller coaster is built that is \( -\frac { 3 }{ 5 }\) times the height of the existing coaster, what will be the height of the new roller coaster?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-42

Question. 149 Here is a table which gives the information about the total rainfall for several months compared to the average monthly rains of a town. Write each decimal in the form of rational number \( -\frac { p }{ q }\) .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-43
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-44
ncert-exemplar-problems-class-8-mathematics-rational-numbers-45

Question. 150 The average life expectancies of males for several states are shown in the table. Express each decimal in the form \( -\frac { p }{ q }\) and arrange the states from the least to the greatest male life expectancy.
State-wise data are included below; more indicators can be found in the “FACTFILE” section on the homepage for each state.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-46
Source Registrar General of India (2003) SRS Based Abridged Lefe Tables. SRS Analytical Studies, Report No. 3 of 2003, New Delhi: Registrar General of India.
The data are for the 1995-99 period; states subsequently divided are therefore included in their pre-partition states (Chhatisgarh in MP, Uttaranchal in UP and Jharkhand in Bihar)
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-47
Arrangement of the states from the least to the greatest male life expectancy, Haryana, Tamil Nadu, West Bengal, Karnataka, Gujarat, AndhraPradesh, Bihar, Rajasthan, Uttar Pradesh, Orissa, Assam, Madhya Pradesh.

Question. 151 A skirt that is 35 \( \frac { 7 }{ 8 }\) cm long has a hem of 3\( \frac { 1 }{ 8 }\) cm. How tong will the skirt . be if the hem is let down?
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-1

Question. 152 Manavi and Kuber each receives an equal allowance. The table shows the fraction of their allowance each deposits into his/her saving account and the fraction each spends at the mall. If allowance of each is Rs 1260, find the amount left with each.
ncert-exemplar-problems-class-8-mathematics-rational-numbers-2
Solution .
ncert-exemplar-problems-class-8-mathematics-rational-numbers-3
ncert-exemplar-problems-class-8-mathematics-rational-numbers-4

NCERT Exemplar Solutions

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Rational Numbers Class 8 Notes Maths Chapter 1

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CBSE Class 8 Maths Notes Chapter 1 Rational Numbers Pdf free download is part of Class 8 Maths Notes for Quick Revision. Here we have given NCERT Class 8 Maths Notes Chapter 1 Rational Numbers.

CBSE Class 8 Maths Notes Chapter 1 Rational Numbers

Rational Number: A number is called rational if we can write the number in the form of \(\frac { p }{ q }\), where p and q are integers and q ≠ 0
i.e., 1 = \(\frac { 1 }{ 1 }\), 2 = \(\frac { 2 }{ 1 }\), 0 = \(\frac { 0 }{ 1 }\) and \(\frac { 5 }{ 8 }\) , \(\frac { -3 }{ 14 }\) , \(\frac { 7 }{ -15 }\) are all rational numbers.

Between two rational numbers x and y, there exists a rational number \(\frac { x+y }{ 2 }\)

We can find countless rational numbers between two rational numbers.

\(\frac { -x }{ y }\) is called the additive inverse of \(\frac { x }{ y }\) and vice-versa.

\(\frac { y }{ x }\) is called the multiplicative inverse or reciprocal of \(\frac { x }{ y }\).

Rational number 0 is the additive identity for all rational numbers because a number does not change when 0 is added to it.

Rational number 1 is the multiplicative identity for all rational numbers because on multiplying a rational number with 1, its value does not change.

Rational numbers can be represented on a number line.

Properties on Rational Numbers
(i) Closure Property
Rational numbers are closed under :

Addition
Rational Numbers Class 8 Notes Maths Chapter 1 1
which is a rational number.

Subtraction
Rational Numbers Class 8 Notes Maths Chapter 1 2
are rational numbers.

Multiplication:
Rational Numbers Class 8 Notes Maths Chapter 1 3
Rational Numbers Class 8 Notes Maths Chapter 1 4
are rational numbers.

Rational numbers are closed under addition subtraction and multiplication.

Division : eq. \(\frac { -3 }{ 5 } \div \frac { 2 }{ 3 } =\frac { -9 }{ 10 }\), which is also a rational number. For any rational number a, a ÷ 0 is not defined. So, rational number are not closed under division.
However, if we exclude zero then the rational numbers are closed under division.

(ii) Commutativity:
Addition: Two rational numbers can be added in any order, i.e., commutativity holds for rational numbers under addition, i.e., for any two rational number a and b, a + b = b + a.
Rational Numbers Class 8 Notes Maths Chapter 1 5

Subtraction:
Rational Numbers Class 8 Notes Maths Chapter 1 6
Hence, subtraction is not associative for rational numbers.

(iii) Multiplication: Multiplication is commutative for rational numbers. In general, a × b = b × a, for any two rational numbers a and b.
Rational Numbers Class 8 Notes Maths Chapter 1 7

Division:
Rational Numbers Class 8 Notes Maths Chapter 1 8
Hence, division is not Cumulative for rational numbers.

(iii) Associativity:
Addition:
Rational Numbers Class 8 Notes Maths Chapter 1 9
So, addition is associative for rational numbers, i.e., for any three rational numbers a, b and c, a + (b + c) = (a + b) + c.

Subtraction:
Rational Numbers Class 8 Notes Maths Chapter 1 10
Hence, subtraction is not associative for rational numbers.

Multiplication:
Rational Numbers Class 8 Notes Maths Chapter 1 11
So, multiplication is associative for rational number, i.e., for any three rational numbers a, b and c, a × (b × c) = (a × b) × c.

Division:
Rational Numbers Class 8 Notes Maths Chapter 1 12
Hence, the division is not associative for rational numbers.

Distributivity of multiplication over addition for rational number : For all rational numbers a, b and c, a(b + c) = ab + ac
Rational Numbers Class 8 Notes Maths Chapter 1 13

Distributivity of multiplication over subtraction for rational number: For any three rational numbers a, b and c, a (b – c) = ab – ac
eg. Let \(\frac { 1 }{ 2 }\), \(\frac { -2 }{ 5 }\) an \(\frac { -3 }{ 10 }\) are any three rational numbers, then
Rational Numbers Class 8 Notes Maths Chapter 1 14

The numbers 1, 2, 3, 4, ………. are called natural numbers.

If we add 0 to the collection of natural numbers, what we get is called the collection of whole numbers.

Thus, 0, 1, 2, 3, 4, are whole numbers.

Natural numbers are also known as positive integers. If we put a negative sign before each positive integer, we get negative integers. Thus, -1, -2,
-3, -4, ………. are negative integers. A number of the form \(\frac { p }{ q }\), where p and q are integers and q ≠ 0 is called a rational number. All the above types of numbers are needed to solve various types of simple algebraic equations.

Properties of Rational Numbers
The list of properties of rational numbers can be given as follows:

  • Closure
  • Commutativity
  • Associativity
  • The role of zero (0)
  • The role of 1
  • Negative of a number
  • Reciprocal
  • Distributivity of multiplication over addition for rational numbers.

Distributivity of Multiplication Over Addition for Rational Numbers
For all rational numbers a, b and c,
a(b + c) = ab + ac
a(b – c) = ab – ac.

Representation of Rational Numbers on the Number Line
Rational Numbers Class 8 Notes Maths Chapter 1 15

  • We draw a line.
  • We mark a point O on it and name it 0. Mark a point to the right of 0. Name it 1. The distance between these two points is called unit distance.
  • Mark a point to the right of 1 at unit distance and name it 2.
  • Proceeding in this manner, we can mark points 3, 4, 5,
  • Similarly we can mark – 1, – 2, – 3, – 4, – 5, ……… to the left of 0. This line is called the number line.
  • This line extends indefinitely on both sides.

The positive rational numbers are represented by points on the number line to the right of O whereas the negative rational numbers are represented by points on the number line to the left of O.

Any rational number can be represented on this line. The denominator of the rational number indicates the number of equal parts into which the first unit has been divided whereas the numerator indicates as to how many of these parts are to be taken into consideration.

Rational Numbers Between Two Rational Numbers
We can find infinitely many rational numbers between any two given rational numbers. We can take the help of the idea of the mean for this purpose.

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Rational Numbers Class 7 Extra Questions Maths Chapter 9

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Rational Numbers Class 7 Extra Questions Maths Chapter 9

Extra Questions for Class 7 Maths Chapter 9 Rational Numbers

Rational Numbers Class 7 Extra Questions Very Short Answer Type

Question 1.
Find three rational numbers equivalent to each of the following rational numbers.
(i) \(\frac { -2 }{ 5 }\)
(ii) \(\frac { 3 }{ 7 }\)
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q1

Question 2.
Reduce the following rational numbers in standard form.
(i) \(\frac { 35 }{ -15 }\)
(ii) \(\frac { -36 }{ -216 }\)
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q2

Question 3.
Represent \(\frac { 3 }{ 2 }\) and \(\frac { -3 }{ 4 }\) on number lines.
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q3

Question 4.
Which of the following rational numbers is greater?
(i) \(\frac { 3 }{ 4 }\), \(\frac { 1 }{ 2 }\)
(ii) \(\frac { -3 }{ 2 }\), \(\frac { -3 }{ 4 }\)
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q4

Question 5.
Find the sum of
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q5
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q5.1

Question 6.
Subtract:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q6
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q6.1
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q6.2

Question 7.
Find the product:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q7
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q7.1

Rational Numbers Class 7 Extra Questions Short Answer Type

Question 8.
If the product of two rational numbers is \(\frac { -9 }{ 16 }\) and one of them is \(\frac { -4 }{ 15 }\), find the other number.
Solution:
Let the required rational number be x.
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q8

Question 9.
Arrange the following rational numbers in ascending order.
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q9
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q9.1

Question 10.
Insert five rational numbers between:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q10
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q10.1
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q10.2

Rational Numbers Class 7 Extra Questions Long Answer Type

Question 11.
Evaluate the following:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q11
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q11.1
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q11.2

Question 12.
Subtract the sum of \(\frac { -5 }{ 6 }\) and -1\(\frac { 3 }{ 5 }\) from the sum 2\(\frac { 2 }{ 3 }\) and -6\(\frac { 2 }{ 5 }\).
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q12

Question 13.
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q13
Solution:
We have
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q13.1

Question 14.
Divide the sum of -2\(\frac { 15 }{ 17 }\) and 3\(\frac { 5 }{ 34 }\) by their difference.
Solution:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q14

Question 15.
During a festival sale, the cost of an object is ₹ 870 on which 20% is off. The same object is available at other shops for ₹ 975 with a discount of 6\(\frac { 2 }{ 3 }\) %. Which is a better deal and by how much?
Solution:
The cost of the object = ₹ 870
Discount = 20% of ₹ 870 = \(\frac { 20 }{ 100 }\) × 870 = ₹ 174
Selling price = ₹ 870 – ₹ 174 = ₹ 696
The same object is available at other shop = ₹ 975
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q15
Selling price = ₹ 975 – ₹ 65 = ₹ 910
Since ₹ 910 > ₹ 696
Hence, deal at first shop is better and by ₹ 910 – ₹ 696 = ₹ 214

Rational Numbers Class 7 Extra Questions Higher Order Thinking Skills (HOTS) Type

Question 16.
Simplify:
21.5 ÷ 5 – \(\frac { 1 }{ 5 }\) of (20.5 – 5.5) + 0.5 × 8.5
Solution:
Using BODMAS rule, we have
21.5 ÷ 5 – \(\frac { 1 }{ 5 }\) of (20.5 – 5.5) + 0.5 × 8.5
= 21.5 ÷ 5 – \(\frac { 1 }{ 5 }\) of 15 + 0.5 × 8.5
= 21.5 × \(\frac { 1 }{ 5 }\) – \(\frac { 1 }{ 5 }\) × 15 + 0.5 × 8.5
= 4.3 – 3 + 4.25
= 4.3 + 4.25 – 3
= 8.55 – 3
= 5.55

Question 17.
Simplify:
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q17
Solution:
Using BODMAS rule, we have
2.3 – [1.89 – {3.6 – (2.7 – 0.77)}]
= 2.3 – [1.89 – {3.6 – 1.93}]
= 2.3 – [1.89 – 1.67]
= 2.3 – 0.22
= 2.08

Question 18.
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q18
Solution:
Using BODMAS rule, we have
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q18.1
Rational Numbers Class 7 Extra Questions Maths Chapter 9 Q18.2

Extra Questions for Class 7 Maths

NCERT Solutions for Class 7 Maths

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Rational Numbers Class 7 Notes Maths Chapter 9

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CBSE Class 7 Maths Notes Chapter 9 Rational Numbers Pdf free download is part of Class 7 Maths Notes for Quick Revision. Here we have given NCERT Class 7 Maths Notes Chapter 9 Rational Numbers.

CBSE Class 7 Maths Notes Chapter 9 Rational Numbers

The numbers used for counting objects are called counting numbers or natural numbers. These are: 1, 2, 3, 4, ………

If we include 0 to natural numbers, we get whole numbers. Thus, 0, 1, 2, 3, 4, ….. are whole numbers.

If we include the negatives of natural numbers to the whole numbers, we get integers. Thus, ….., -3, -2, -1, 0, 1, 2, 3, …… are integers.
We see that we have extended the number system from natural numbers to whole numbers and then from whole numbers to integers.

The numbers of the form \(\frac { numerator }{ denominator }\) where the numerator is either 0 or a positive integer and the denominator is a positive integer, are called fractions.

We compare two fractions by finding their equivalent forms. We have studied all the four basic operations of addition, subtraction, multiplication, and division on them. In this chapter, we shall further extend the number system by introducing rational numbers.

Need for Rational Numbers
There are many situations which involve fractional numbers. To include such numbers, we need to extend our number system by introducing rational numbers.

What are Rational Numbers?
A number of the form \(\frac { p }{ q }\) where p and q (≠0) are integers, is called a rational number.

Numerator and Denominator
In \(\frac { p }{ q }\), the integer p is the numerator, and the integer q (≠0) is the denominator.
Thus in \(\frac { -3 }{ 7 }\), the numerator is -3 and the denominator is 7.

Equivalent Rational Numbers
If we multiply the numerator and denominator of a rational number by the same non-zero integer, we obtain another rational number equivalent to the given rational number.

Positive and Negative Rational Numbers
A rational number whose numerator and denominator both are positive integers is called a positive rational number.
A rational number, whose numerator is a negative integer and denominator is a positive integer, is called a negative rational number. Similarly, if the numerator is positive integer and denominator is a negative integer; is also a negative rational number.

Rational Numbers on a Number Line
Positive rational numbers are marked on the right of 0 on the number line whereas negative rational numbers are marked on the left of 0 on the number line.
The method of representation is the same as the method of representation of fractions on the number line.

Rational Numbers in Standard Form
A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and the denominator have no common factor other than 1. Note that the negative sign occurs only in the numerator.
A rational number in standard form is said to be in its lowest form.

Reduction of a Rational Number to its Lowest Form
To reduce a rational number to its standard form (or lowest form), we divide its numerator and denominator by their HCF ignoring the negative sign, if any.
However, if there is a negative sign in the denominator, we divide by -HCF’.

Comparison of Rational Numbers
Two positive rational numbers can be compared exactly as we compare two fractions.
Two negative rational numbers can be compared by ignoring their negative signs and then reversing the order.
Comparison of a negative and a positive rational number is obvious as a negative rational number is always less than a positive rational number.

Rational Numbers Between Two Rational Numbers
There exist an unlimited number of rational numbers between any two rational numbers.

Operations on Rational Numbers
Addition
Addition of two rational numbers with same denominators: Two rational numbers with the same denominators can be added by adding their numerators, keeping the denominator same.

Addition of two rational numbers with different denominators: As in the case of fractions, we first find the LCM of the two denominators. Then we find the rational numbers equivalent to the given rational numbers with this LCM as the denominator. Now, we add the two rational numbers as in (A).

Additive Inverse
The additive inverse of the rational number \(\frac { p }{ q }\) is –\(\frac { p }{ q }\)

Subtraction
While subtracting two rational numbers, we add the additive inverse of the rational number to be subtracted to the other rational number.

Multiplication
Multiplication of a rational number by a positive integer:
While multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged.

Multiplication of rational number by a negative integer:
While multiplying a rational number by a negative integer, we multiply the numerator by that integer, keeping the denominator unchanged.

Multiplication of two rational numbers (none of which is an integer):
Based on the above observations,
So, as done in fractions we multiply two rational numbers as follows:

  • Step 1. Multiply the numerators of the two rational numbers.
  • Step 2. Multiply the denominators of the two rational numbers.
  • Step 3. Write the product as \(\frac { Result\quad of\quad Step1 }{ Result\quad of\quad Step2 }\)

Division
The reciprocal of the rational number \(\frac { p }{ q }\) is \(\frac { q }{ p }\)
To divide one rational number by other rational number, we multiply one rational number by the reciprocal of the other.

Product of Reciprocals
The product of a rational number with its reciprocal is always 1.

A rational number is defined as a number that can be expressed in the form \(\frac { p }{ q }\), where p and q are integers and q ≠ 0.
Rational Numbers Class 7 Notes Maths Chapter 9 1

Rational Numbers include integers and fractions.

In \(\frac { p }{ q }\), p is the numerator and q is the denominator.

Zero is a rational number. We can write \(\frac { 0 }{ 1 }\).

By multiplying the numerator and denominator of a rational number by the same non – zero integer, we obtained another rational number equivalent to the given rational number.
Rational Numbers Class 7 Notes Maths Chapter 9 2

If both the numerator and denominator are either positive or negative integers, then they are said to be a positive rational number.
Rational Numbers Class 7 Notes Maths Chapter 9 3

A rational number is said to be negative if its numerator and denominator are such that one of them is a positive integer and the other is a negative integer.
Rational Numbers Class 7 Notes Maths Chapter 9 4

Zero is neither positive nor negative rational numbers.
Representation of Rational Numbers on a Number line
Rational Numbers Class 7 Notes Maths Chapter 9 5

To reduce the rational number to its standard form, we divide its numerator and denominator by their HCF ignoring the negative sign.
Rational Numbers Class 7 Notes Maths Chapter 9 6

To compare two negative rational numbers, we compare them ignoring their negative signs and then reverse the order.
Rational Numbers Class 7 Notes Maths Chapter 9 7

We can find an unlimited number of rational numbers between any two rational numbers.
While adding rational numbers with same denominators, we add the numerators keeping the denominators same.
Rational Numbers Class 7 Notes Maths Chapter 9 8

While subtracting two rational numbers, we add the additive inverse of the rational, number that is being Subtracted, to the other rational number.
Rational Numbers Class 7 Notes Maths Chapter 9 9

While multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged.
Rational Numbers Class 7 Notes Maths Chapter 9 10

Product of reciprocals is always equal to 1.
Rational Numbers Class 7 Notes Maths Chapter 9 11

To divide one rational number by the other non – zero rational number, we multiply the rational number by the reciprocal of the other.
Rational Numbers Class 7 Notes Maths Chapter 9 12

We hope the given CBSE Class 7 Maths Notes Chapter 9 Rational Numbers Pdf free download will help you. If you have any query regarding NCERT Class 7 Maths Notes Chapter 9 Rational Numbers, drop a comment below and we will get back to you at the earliest.

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What are Rational Numbers?

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If you are willing to know What are Rational Numbers and general representation of it have a look at the further modules. The number that can be expressed in the form of a fraction a/b where a, b are integers and the denominator b is non zero is called a rational number.

You can also say that a Rational Number is a number that can be expressed as the quotient of two integers having the condition where the divisor is non zero.

Numerator and Denominator: If a/b is a rational number then integer a is the numerator and b is the denominator.

Examples: 3/2, 8/5, -14/3, -11/5 are all Rational Numbers as they have integers in numerators and denominators and denominators are non zero.

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Is Every Rational Number a Natural Number?

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Every Natural Number is a Rational Number but it’s not the same in the case of Rational Numbers. There is no such thing when it comes to a Rational Number it may or may not be a Natural Number.

We know that 1 = 1/1, 2 = 2/1, ….

In fact, we can say a natural number n can be expressed as n = n/1 which is nothing but the quotient of two integers. Therefore, every natural number is a Rational Number.

On the other hand, 5/2, 3/5, 2/7, 4/20, etc. are all Rational Numbers but aren’t natural numbers.

Therefore, every natural number is a Rational Number but a Rational Number need not be a Natural Number.

Determine Whether the Following Rational Numbers are Natural Numbers or Not

(i) 5/2

5/ 2 is not a natural number.

(ii) 8/2

8/2 is a natural number as on simplifying it we get the result as 4 which is a natural number.

(iii) -20/5

-20/5 isn’t a natural number as on further simplifying we get the result -4 which is an integer but not a natural number.

(iv) -6/-3

-6/-3 is a natural number as we will get the result 2 on simplification which is a natural number.

(v) 1/5

1/5 is not a natural number.

(vi) 0/3

0/3 is not a natural number since 0/3 =0 and 0 is not a natural number.

(vii) 5/5

5/5 is a natural number on simplifying to its lowest term we get 1/1 = 1 and 1 is a natural number.

(viii) 27/45

27/45 is not a natural number as we get 3/5 on reducing to its lowest term which is a rational number but not a natural number.

Thus, by looking after the instances above we can state that Not every Rational Number is a Natural Number.

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Is Zero a Rational Number?

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Know whether zero falls under Rational Numbers or not and the statements supporting it here. Yes, Zero is a Rational Number and you will have clarity on it by the end. As we can write the Integer 0 in any of the below forms.

For instance, 0/1, 0/-1, 0/2, 0/-2, 0/3, 0/-3, 0/4, 0/-4 …..

In other words, we can express as 0 = 0/b where b is a non zero integer.

Thus, you can write 0 as a/b = 0 where a is 0 and the denominator b is a non- zero integer.

Therefore, 0 is a Rational Number.

Examples

(i) 0/6

0/6 is a rational number as we have the denominator non- zero integer.

(ii) 0/-2

0/-2 is a rational number since -2 is an integer and is non zero.

(iii) 0/10

0/10 is a rational number since we have 12 in the denominator which is a non zero integer.

Thus, the above instances prove that 0 is a Rational Number.

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Is Every Rational Number an Integer?

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Every Integer is a Rational Number but a Rational Number need not be an Integer. Check out the statements, examples supporting whether or not All Rational Numbers are Integers.

We know 1 = 1/1, 2 = 2/1, 3 = 3/1 ……..

Also, -1 = -1/1, -2 = -2/1, -3 = -3/1 ……..

You can also express integer a in the form of a/1 which is also a Rational Number.

Hence, every integer is clearly a Rational Number.

Clearly, 5/2,-4/3, 3/7, etc. are all Rational Numbers but not Integers.

Therefore, every integer is a Rational Number but a Rational Number need not be an Integer. Check out the following sections and get a complete idea of the statement.

Determine whether the following Rational Numbers are Integers or not

(i) 3/5

3/5 is not an Integer and we can’t express it other than a fraction form or decimal value.

(ii) 6/3

6/3 is an integer. On simplifying 6/3 to its lowest form we get 6/3 = 2/1 which is an integer.

(iii) -3/-3

-3/-3 is an integer. On reducing -3/-3 to its reduced form we get -1/-1 =1 which is an integer.

(iv) -13/2

-13/2 is not an integer and we can’t express it other than a fraction form or decimal value.

(v) -36/9

-36/9 is an integer as we get the reduced form -36/9=-4 which is an integer.

(vi) 47/-9

47/-9 is not an integer and we can’t express it other than fraction form or decimal value.

(vii) -70/-20

-70/-20 is not an integer and we can’t express it other than fraction form or decimal value.

(viii) 1000/-10

1000/-10 is an integer as we get 1000/-10 = -100 on reducing to its lowest form and -100 is an integer.

From the above instances, we can conclude that Not Every Rational Number is an Integer.

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Is Every Rational Number a Fraction?

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Every Fraction is a Rational Number however a Rational Number need not be a Fraction. Refer to the entire article to know whether or not All Rational Numbers are Fractions.

Let us Consider a/b to be a fraction where a, b are natural numbers. We know every natural number is an integer thus a, b are integers too. Therefore the fraction a/b is the quotient of two integers given that b ≠ 0.

Thus, a/b is a Rational Number. We do have instances where a/b is a rational number but not a fraction. To help you we have taken an example.

4/-3 is a Rational Number but not a fraction as the denominator is not a natural number.

Mixed Fraction consisting of both Integer Part and Fractional Part can be expressed as an Improper Fraction, which is a quotient of two integers. Hence, we can say every Mixed Fraction is a Rational Number. Thus, Every Fraction is a Rational Number.

Determine whether the following rational numbers are fractions or not

(i) 2/3

2/3 is a Fraction as both the numerator 2 and denominator 3 are natural numbers.

(ii) 3/4

3/4 is a Fraction as both the numerator 3 and denominator 4 are natural numbers.

(iii) -6/-2

-6/-2 is not a fraction as the numerator -6 and denominator -2 are not natural numbers.

(iv) -15/9

-15/9 is not a fraction since the numerator -15 is not a natural number.

(v) 36/-4

36/-4 is not a fraction since the numerator -36 is not a natural number.

(vi) 45/1

45/1 is a Fraction since both the numerator 45 and denominator 1 are natural numbers.

(vii) 0/5

0/5 is not a reaction since the numerator 0 is not a natural number.

(viii) 2/10

2/10 is a Fraction as the numerator 2 and denominator 10 are natural numbers.

By referring to the above instances we can infer that Not Every Rational Number is a Fraction.

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Positive Rational Number

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In this article, you will learn about Positive Rational Numbers. Get to know about solved examples and explaining all about how come they are called Positive Rational Numbers. A Rational Number is said to be positive if both the numerator and denominator are either positive integers or negative integers. You can also say that a rational number is positive if both numerator and denominator are of the same sign.

1/6, 2/7, -9/-11, -5/-13, 8/12 are positive rationals, but 6/-5, -3/11, -8/7, 9/-23 are not positive rationals.

Is every natural number a positive rational number?

We know 1 = 1/1, 2 = 2/1, 3 = 3/1, 4 = 4/1 and ……..

Any natural number n can be written as n/1 where n, 1 are positive integers.

Therefore, every natural number is a positive rational number. Do remember Rational Number 0 is neither positive nor negative.

Determine whether the following rational numbers are positive or not

(i) -7/3

-7/ 3 is not a positive rational number as both the numerator and denominator are of opposite sign.

(ii) -9/-11

-9/-11 is a positive rational number as both the numerator and denominator are of the same sign.

(iii) 11/19

11/19 is a positive rational number since both numerator and denominator are positive integers.

(iv) 21/-7

21/-7 is not a positive rational number as both the numerator and denominator are of opposite sign.

(v) -105/7

-105/7 is not a positive rational number as both numerator and denominator are of opposite sign.

(vi) 25/31

25/31 is a positive rational number since both numerator and denominator are positive integers.

(vii) -6/5

-6/5 is not a positive rational number as both the numerator and denominator are of opposite sign.

(viii) 21/-25

21/-25 is not a positive rational number as both numerator and denominator are of opposite sign.

Thus, we can say that a rational number is positive if it has both numerator and denominator are of the same sign.

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Negative Rational Number

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Get a Complete Idea of Negative Rational Numbers from this article. You can see the conditions for Negative Rational Numbers along with a few examples.

A Rational Number is said to be negative if the numerator and denominator are of opposite sign i.e. any one of them is a positive integer and the other is a negative integer. You can also say that a Rational Number is Negative if the numerator and denominator are of opposite signs.

All the Rational Numbers -1/7, 4/-5, -25/11, 10/-19, -13/23 are negative. Rational Numbers -11/-14, 2/3, -3/-4, 1/2 are not negative.

Is every negative integer a negative rational number?

We know -1 = -1/1, -2 = -2/1, -3 = -3/1, -4 = -4/1 ……

We can express negative integer n in the form of n/1 where n is a negative integer and 1 is a positive integer.

Thus, every negative integer is a negative rational number. On the other hand, Rational Number 0 is neither positive nor negative.

Determine whether the following rational numbers are negative or not?

(i) 3/(-6)

3/(-6) is a negative rational since the denominator and numerator are having opposite signs.

(ii) (-1)/(-4)

(-1)/(-4) is not a negative rational as both the numerator and denominator are having the same sign.

(iii) 11/23

11/23 is not a negative rational since both the numerator and denominator are of the same sign.

(iv) 9/-14

9/-14 is a negative rational since both the numerator and denominator are of opposite signs.

(v) (-64)/(-8)

(-64)/(-8) is not a negative rational as both the numerator and denominator are of the same sign.

(vi) 20/24

20/24 is not a negative rational as you have both the numerator and denominator of the same sign.

(vii) (-13)/39

(-13)/39 is a negative rational since we have both the numerator and denominator of opposite signs.

(viii) (-31)/7

(-31)/7 is a negative rational since we have both the numerator and denominator of opposite signs.

Thus, from the above examples, we can say that a negative rational number is the one that has both the numerator and the denominator of the opposite sign.

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Equivalent Rational Numbers

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In this article of ours, you will learn how to find Equivalent Rational Numbers by multiplication and division. Get to see solved examples in the coming modules.

Equivalent Rational Numbers by Multiplication

Let suppose a/b is a rational number and m is a non-zero integer then (a*m)/(b*m) is a rational number equivalent to a/b.

For instance, 16/20, 40/50, -56/-70, -96/-120 are equivalent fractions and are equal to the rational number 4/5.

On multiplying the numerator and denominator of a fraction with the same integer the fraction value doesn’t change.

Example: Fractions 4/8 and 16/32 are equivalent because the numerator and denominator can be obtained by multiplying with each of them with 4.

Also, -4/5 = -4*(-1)/5*(-1) = -4*(-2)/5*(-2) = -4*(-3)/5*(-3) and so on ……

If the denominator of a rational number is a negative integer then by using the above-mentioned property we can convert it to positive by multiplying the numerator and denominator by -1.

Example: 7/-5 = 7*(-1)/-5*(-1) = -7/5

Equivalent Rational Numbers by Division

If a/b is a rational number and m is the common divisor of a, b then (a÷m)/ (b÷m) is a rational number equivalent to a/b.

Rational Numbers -24/-30, -28/-35, 40/50, 60/75 are equivalent to the rational numbers 4/5.

24/32 = (24÷8)/(32÷8) = 3/4

Solved Examples

1. Find the Two Rational Numbers Equivalent to 4/7?

Solution:

4/7 = (4*4)/(7*4) = 16/28

4/7 = (4*7)/(7*7) = 28/49

Thus, the two rational numbers equivalent to 4/7 are 16/28 and 28/49.

2. Determine the smallest equivalent rational number of 100/125?

Solution:

100/125 = (100÷5)/(125÷5) = 20/25 = (20÷5)/(25÷5) = 4/5

Thus, the Equivalent Rational Number of 100/125 is 4/5.

3. Write down the following rational numbers with a positive denominator 4/-9, 11/-22, -17/-3?

Solution:

4/-9 = 4*(-1)/-9*(-1) = -4/9

11/-22 = 11*(-1)/-22*(-1) = -11/22

-17/-3 = -17*(-1)/-3*(-1) = 17/3

Therefore Rational Numbers 4/-9, 11/-22, -17/-3 changed with a positive denominator are -4/9, -11/22, 17/3.

4. Express -4/7 as a Rational Number with the numerator

(i) -16 (ii) 24

Solution:

(i) In order to make -4 as a rational number having the numerator -16 we first need to find a number when multiplied by results in -16.

Clearly, such number is (-16 )÷ (-4) = 4

Multiplying both the numerator and denominator with 4 we get

-4/7 = (-4*4)/(7*4) = -16/28

(ii) In order to make -4 as a rational number having the numerator 24 we first need to find a number when multiplied by results in 24.

Clearly, such number is (24 )÷ (-4) = -6

Multiplying both the numerator and denominator with -6 we get

-4/7 = (-4*-6)/(7*-6) = 24/-42

All the examples listed above are for Equivalent Rational Numbers.

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Rational Numbers | Definition, Types, Properties, Standard Form of Rational Numbers

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In Maths, Rational Numbers sound similar to Fractions and they are expressed in the form of p/q where q is not equal to zero. Any fraction that has non zero denominators is called a Rational Number. Thus, we can say 0 also a rational number as we can express it in the form of 0/1, 0/2 0/3, etc. However, 1/0, 2/0 aren’t rational numbers as they give infinite values.

Continue reading further modules to learn completely about Rational Numbers. Get to know about Types of Rational Numbers, Difference Between Rational and Irrational Numbers, Solved Examples, and learn how to Identify Rational Numbers, etc. In order to represent Rational Numbers on a Number Line firstly change them into decimal values.

Definition of Rational Number

Rational Number in Mathematics is defined as any number that can be represented in the form of p/q where q ≠ 0. On the other hand, we can also say that any fraction fits into the category of Rational Numbers if bot p, q are integers and the denominator is not equal to zero.

How to Identify Rational Numbers?

You need to check the following conditions to know whether a number is rational or not. They are as follows

  • It should be represented in the form of p/q, where q ≠ 0.
  • Ratio p/q can be further simplified and expressed in the form of a decimal value.

The set of Rational Numerals include positive, negative numbers, and zero. It can be expressed as a Fraction.

Examples of Rational Numbers

p q p/q Rational
20 4 20/4 =5 Rational
2 2000 2/2000 = 0.001 Rational
100 10 100/10 = 10 Rational

Types of Rational Numbers

You can better understand the concept of sets by having a glance at the below diagram.

Rational Numbers

  • Real numbers (R) include All the rational numbers (Q).
  • Real numbers include the Integers (Z).
  • Integers involve Natural Numbers(N).
  • Every whole number is a rational number as every whole number can be expressed in terms of a fraction.

Standard Form of Rational Numbers

A Rational Number is said to be in its standard form if the common factors between divisor and dividend is only one and therefore the divisor is positive.

For Example, 12/24 is a rational number. It can be simplified further into 1/2. As the Common Factors between divisor and dividend is one the rational number 1/2 is said to be in its standard form.

Positive and Negative Rational Numbers

Positive Rational Numbers Negative Rational Numbers
If both the numerator and denominator are of the same signs. If numerator and denominator are of opposite signs.
All are greater than 0 All are less than 0
Example: 12/7, 9/10, and 3/4 are positive rational numbers Example: -2/13, 7/-11, and -1/4 are negative rational numbers

Arithmetic Operations on Rational Numbers

Let us discuss how to perform basic operations i.e. Arithmetic Operations on Rational Numbers. Consider p/q, s/t as two rational numbers.

Addition: Whenever we add two rational numbers p/q, s/t we need to make the denominator the same. Thus, we get (pt+qs)/qt.

Ex: 1/3+3/4 = (4+3)/12 = 7/12

Subtraction: When it comes to subtraction between rational numbers p/q, s/t we need to make the denominator the same and then subtract.

Ex: 1/2-4/3 = (3-8)/6 = -5/6

Multiplication: While Multiplying Rational Numbers p/q, s/t simply multiply the numerators and the denominators of the rational numbers respectively. On multiplying p/q with s/t then we get (p*s)/(q*t)

Ex: 1/3*4/2=4/6

Division: Division of p/q & s/t is represented as (p/q)÷(s/t) = pt/qs

Ex: 1/4÷4/3 =1*3/4*4 = 3/16

Properties of Rational Numbers

  • If we add a zero to a Rational Number you will get the Rational Number Itself.
  • Addition, Subtraction, Multiplication of a Rational Number yields in a Rational Number.
  • Rational Number remains the same on multiplying or dividing both the numerator and denominator with the same factor.

There are few other properties of rational numbers and they are given as under

  • Closure Property
  • Commutative Property
  • Associative Property
  • Distributive Property
  • Identity Property
  • Inverse Property

Representation of Rational Numbers on a Number Line

Number Line is a straight line diagram on which each and every point corresponds to a real number. As Rational Numbers are Real Numbers they have a specific location on the number line.

Rational Numbers Vs Irrational Numbers

There is a difference between Rational Numbers and Irrational Numbers. Fractions with non zero denominators are called Rational Numbers. All the numbers that are not Rational are Called Irrational Numbers. Rational Numbers can be Positive, Negative, or Zero. To specify a negative Rational Number negative sign is placed in front of the numerator.

When it comes to Irrational Numbers you can’t write them as simple fractions but can represent them with a decimal. You will endless non-repeating digits after the decimal point.

Pi (π) = 3.142857…

√2 = 1.414213…

Solved Examples

Example 1.

Identify whether Mixed Fraction 1 3/4 is a Rational Number or Not?

Solution: The Simplest Form of Mixed Number 1 3/4 is 7/4

Numerator = 7 which is an integer

Denominator = 4 which is an integer and not equal to 0.

Thus, 7/4 is a Rational Number.

Example 2.

Determine whether the given numbers are rational or irrational?

(a) 1.45 (b) 0.001 (c) 0.15 (d) 0.9 (d) √3

Solution:

Given Numbers are in Decimal Format and to find out whether they are rational or not we need to change them into fraction format i.e. p/q. If the denominator is non zero then the number is rational or else irrational.

Decimal Number Fraction Rational Number
1.45 29/20 Yes
0.001 1/1000 Yes
0.15 3/20 Yes
0.9 9/10 Yes
√ 3 1.732… No

FAQs on Rational Numbers

1. How to Identify a Rational Number?

If the Number is expressed in the form of p/q where p, q are integers and q is non zero then it called a Rational Number.

2. Is 5 a Rational Number?

Yes, 5 is a Rational Number as it can be expressed in the form of 5/1.

3. What do we get on adding zero to a Rational Number?

On Adding Zero to a Rational Number, you will get the Same Rational Number.

4. What is the difference between Rational and Irrational Numbers?

Rational Numbers are terminating decimals whereas Irrational Numbers are Non-Terminating Decimals.

 

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Equivalent Form of Rational Numbers

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Refer to the complete article and get an idea of how to find the Equivalent Form of Rational Numbers i.e. expressing the given rational number in different forms. Furthermore, you can witness the equivalent form of rational numbers having the common denominator.

1. Express Rational Number -48/60 with the denominator 5?

Solution:  

In order to express the Rational Number -48/60 with denominator 5 firstly figure out the number which gives 5 when 60 is divided by it.

It is clearly evident that such a number = (60÷5) = 12

Dividing the numerator and denominator of -48/60 by 12 we get

-48/60 = (-48÷12)/(60÷12)=-4/5

Therefore, expressing the Rational Number -48/60 with the denominator 5 is -4/5.

2. Fill in the blanks with the suitable number in the numerator 4/-7 = ...../-35 ...../-77

Solution:

4/-7 = 4*7/-5*7 = 28/-35

4/-7 = 4*11/-7*11 = 44/-77

3. Find an equivalent form of the rational numbers 3/9 and 4/6 having a common denominator?

Solution:

It is evident that the denominator is the LCM of 9, 6

LCM of 9, 6 is 18

Now, 18 ÷ 9 = 2, 18 ÷ 6 = 3

Therefore 3/9 = 3*2/9*2 = 6/18

4/6 = 4*3/6*3 = 12/18

Given rational numbers 3/9, 4/6 with common denominators are 6/18, 12/18.

4. Find an equivalent form of rational numbers 4/3, 7/5, 10/12 having a common denominator?

Solution:

We need to convert given rational numbers into equivalent rational numbers having the common denominator.

LCM of 3, 5, 12 is 60

60 ÷ 3 = 20, 60 ÷ 5 = 12, 60 ÷ 12 = 5

Therefore 4/3 = 4*20/3*20 = 80/60

7/5 = 7*12/5*12 = 84/60

10/12 = 10*5/12*5 = 50/60

Hence the given rational numbers with common denominator are 80/60, 84/60, 50/60.

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Rational Number in Different Forms

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Let us learn how to find Rational Numbers in Different Forms using the properties in expressing a rational number. Get to see solved examples along with clear-cut explanations and learn how to express rational number in different forms.

1. Express Rational Number -4/10 as a rational number with the denominator 20?

Solution:

In Order to express -4/10 as a rational number with denominator 20, we first need to figure out the number that when multiplied by 10 results in 20.

Clearly, the number is 20 ÷ 10 = 2

Multiplying the numerator and denominator -4/10 with 2 we get

-4/10 = -4*2/10*2 = -8/20

Expressing Rational Number -4/10 with the denominator 20 we get -8/20.

2.  Express – 2/10 as a rational number with denominator -30?

Solution:

In order to express the rational number -2/10 with denominator -30, we need to find out the number that when multiplied by 10 results in -30.

It is clear the number is 10 ÷ -30 = -3

Multiplying both the numerator and denominator -2/10 with -3 we get

-2/10 = (-2*-3)/(10*-3) = 6/-30

3. Express 52/-39 as a Rational Number having the denominator 3?

Solution:

To express 52/-39 as a rational number having the denominator 3 we need to find the number which gives 3 when -39 is divided by it.

Clearly, the number is -39 ÷ 3 = -13

52/-39 = (52÷ -13)/(-39 ÷ -13) = -4/3

Thus, 52/-39 as a Rational Number having the denominator 3 is -4/3.

4. Fill up the blanks with the appropriate number in the denominator 6/12 = 36/….=-66/…..?

We have 36 ÷ 6 = 6

Therefore, 6/ 12 = 6*6/12*6 = 36/72

Similarly, we have -66 ÷ 6 = -11

Therefore, 6/ 12 = 6*-11/12*-11

=-66/-132

Thus, 6/12 = 36/72 =-66/-132

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