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RS Aggarwal Solutions

RS Aggarwal Class 7 Solutions Chapter-4 Rational Numbers

RS Aggarwal Class 7 Solutions Chapter-4 Rational Numbers

Q: 1. What defines a rational number in its standard form as per the solutions for RS Aggarwal Class 7 Chapter 4?

A rational number is considered to be in its standard form when its denominator is a positive integer, and the numerator and denominator share no common factor other than 1. For instance, the rational number 3/5 is in standard form, whereas 6/10 is not because it can be simplified further to 3/5.

Q: 6. Why is finding the LCM a necessary first step when adding or subtracting rational numbers with unlike denominators?

Finding the Least Common Multiple (LCM) is essential because it allows us to convert the rational numbers into 'like' fractions—fractions that share a common denominator. We can only add or subtract parts of a whole accurately when the parts are of the same size. The LCM provides this common sizing, ensuring that the numerators we add or subtract represent portions of the same-sized unit.

Q: 8. How does the 'density property' of rational numbers explain what lies between any two rational numbers on a number line?

The density property states that between any two distinct rational numbers, you can always find another rational number. In fact, you can find infinitely many. For example, to find a number between 1/2 and 3/4, you can find their average: [(1/2) + (3/4)] / 2 = (5/4) / 2 = 5/8. This process can be repeated endlessly, proving that the rational numbers are densely packed along the number line.

Class 7 RS Aggarwal Chapter-4 Rational Numbers Solutions - Free PDF Download

Vedantu provides the solutions of RS Aggarwal Class 7 Math Chapter 4. The topic of Chapter 4 of Mathematics is a Rational Number. The Chapter Rational Number in Class 7 RS Aggarwal deals with the concepts involving the definitions and properties of the rational number. The way the solutions are provided by Vedantu is easily understandable. The stepwise solutions of every sum of the Chapter give the student an enhanced insight into the Chapter. You can download the PDF of the solutions of Chapter 4 for free from Vedantu.

Every NCERT Solution is provided to make the study simple and interesting on Vedantu. Vedantu is No.1 Online Tutoring Company in India Provides you Free PDF download of NCERT Math Class 7 solved by Expert Teachers as per NCERT (CBSE) Book guidelines. All Chapter wise Questions with Solutions to help you to revise complete Syllabus and Score More marks in your examinations. You can also register Online for NCERT Class 7 Science tuition on Vedantu to score more marks in your examination.

Introduction

Rational Numbers

A rational number is one that may be represented in the following way:

x/y, where x and y are integers and y is not equal to 0.
We utilize various quantities in our daily lives that are not whole numbers but can be expressed as fractions p/q. As a result, we require rational numbers.

Rational Numbers with Equivalents

By multiplying or dividing the numerator and denominator of a rational number by the same non-zero integer, we receive another rational number that is equivalent to the original rational number. Equivalent fractions are what they're called.

Standardized Rational Numbers

If the denominator is a positive integer and the numerator and denominator have no common factor other than 1, the rational number is said to be in standard form.

LCM

The smallest number (≠0) that is a multiple of both is the least common multiple (LCM) of two numbers.

Example: LCM of 5 and 6 can be calculated as shown below:

0, 5, 10, 15, 20, 25, 30, 35 are multiples of 5.

0, 6, 12, 18, 24, 30, 36 are multiples of 6.

30 is the L.C.M. of 5 and 6.

Rational Numbers in the Interval Between Two Rational Numbers

Any two rational numbers can have an endless number of rational numbers between them.
List some rational numbers between 35 and 13 as an example.

The L.C.M. of 5 and 3 equals 15.

Rational Numbers and Their Properties

Property of Closure

A rational number is a sum, difference, and product of two rationals. As a result, rational numbers are closed when they are added, subtracted, or multiplied, but not when they are divided.

Property of Commutativity

x*y=y*x for any two rational numbers x and y.
When it comes to addition and multiplication, rational numbers are commutative, but not when it comes to subtraction and division.

Property of Association

(a*b) c=a (b*c) for any three rational numbers a, b, and c.
When it comes to rational numbers, addition and multiplication are associative, but subtraction and division are not.

Reciprocals and Negatives

Positive and negative rational numbers are the two types of rational numbers.

A positive rational number is one in which the numerator and denominator are both positive integers or negative integers.
A negative rational number is one in which either the numerator or denominator is a negative integer.

When the product of two rational integers is 1, they are referred to as each others’ reciprocals.

Note that a rational number's product with its reciprocal is always 1.

We have provided step by step solutions for all exercise questions given in the PDF of Class 7 RS Aggarwal Chapter-4 Rational Numbers. All the Exercise questions with solutions in Chapter-4 Rational Numbers are given below:

At Vedantu, students can also get Class 7 Math Revision Notes, Formula and Important Questions and also students can refer to the complete Syllabus for Class 7 Math, Sample Paper and Previous Year Question Paper to prepare for their exams to score more marks.

RS Aggarwal Solutions Class 7 Chapter 4 (Rational Numbers)

First of all, to attempt to solve the questions of this Chapter a student must know,

What is a Rational Number?

A Rational Number is defined as a number that can be expressed in the form of p/q where q ≠ 0. For example 1/3, 50/34, 72/99, etc. A rational number can be positive, negative, and zero and can also be expressed in the form of a fraction. Therefore, all the whole numbers are rational numbers.

Properties of Rational Number

Closure Property: A+B=B+A

A*B=B*A

Associative Property: A+(B+C)=(A+B)+C

(A*B)*C= A*(B*C)

Distributive Property: A*(B+C)=A*B+A*C

Standard Form of Rational Numbers

A rational number is said to be standard when the denominators and numerators of a rational number have only a common factor of 1.

For example; 24/120, a rational number whose numerator and denominator are 24 and 120 respectively, have more than 1 common factor. Now when 24/48 is simplified, we get ½ which is in standard form. ½ is in standard form since 1 and 2 have no common factor other than 1.

Difference Between Positive and Negative Rational Numbers

Positive Rational Numbers

Negative Rational Numbers

Definition: A Rational number is said to be positive if both the numerator and denominator have the same signs.

Examples: 13/72, 1/4, 25/50

Definition: A Rational number is said to be negative if the numerator and denominator have opposite signs.

Examples : -13/72 , -1/4 , -25/50

Range: Greater than 0

Range: Less than 0

The Multiplicative Inverse of Rational Numbers

The reciprocal of a given rational number is the multiplicative inverse of that rational number. Therefore, the multiplication of the rational number and the multiplicative inverse should always be equal to 1.

Example: Find the multiplicative inverse of 7/23.

Solution:

Reciprocal of 7/23 = 23/7

Now, 7/23*23/7=1

Therefore, 23/7 is the multiplicative inverse of 7/23.

Additive Inverse of Rational Number

The additive inverse of a rational number is the number that when added to the rational number gives zero. Therefore, the additive inverse of a rational number is negative of that rational number.

Example: Find the additive inverse of -4/13.

Solution: Additive inverse = - ( - 4/13) = 4/13

-4/13+4/13=0

Therefore, 4/13 is the additive inverse of -4/13.

Difference Between Rational and Irrational Numbers

Rational Numbers	Irrational Numbers
Definition: A rational number is defined as a number that can be expressed in the form of p/q where q ≠ 0.	Definition: A rational number is defined as a number that cannot be expressed in the form of p/q.
It includes only the decimals that are finite and are recurring.	It includes the number that is non-terminating or non-recurring.
Example: 10/13, 1.4444, 1.12346….	Example: √ 55, √ 5, √ 34

Preparation Tips

First, go through the Chapter thoroughly and cover all the topics.
Note down all the important formulae and try to learn them.
Start solving the exercise questions without any guidance from anyone.
Try the questions at least 3 times if you are unable to solve them.
Seek help from Vedantu’s solution to check your answers and solve the ones that you were unable to solve.

FAQs on RS Aggarwal Class 7 Solutions Chapter-4 Rational Numbers

1. What defines a rational number in its standard form as per the solutions for RS Aggarwal Class 7 Chapter 4?

A rational number is considered to be in its standard form when its denominator is a positive integer, and the numerator and denominator share no common factor other than 1. For instance, the rational number 3/5 is in standard form, whereas 6/10 is not because it can be simplified further to 3/5.

2. What is the correct method for adding two rational numbers with different denominators, as explained in Chapter 4?

To add rational numbers with different denominators, follow these steps: First, find the Least Common Multiple (LCM) of the denominators. Next, convert each rational number into an equivalent fraction that has the LCM as its denominator. Finally, add the numerators of these new, like-fractions, keeping the common denominator. For example, to add 1/4 and 2/3, the LCM is 12. The equivalent fractions are 3/12 and 8/12, and their sum is (3+8)/12 = 11/12.

3. How do the RS Aggarwal solutions for Class 7 explain the subtraction of one rational number from another?

Subtracting a rational number is explained as the process of adding its additive inverse. To subtract c/d from a/b, you calculate a/b + (-c/d). The additive inverse of any rational number p/q is simply -p/q. If the denominators are not the same, you must first find their LCM to create like-fractions before performing the addition.

4. What is the step-by-step process for representing a rational number like -7/4 on a number line?

To represent -7/4 on a number line, you should first convert it to a mixed fraction, which is -1 3/4. This tells you the number lies between the integers -1 and -2. Next, divide the segment on the number line between -1 and -2 into four equal parts. The third mark to the left of -1 represents the position of -7/4.

5. Is every integer also a rational number? Explain how this is addressed in Class 7 Maths Chapter 4.

Yes, every integer is a rational number. A rational number is defined as any number that can be written in the form p/q, where p and q are integers and q is not zero. Any integer, such as 8, can be expressed as 8/1. In this case, p=8 and q=1, fulfilling the definition. Similarly, a negative integer like -5 can be written as -5/1, confirming it is also a rational number.

6. Why is finding the LCM a necessary first step when adding or subtracting rational numbers with unlike denominators?

Finding the Least Common Multiple (LCM) is essential because it allows us to convert the rational numbers into 'like' fractions—fractions that share a common denominator. We can only add or subtract parts of a whole accurately when the parts are of the same size. The LCM provides this common sizing, ensuring that the numerators we add or subtract represent portions of the same-sized unit.

7. What is the fundamental difference between the additive inverse and the multiplicative inverse of a rational number?

The primary difference is the mathematical operation and the identity element they relate to.

The additive inverse of a number p/q is -p/q. When added together, they result in 0 (the additive identity). Example: 3/7 + (-3/7) = 0.
The multiplicative inverse (or reciprocal) of a non-zero number p/q is q/p. When multiplied, they result in 1 (the multiplicative identity). Example: (3/7) × (7/3) = 1.

In simple terms, one is used for 'undoing' addition, and the other for 'undoing' multiplication.

8. How does the 'density property' of rational numbers explain what lies between any two rational numbers on a number line?

The density property states that between any two distinct rational numbers, you can always find another rational number. In fact, you can find infinitely many. For example, to find a number between 1/2 and 3/4, you can find their average: [(1/2) + (3/4)] / 2 = (5/4) / 2 = 5/8. This process can be repeated endlessly, proving that the rational numbers are densely packed along the number line.

9. What is a common mistake students make when solving division problems with rational numbers?

A common mistake is to divide the numerators and denominators directly, similar to multiplication. The correct method for dividing a rational number a/b by c/d is to multiply a/b by the reciprocal (multiplicative inverse) of c/d. The correct calculation is (a/b) ÷ (c/d) = (a/b) × (d/c). Forgetting to flip the second fraction before multiplying is a frequent error.

10. How do you correctly compare a negative rational number with a positive one, for example, -5/2 and 1/8?

When comparing a negative and a positive rational number, the rule is absolute: any positive rational number is always greater than any negative rational number. Therefore, without needing to find a common denominator or compare the numerical values, we know that 1/8 is greater than -5/2 simply because 1/8 is positive and -5/2 is negative.