CBSE Class 8 Maths Important Questions Chapter 1 - Rational Numbers

Very Short Answer Questions                                                                                   1 Marks

1. The additive inverse of  $\dfrac{3}{4}$ is ____ 

Ans: The Additive inverse of any number is a number that can be added to a number to get $0$.

Hence, the additive inverse of $\dfrac{\text{3}}{\text{4}}$ is $\text{-}\dfrac{\text{3}}{\text{4}}$.

2. Multiplicative inverse of is $\dfrac{1}{8}$ 

Ans: Multiplicative inverse of any number is a number that can be multiplied to a number to get $1$.

The multiplicative inverse of  $\dfrac{\text{p}}{\text{q}}$  is  $\dfrac{\text{q}}{\text{p}}$ .

Hence, the multiplicative inverse of  $\dfrac{1}{8}$  is  $8$.

3. A Rational number between $3$  and $4$  is ____ 

Ans: The rational number of any number can be calculated as;

$\dfrac{\left( \text{p+q} \right)}{\text{2}}$

Hence, the required rational number is;

$\dfrac{\left( 3+4 \right)}{2}=\dfrac{7}{2}$

Hence,  $\dfrac{7}{2}$ is a rational number between $3$  and $4$.

4. Reciprocal of $-2$  is ___ 

Ans: Reciprocal of any number will be the inverse of that number.

That is, the reciprocal of  $a$ is  $\dfrac{1}{a}$ .

Hence, the reciprocal of $-2$  is $-\dfrac{1}{2}$ .

5. Zero has ____ reciprocal 

Ans: The product of any number with $0$ will be  $0$  and hence, zero has no reciprocal.

6. Reciprocal of a negative number is _______ 

Ans: The product of any negative number with any number will be negative and hence, the reciprocal of a negative number will be negative.

7. Whole numbers start from______ 

Ans: The numbers that start from  $0$, are called whole numbers.

8. There are _____ rational numbers between $3$  and $4$. 

Ans: Infinite rational numbers are between $3$  and  $4$.

9. What are the multiplicative and additive identities of rational numbers?

Ans: 0 and 1 are the additive and multiplicative identity of rational numbers respectively.

10. Write the additive inverse of $\dfrac{19}{-6}$ and $-\dfrac{2}{3}$

Ans: $\dfrac{19}{-6}$ and $-\dfrac{2}{3}$ = $\dfrac{2}{3}$

11. Write the multiplicative inverse of $-\dfrac{13}{19}$ and $-7$

Ans: $-\dfrac{13}{19} = -\dfrac{19}{13}$ and $-7 = -\dfrac{1}{7}$

12. Mention a rational number which has no reciprocal.

Ans: A rational number “0” has no reciprocal or multiplicative inverse.

13. Mention any 4 rational numbers which are less than 5.

Ans: -1, 1, 2, and 3.

Short Answer Questions                                                                                             2 Marks 

14. Sum of two numbers is  $\dfrac{3}{4}$, one of the number is $\dfrac{1}{8}$ .Find the other one.

Ans: Let the other number be $\text{x}$.

The sum of $\text{x}$ and $\dfrac{1}{8}$ is $\dfrac{3}{4}$ .  Therefore, we have

\[\begin{align} & \Rightarrow \text{x}+\dfrac{1}{8}=\dfrac{3}{4} \\ & \Rightarrow \text{x}=\dfrac{3}{4}-\dfrac{1}{8} \\ & \Rightarrow \text{x}=\dfrac{\left( 3\times 2 \right)-1}{8} \\ \end{align}\] Since, $[\text{LCM}=8]$ Therefore, $\begin{align} & \Rightarrow \text{x}=\dfrac{6-1}{8} \\ & \Rightarrow \text{x}=\dfrac{5}{8} \\ \end{align}$

Therefore, the sum of  $\dfrac{5}{8}$ and $\dfrac{1}{8}$ is $\dfrac{3}{4}$.

15. Simplify $\left( \dfrac{-8}{13} \right)+\left( \dfrac{-3}{26} \right)$ 

Ans: The given dfraction is;

$\dfrac{-8}{13}-\dfrac{3}{26}$

By simplifying above dfraction, we get

$\Rightarrow \dfrac{\left( -8\times 2 \right)-3}{26}$ 

Since, $[\text{LCM}=26]$

Therefore,

$\begin{align} & \Rightarrow \dfrac{-16-3}{26} \\ & \Rightarrow -\dfrac{19}{26} \\ \end{align}$

Therefore, $\left( \dfrac{-8}{13} \right)+\left( \dfrac{-3}{26} \right)=-\dfrac{19}{26}$ .

16. What number to be multiplied with $\dfrac{1}{4}$ so as to get the product as $-\dfrac{5}{16}$ 

Ans: Let the number be $\text{x}$ .

The product can be written as;

$\Rightarrow \dfrac{1}{4}\times \text{x}$ and,

Product of $\text{x}$ and  $\dfrac{1}{4}$ is $-\dfrac{5}{16}$ .

Therefore, we have

\[\begin{align} & \Rightarrow -\dfrac{5}{16}=\dfrac{1}{4}\times \text{x} \\ & \Rightarrow \text{x}=4\times \left( -\dfrac{5}{16} \right) \\ \end{align}\]

Therefore, we get

$\Rightarrow \text{x}=-\dfrac{5}{4}$

Hence, $-\dfrac{5}{4}$ is the number to be multiplied with $\dfrac{1}{4}$ so as to get the product as  $-\dfrac{5}{16}$.

17. Represent $-\dfrac{2}{7}$ on the number line.

Ans: $-\dfrac{2}{7}$ is a rational number.

$-\dfrac{2}{7}$ on a Number Line, can be represented as follows;

18. Divide $\dfrac{1}{2}$by $\left[ \dfrac{-1}{3}+\dfrac{2}{5} \right]$ 

Ans: The given fraction is;

$\left[ \dfrac{-1}{3}+\dfrac{2}{5} \right]$

By simplifying the above fraction, we get

\[\begin{align} & \Rightarrow -\dfrac{1}{3}+\dfrac{2}{5} \\ & \Rightarrow \dfrac{\left( -1.5 \right)+\left( 2.3 \right)}{15} \\ \end{align}\]

Since, $[\text{LCM}=15]$ , therefore

\[\begin{align} & \Rightarrow \dfrac{-5+6}{15} \\ & \Rightarrow \dfrac{1}{15} \\ \end{align}\]

On dividing  $\dfrac{1}{2}$ by $\dfrac{1}{15}$ , we get

\[\begin{align} & \Rightarrow \dfrac{1}{2}\div \dfrac{1}{15} \\ & \Rightarrow \dfrac{1}{2}\times \dfrac{15}{1} \\ & \Rightarrow \dfrac{15}{2} \\ \end{align}\]

19. Find three rational number between $-4$and $4$ Represent them on line.

 Ans: The three rational numbers between $-4$  and $4$ are $-2,-1,1$ .

On a number line, the mentioned rational numbers are represented as follows:

20. Define by example of addition 

(a) Associative Property 

Ans: Associative Property can be stated as while addition or multiplication of any two grouped numbers, the interchange of those numbers won’t get affected on resultant addition or multiplication.

That is,

$\Rightarrow \text{a+b = b+a}$

The following example shows how the associative property gets used to solve it;

We can solve  $9+7$ by using Associative Property; we get

$\begin{align} & \Rightarrow \left( 7+9 \right)=\left( 9+7 \right) \\ & \Rightarrow 7+9=16 \\ \end{align}$ and

$\Rightarrow 9+7=16$ 

Therefore, according to the property, we can add or multiply regardless of how the numbers are grouped.

Short Answer Questions                                                                                             3 Marks

21. Simplify $\left[ \dfrac{6}{7}+\dfrac{3}{8}-\dfrac{1}{2} \right]\dfrac{4}{3}$   and find its reciprocal. 

Ans: Reciprocal of any number will be the inverse of that number.

Hence, by solving above dfraction, we get

$\Rightarrow \left[ \dfrac{6\times 8+3\times 7-1\times 28}{56} \right]\dfrac{4}{3}$ 

(Since the LCM of $7,8$ and $2$ is $56$ )

\[\begin{align} & \Rightarrow \left[ \dfrac{48+21-28}{56} \right]\times \dfrac{4}{3} \\ & \Rightarrow \dfrac{41}{56}\times \dfrac{4}{3} \\ & \Rightarrow \dfrac{41}{42} \\ \end{align}\]

Therefore,

Reciprocal of $\dfrac{41}{42}$ is 

$\Rightarrow \dfrac{1}{\dfrac{41}{42}}=1\times \dfrac{42}{41}$ 

\[\Rightarrow \dfrac{42}{41}\] 

Hence, reciprocal of  $\left[ \dfrac{6}{7}+\dfrac{3}{8}-\dfrac{1}{2} \right]\dfrac{4}{3}$ is \[\dfrac{42}{41}\] .

22. Find three Rational Number between $3$ and $4$. Represent them on the Number line.

Ans: $3$  can be written as 

\[\begin{align} & \Rightarrow 3\times \dfrac{10}{10} \\ & \Rightarrow \dfrac{30}{10} \\ \end{align}\] $4$ can be written as \[\begin{align} & \Rightarrow 4\times \dfrac{10}{10} \\ & \Rightarrow \dfrac{40}{10} \\ \end{align}\]

Hence, the three Rational Numbers are  $\dfrac{31}{10},\dfrac{32}{10},\dfrac{33}{10}$ .

$\dfrac{31}{10},\dfrac{32}{10},\dfrac{33}{10}$ these numbers can be represented on a Number line, are as follows;

23. Use appropriate property and find $-\dfrac{1}{6}\times \dfrac{4}{7}+\dfrac{1}{2}-\dfrac{3}{7}\times \dfrac{1}{6}$ 

Ans: The given dfraction is;

$-\dfrac{1}{6}\times \dfrac{4}{7}+\dfrac{1}{2}-\dfrac{3}{7}\times \dfrac{1}{6}$ 

By using associative property $\left( \text{a+b = b+a} \right)$ , we can be simplifying the above dfraction as follows;

\[\begin{align} & \Rightarrow -\dfrac{1}{6}\times \dfrac{4}{7}-\dfrac{3}{7}\times \dfrac{1}{6}+\dfrac{1}{2} \\ & \Rightarrow -\dfrac{1}{6}\times \dfrac{4}{7}-\dfrac{1}{6}\times \dfrac{3}{7}+\dfrac{1}{2} \\ & \Rightarrow -\dfrac{1}{6}\left[ \dfrac{4}{7}+\dfrac{3}{7} \right]+\dfrac{1}{2} \\ & \Rightarrow -\dfrac{1}{6}\times \dfrac{7}{7}+\dfrac{1}{2} \\ \end{align}\]

Therefore,

\[\begin{align} & \Rightarrow \dfrac{-1+3}{6} \\ & \Rightarrow \dfrac{2}{6} \\ & \Rightarrow \dfrac{1}{3} \\ \end{align}\]

Hence,  $-\dfrac{1}{6}\times \dfrac{4}{7}+\dfrac{1}{2}-\dfrac{3}{7}\times \dfrac{1}{6}=\dfrac{1}{3}$ .

24. Find the multiplicative inverse of following  

1. \[\dfrac{1}{6}\] 

Ans: Multiplicative inverse of any number is a number which can be multiplied to a number to get $1$ .

Hence, the multiplicative inverse of  $\dfrac{1}{6}$ is

\[\begin{align} & \Rightarrow \dfrac{1}{6}\times \text{x}=1 \\ & \Rightarrow \text{x}=6 \\ \end{align}\]

Therefore, the multiplicative inverse of  $\dfrac{1}{6}$ is $6$ .

2. $-\dfrac{3}{8}$ 

Ans: Multiplicative inverse of any number is a number which can be multiplied to a number to get $1$ .

Hence, the multiplicative inverse of  $\dfrac{-3}{8}$ is

$\begin{align} & \Rightarrow -\dfrac{3}{8}\times \text{x}=1 \\ & \Rightarrow \text{x}=-\dfrac{8}{3} \\ \end{align}$

Therefore, the multiplicative inverse of  $-\dfrac{3}{8}$  is $-\dfrac{8}{3}$.

3. $\dfrac{4}{19}$ 

Ans: Multiplicative inverse of any number is a number which can be multiplied to a number to get $1$ .

Hence, the multiplicative inverse of  $\dfrac{4}{19}$ is

\[\begin{align} & \Rightarrow \dfrac{4}{19}\times \text{x}=1 \\ & \Rightarrow \text{x}=\dfrac{19}{4} \\ \end{align}\]

Therefore, the multiplicative inverse of  $\dfrac{4}{19}$  is $\dfrac{19}{4}$ .

Long Answer Questions                                                                       4 or 5 Marks

25. Find three Rational number between $\dfrac{3}{6}$ and $\dfrac{3}{4}$ 

Ans: The rational number of any number can be calculated as;

$\dfrac{\left( \text{p+q} \right)}{\text{2}}$

Hence, the mean of two Rational numbers is a Rational number.

For given numbers;

$\dfrac{3}{6}=\dfrac{1}{2}$ and  $\dfrac{3}{4}$ 

Mean can be calculated as;

\[\begin{align} & \Rightarrow \dfrac{\dfrac{1}{2}+\dfrac{3}{4}}{2} \\ & \Rightarrow \dfrac{\dfrac{5}{4}}{2} \\ & \Rightarrow \dfrac{5}{8} \\ \end{align}\]

Since, $\dfrac{1}{2}<\dfrac{5}{8}<\dfrac{3}{4}$ ;

Now Mean of $\dfrac{1}{2}$ and $\dfrac{5}{8}$ 

Mean is,

\[\begin{align} & \Rightarrow \dfrac{\dfrac{1}{2}+\dfrac{5}{8}}{2} \\ & \Rightarrow \dfrac{\dfrac{9}{8}}{2} \\ & \Rightarrow \dfrac{9}{16} \\ \end{align}\] 

Since, \[\dfrac{1}{2}<\dfrac{9}{16}<\dfrac{5}{8}\]

Mean of $\dfrac{5}{8}$ and $\dfrac{3}{4}$ 

Mean is,

\[\begin{align} & \Rightarrow \dfrac{\dfrac{5}{8}+\dfrac{3}{4}}{2} \\ & \Rightarrow \dfrac{5+6}{8\times 2} \\ & \Rightarrow \dfrac{11}{16} \\ \end{align}\]

$\therefore \dfrac{5}{8}<\dfrac{11}{16}<\dfrac{3}{4}$

Hence, $\dfrac{9}{16},\dfrac{5}{8},\dfrac{11}{16}$ are the three rational numbers are between $\dfrac{3}{6}$ and $\dfrac{3}{4}$ .

26.  

1. Reciprocal of $-\dfrac{1}{2}$ 

Ans: Reciprocals of any number are going to be the inverse of that number.

Hence, the reciprocal of $-\dfrac{1}{2}$ is;

$\dfrac{1}{-\dfrac{1}{2}}=-2$

Therefore, Reciprocal of $-\dfrac{1}{2}$ is  $-2$ .

2. Additive inverse of $\dfrac{4}{9}$

Ans: Additive inverse of any number is a number which can be added to a number to get $0$ .

For the given number, we get

$\Rightarrow \dfrac{4}{9}+x=0$ 

$\Rightarrow $       $x=-\dfrac{4}{9}$

Therefore, an additive inverse of $\dfrac{4}{9}$ is $-\dfrac{4}{9}$ .

3. Multiplicative inverse of $\left[ \dfrac{1}{6}+\dfrac{4}{9} \right]\times \dfrac{4}{3}$ 

Ans:  Multiplicative inverse of any number is a number which can be multiplied to a number to get $1$ .

For given number, we get

 $\begin{align} & \Rightarrow \left[ \dfrac{1}{6}+\dfrac{4}{9} \right]\times \dfrac{4}{3} \\ & \Rightarrow \dfrac{(1\times 3+2\times 4)}{18}\times \dfrac{4}{3} \\ & \Rightarrow \dfrac{(3+8)}{18}\times \dfrac{4}{3} \\ \end{align}$ Therefore, $\begin{align} & \Rightarrow \dfrac{11}{18}\times \dfrac{4}{3} \\ & \Rightarrow \dfrac{22}{27} \\ \end{align}$

Hence, multiplicative inverse of $\left[ \dfrac{1}{6}+\dfrac{4}{9} \right]\times \dfrac{4}{3}$ is  $\dfrac{27}{22}$ .

27. Match the correct 

(a) Whole number

(i) 

(b) Natural number 

(ii)

(c) Integer 

(iii)

(d) Rational Number

(iv)

Ans: The correct pairs are as follows;

(a) (ii) 

(b) (iii) 

(c) (i) 

(d) (iv) 

Whole numbers starts from $0$ .

Natural numbers are positive numbers and starts from $1$.

Integers will be positive, negative or $0$ numbers.

Rational numbers are the dfractional numbers.

Subtopics Covered in Detail in Chapter 1 - Rational Numbers

The subtopics that are covered in detail in the chapter are:

1. Properties of Rational Numbers.

• Closure property

• Commutative property

• Associativity property

• The role of 0

• The role of 1

• The negative of a number

• Reciprocal

• Distributivity of multiplication over addition for rational numbers

2. Representation of rational numbers on the Number Line.

3. Rational numbers between two rational numbers.

Rational numbers are of the type p/q where p and q are integers and q ≠ 0. Simply rational numbers can be defined as fractions that can be represented on the number line. Let us look at the concepts that govern rational numbers.

Properties of Rational Numbers

The properties we are going to discuss are basic mathematical properties that are applicable to integers as well. Let us discuss the properties one by one:

• Closure Property

The result of addition, multiplication and subtraction between any two rational numbers is also a rational number. This property is true for all mathematical properties except division because division by 0 is not defined. So divisions other than 0 are included in the closure property.

• Commutative Property

The commutative property is true only for addition and multiplication of rational numbers. For any two rational numbers:

Commutative property of addition:  $a + b = b + a$

$\dfrac{2}{5} + \dfrac{3}{5} = \dfrac{3}{5} + \dfrac{2}{5}$

Commutative property of multiplication: $a \times b = b \times a$

$\dfrac{3}{7} \times \dfrac{2}{7} = \dfrac{2}{7} \times \dfrac{3}{7}$

• Associative Property

Rational numbers follow the property of association only for addition and multiplication.

Let a, b and c be three rational numbers:

Associative property for addition: $a + (b + c) = (a + b) + c$

Associative property for multiplication: $a \times (b \times c) = (a \times b) \times c$

• Distributive Property

Rational numbers follow distributive property for addition and multiplication only. For any three rational numbers p,q and r. 

$p \times (q + r) = (p \times q) + (p \times r)$

• Identity Property

The additive identity for any rational number is 0 and the multiplicative identity is 1.

• Inverse Property

For a given rational number $\dfrac{x}{y}$,

Additive inverse: $-\dfrac{x}{y}$

Multiplicative inverse: $\dfrac{y}{x}$

Representation of Rational Numbers on the Number Line

Rational numbers can also be represented on the number line just like other whole numbers or integers. 0 represents the origin. Negative rational numbers are present to the left and the positive rational numbers are located on the right side of the number line.

Representation of a rational number can be categorised into:

• Proper Fraction: When the value of the numerator is less than the value of the denominator, the fraction is called the proper fraction and the value is less than one.

• Improper Fraction: In case of the improper fraction, the value of the numerator is greater than the value of the denominator. First, we convert the given fraction into a mixed fraction to get a better picture of where the number exists on the number line.

To represent the rational number we divide the distance between the two whole numbers into the number of subunits denoted by the denominator.

The number can also be represented using the process of successive magnification where we convert the rational number into decimal expansion and represent the decimal places successively.

Rational Number Between Two Rational Numbers

The different methods using which rational numbers between two rational numbers can be found are:

1. The basic method that is used to find the number between two rational numbers is dividing their sum by 2.

A rational number between 5 and 6 can be calculated by

$\dfrac{5 + 6}{2} = \dfrac{11}{2}$

$\dfrac{11}{2}$ is a rational number between 5 and 6.

2. For numbers with a common denominator, rational numbers between them can be easily found.

Rational numbers between $\dfrac{4}{9}$ and $\dfrac{8}{9} are $\dfrac{5}{9}$, $\dfrac{6}{9}$, $\dfrac{7}{9}$.

Rational numbers between $-\dfrac{2}{3}$ and $\dfrac{2}{3}$ are: $-\dfrac{1}{3}, 0, \dfrac{1}{3}$

3. In the case of different denominators, the first step is to make the denominators of the fraction similar before finding the rational numbers between them.

To find the rational number between $\dfrac{1}{5}$ and $\dfrac{14}{15}$, make the denominators of both the rational numbers equal.

So, $\dfrac{1 \times 3}{5 \times 3} = \dfrac{3}{15}$

The rational numbers between them are$\dfrac{4}{15}, \dfrac{5}{15}, \dfrac{6}{15}, \dfrac{7}{15} …. \dfrac{12}{15}, \dfrac{13}{15}$ etc.

List of Important Class 8 Maths Rational Numbers Important Questions

The pdf class 8 maths chapter 1 important questions contains different types of questions that cover all the sub-topics of the entire chapter. Important questions from each topic are included in the pdf to give the students a clear and logical understanding of the chapter. Some of the important questions that are commonly asked in the examination from this chapter are:

• Find the rational number between $\dfrac{2}{3}$ and $\dfrac{4}{5}$.

• Determine the rational numbers lying between $-\dfrac{1}{2}$ and $\dfrac{1}{2}$.

• Simplify: $\dfrac{3}{4} + \dfrac{(-5)}{8}$.

• Simplify: $-\dfrac{2}{3} - \dfrac{4}{5}$.

• Solve: $\dfrac{3}{7} \times \dfrac{5}{6}$.

• Solve: $-\dfrac{2}{5} \div \dfrac{3}{4}$.

• Determine whether the following numbers are rational or irrational: $\sqrt{16}, \sqrt{5}, -3.25, 0.8888 …$

• Find the rational number between 1.5 and 1.6.

• Simplify: $-\dfrac{7}{9} + \dfrac{(-4)}{9} - \dfrac{1}{3}$.

• Find the value of 'x' if $\dfrac{2}{3} \div x = \dfrac{4}{5}$.

5 Important Topics of Class 8 Maths Chapter 1 You Shouldn’t Miss!

Benefits of Class 8 Chapter 1 Maths Important Questions

• Practising important questions helps you understand rational numbers and their properties. This strong foundation is important for learning more difficult maths concepts later.

• Regular practice improves your problem-solving skills by exposing you to different types of questions. This helps you find solutions more efficiently and accurately.

• Focusing on these key questions is great for exam preparation. It ensures you cover all topics that might be on your tests, increasing your chances of getting better grades.

• Working through these questions builds your confidence. The more you practice, the more comfortable you will feel when facing tough problems during exams.

• Important questions help you find areas where you need more help. This lets you focus your study efforts where they are needed most.

• These questions provide a structured way to learn, helping you organize your study time. This method lets you track your progress and make sure you cover everything.

• Practising these questions improves your time management skills, which are important for exams. By learning to solve problems quickly, you can finish tests on time.

• Understanding rational numbers gives you a good base for more advanced topics in maths. This will make it easier to learn subjects like algebra and geometry later on.

• Engaging with important questions makes learning maths more interesting. This active approach keeps you motivated and makes studying more enjoyable.

Conclusion

Practising important questions from Class 8 Maths Chapter 1 on Rational Numbers is essential for mastering the topic. These questions help you understand the concepts better and prepare effectively for exams. With regular practice, you can improve your problem-solving skills and build confidence. Remember, a strong understanding of rational numbers will serve you well in your future maths studies. By using the resources and support available at Vedantu, you can enhance your learning experience and achieve success in your exams. Keep practising and stay focused!

Important Study Materials for Class 8 Maths Chapter 1

CBSE Class 8 Maths Important Questions for All Chapters

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