CBSE Class 8 Mathematics Chapter 1: Rational Numbers NCERT Solutions

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NCERT Solutions PDF Class 8 PDF

CBSE Class 8 Mathematics Chapter 1 NCERT Solutions introduces students to the world of Rational Numbers. This chapter delves into the fundamental properties of rational numbers, such as the commutative, associative, and distributive laws, which are crucial for performing operations with these numbers. Students will learn how to represent rational numbers on a number line and find rational numbers between any two given rational numbers. The solutions provide clear, step-by-step explanations for all exercises, including finding the additive inverse and multiplicative inverse, and verifying identities like -(-x) = x. This resource aims to build a strong foundation in rational numbers, enabling students to confidently tackle problems and excel in their mathematics studies.

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Quick info

BoardCBSE
ClassClass 8
SubjectMathematics
Session2026
LanguageEnglish
TypeNCERT Solutions
ChapterChapter 1

Chapter summary

Chapter 1 of the Class 8 Mathematics NCERT textbook focuses on Rational Numbers. This section provides step-by-step solutions for exercises involving the application of properties like distributivity and associativity to simplify expressions with rational numbers. It also includes problems on finding the additive inverse of given rational numbers and verifying the property -(-x) = x. The solutions aim to build a strong foundation in rational number operations.

Learning outcomes

  • Understand and apply the properties of rational numbers (distributive, associative) to simplify expressions.
  • Calculate the additive inverse of given rational numbers.
  • Verify the identity -(-x) = x for rational numbers.
  • Perform arithmetic operations on rational numbers accurately.
  • Solve problems involving rational numbers using appropriate mathematical properties.

Topics covered

Paper topics

  • Rational Numbers
  • Properties of Rational Numbers
  • Distributive Property
  • Associative Property
  • Commutative Property
  • Additive Inverse
  • Verification of -(-x) = x
  • Arithmetic Operations on Rational Numbers

Important topics

  • Properties of Rational Numbers (Distributive, Associative)
  • Additive Inverse
  • Simplification of Rational Number Expressions
  • Verification of Identities

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Questions and Solutions

Question 1

Using appropriate properties, find the value of the following expressions:
  1. -\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6}
  2. \frac{2}{5} \times \left(\frac{3}{-7}\right) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5}
Solution:

  1. We need to evaluate -\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6}.

    First, we can rearrange the terms using the commutative property of addition to group the terms with the common factor \(\frac{3}{5}\):

    \left(-\frac{2}{3} \times \frac{3}{5}\right) - \left(\frac{3}{5} \times \frac{1}{6}\right) + \frac{5}{2}

    Now, we apply the distributive property, a \times (b - c) = a \times b - a \times c, in reverse. We factor out \(\frac{3}{5}\) from the first two terms:

    \frac{3}{5} \left(-\frac{2}{3} - \frac{1}{6}\right) + \frac{5}{2}

    To subtract the fractions inside the parenthesis, we find a common denominator, which is 6:

    \frac{3}{5} \left(\frac{-2 \times 2}{3 \times 2} - \frac{1}{6}\right) + \frac{5}{2} = \frac{3}{5} \left(\frac{-4}{6} - \frac{1}{6}\right) + \frac{5}{2}

    Perform the subtraction inside the parenthesis:

    \frac{3}{5} \left(\frac{-4 - 1}{6}\right) + \frac{5}{2} = \frac{3}{5} \left(\frac{-5}{6}\right) + \frac{5}{2}

    Now, multiply the fractions:

    \frac{3 \times (-5)}{5 \times 6} + \frac{5}{2} = \frac{-15}{30} + \frac{5}{2}

    Simplify the fraction \(\frac{-15}{30}\) to \(\frac{-1}{2}\):

    -\frac{1}{2} + \frac{5}{2}

    Finally, add the fractions:

    \frac{-1 + 5}{2} = \frac{4}{2} = 2

    Thus, the value of the expression is 2.

  2. We need to evaluate \frac{2}{5} \times \left(\frac{3}{-7}\right) - \frac{1}{6} \times \frac{3}{2} + \frac{1}{14} \times \frac{2}{5}.

    First, simplify the terms and rearrange using the commutative property of addition:

    \frac{2}{5} \times \left(\frac{-3}{7}\right) + \frac{1}{14} \times \frac{2}{5} - \frac{1}{6} \times \frac{3}{2}

    Simplify the multiplication of the last term:

    \frac{1}{6} \times \frac{3}{2} = \frac{3}{12} = \frac{1}{4}

    The expression becomes:

    \frac{2}{5} \times \left(\frac{-3}{7}\right) + \frac{1}{14} \times \frac{2}{5} - \frac{1}{4}

    Now, apply the distributive property by factoring out \(\frac{2}{5}\) from the first two terms:

    \frac{2}{5} \left(\frac{-3}{7} + \frac{1}{14}\right) - \frac{1}{4}

    To add the fractions inside the parenthesis, find a common denominator, which is 14:

    \frac{2}{5} \left(\frac{-3 \times 2}{7 \times 2} + \frac{1}{14}\right) - \frac{1}{4} = \frac{2}{5} \left(\frac{-6}{14} + \frac{1}{14}\right) - \frac{1}{4}

    Perform the addition inside the parenthesis:

    \frac{2}{5} \left(\frac{-6 + 1}{14}\right) - \frac{1}{4} = \frac{2}{5} \left(\frac{-5}{14}\right) - \frac{1}{4}

    Multiply the fractions:

    \frac{2 \times (-5)}{5 \times 14} - \frac{1}{4} = \frac{-10}{70} - \frac{1}{4}

    Simplify the fraction \(\frac{-10}{70}\) to \(\frac{-1}{7}\):

    -\frac{1}{7} - \frac{1}{4}

    To subtract these fractions, find a common denominator, which is 28:

    \frac{-1 \times 4}{7 \times 4} - \frac{1 \times 7}{4 \times 7} = \frac{-4}{28} - \frac{7}{28}

    Perform the subtraction:

    \frac{-4 - 7}{28} = \frac{-11}{28}

    Thus, the value of the expression is \(\frac{-11}{28}\).

Question 2

Write the additive inverse of each of the following rational numbers:
  1. \frac{2}{8}
  2. \frac{-5}{9}
  3. \frac{-6}{-5}
  4. \frac{2}{-9}
  5. \frac{19}{-6}
Solution:

The additive inverse of a rational number \(\frac{a}{b}\) is \(\frac{-a}{b}\), such that their sum is zero: \frac{a}{b} + \left(\frac{-a}{b}\right) = 0.

  1. The additive inverse of \frac{2}{8} is \frac{-2}{8}.
  2. The additive inverse of \frac{-5}{9} is \frac{5}{9}.
  3. First, simplify \frac{-6}{-5} to \frac{6}{5}. The additive inverse of \frac{6}{5} is \frac{-6}{5}.
  4. The additive inverse of \frac{2}{-9} is \frac{-2}{-9}, which simplifies to \frac{2}{9}.
  5. The additive inverse of \frac{19}{-6} is \frac{-19}{-6}, which simplifies to \frac{19}{6}.

Question 3

Verify that -(-x) = x for the following values of x:
  1. x = \frac{11}{15}
  2. x = -\frac{13}{17}
Solution:

  1. Given x = \frac{11}{15}. We need to verify that -(-x) = x.

    Substitute the value of x into the left side of the equation:

    -(-x) = -\left(-\frac{11}{15}\right)

    The negative of a negative number is positive:

    - \left(-\frac{11}{15}\right) = \frac{11}{15}

    Since \frac{11}{15} is equal to the given value of x, we have verified that -(-x) = x for x = \frac{11}{15}.

  2. Given x = -\frac{13}{17}. We need to verify that -(-x) = x.

    Substitute the value of x into the left side of the equation:

    -(-x) = -\left(-\left(-\frac{13}{17}\right)\right)

    First, evaluate the innermost negative sign:

    -\left(-\frac{13}{17}\right) = \frac{13}{17}

    Now, apply the outer negative sign:

    -\left(\frac{13}{17}\right) = -\frac{13}{17}

    Since - \frac{13}{17} is equal to the given value of x, we have verified that -(-x) = x for x = -\frac{13}{17}.

Common mistakes

  • Errors in applying the distributive property.
  • Incorrectly calculating the common denominator for addition/subtraction.
  • Sign errors when finding the additive inverse.
  • Mistakes in simplifying fractions after operations.

Revision tips

  • Review the definitions of rational numbers and their properties.
  • Practice simplifying expressions using the distributive and associative properties.
  • Ensure you understand the concept of additive inverse and how to find it.
  • Work through each example and exercise solution to reinforce understanding.

Practice MCQs

Q1. What is the additive inverse of \(\frac{2}{8}\)?

Q2. Which property is used to rearrange \(-\frac{2}{3} \times \frac{3}{5} + \frac{5}{2} - \frac{3}{5} \times \frac{1}{6}\) as \(-\frac{2}{3} \times \frac{3}{5} - \frac{3}{5} \times \frac{1}{6} + \frac{5}{2}\)?

Q3. What is the result of \(\frac{3}{5}\left(\frac{-2}{3}-\frac{1}{6}\right)\)?

Q4. The additive inverse of \(\frac{-6}{-5}\) is:

Q5. If \(x = \frac{11}{15}\), what is the value of \(-(-x)\)?

Frequently asked questions

What are rational numbers?

Rational numbers are numbers that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). Examples include \(\frac{2}{3}\), \(-5\), and \(0.75\).

How do I find the additive inverse of a rational number?

The additive inverse of a rational number \(\frac{a}{b}\) is \(\frac{-a}{b}\). When you add a number and its additive inverse, the result is always 0. For example, the additive inverse of \(\frac{5}{9}\) is \(\frac{-5}{9}\).

What is the distributive property of rational numbers?

The distributive property states that for any rational numbers \(a, b,\) and \(c\), \(a \times (b + c) = (a \times b) + (a \times c)\). It helps in simplifying expressions by distributing multiplication over addition or subtraction.

How can these NCERT solutions help me prepare for exams?

These solutions provide clear, step-by-step explanations for each problem in Chapter 1. By understanding the methods and practicing the problems, you can strengthen your grasp of rational numbers and improve your problem-solving skills for exams.

What does it mean to verify that -(-x) = x?

Verifying \(-(-x) = x\) means showing that applying the negative sign twice to a number returns the original number. For example, if \(x = \frac{11}{15}\), then \(-(-x) = -\left(-\frac{11}{15}\right) = \frac{11}{15}\), which is equal to \(x\).

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