NCERT Solution for Class 8 Rational Number

A rational number is defined as any number that can be expressed in the form p/q, where q ≠ 0. In simple terms, it represents a fraction with a non-zero denominator, making it one of the fundamental concepts introduced in Chapter 1.

These NCERT solutions act as a reliable resource for students to clear doubts, revise concepts, and practice exercise questions effectively. Each solution is presented step by step to give a deeper understanding of the sub-topics, helping students strengthen their knowledge of rational numbers.

Mastering this chapter is not only important for scoring well in Class 8 Maths but also serves as a strong base for future classes and board examinations.

NCERT Solutions for Class 8 Maths Chapter 1 PDF Download

The NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers cover all the important concepts of this chapter in an easy-to-understand way. Students will learn about the closure property, commutative property, associative property, distributive property, additive inverse, and multiplicative inverse.

These properties of rational numbers are explained with solved examples and exercises. The solutions also show how to check whether these properties are true for different arithmetic operations like addition, subtraction, multiplication, and division of rational numbers.

By practicing these NCERT solutions, students can clear their doubts, understand what is a rational number, and strengthen their basics for higher classes. A free Rational Numbers Worksheet for Class 8 PDF download is also available for revision and offline study.

NCERT Solutions for Class 8 Maths Chapter 1 : All Exercises

NCERT Class 8 Maths Chapter 1: Rational Numbers - Step-by-Step Solutions

Question 1: Using appropriate properties, find:

(a) 2/5 + 3/5

Step 1: Since both fractions have the same denominator, we can add directly. Step 2: Add the numerators: 2 + 3 = 5 Step 3: Keep the same denominator: 5 Step 4: Result = 5/5 = 1

Answer: 1

(b) 7/9 × 3/7

Step 1: Multiply numerators: 7 × 3 = 21 Step 2: Multiply denominators: 9 × 7 = 63 Step 3: Result = 21/63 Step 4: Simplify by dividing both numerator and denominator by their GCD (21) Step 5: 21 ÷ 21 = 1, 63 ÷ 21 = 3

Answer: 1/3

Question 2: Write the additive inverse of each of the following:

(i) 2/8

Step 1: The additive inverse of any number x is -x Step 2: First simplify 2/8 = 1/4 Step 3: Additive inverse = -1/4

Answer: -1/4

(ii) -5/9

Step 1: The additive inverse of -5/9 is -(-5/9) Step 2: This equals +5/9

Answer: 5/9

(iii) -6/-5

Step 1: First simplify -6/-5 = 6/5 (negative divided by negative is positive) Step 2: Additive inverse = -6/5

Answer: -6/5

(iv) 2/-9

Step 1: First simplify 2/-9 = -2/9 Step 2: Additive inverse = -(-2/9) = 2/9

Answer: 2/9

(v) 19/-6

Step 1: First simplify 19/-6 = -19/6 Step 2: Additive inverse = -(-19/6) = 19/6

Answer: 19/6

Question 3: Verify that -(-x) = x for:

(i) x = 11/5

Step 1: Find -x = -(11/5) = -11/5 Step 2: Find -(-x) = -(-11/5) = 11/5 Step 3: Compare with original x = 11/5

Verification: -(-x) = 11/5 = x 

(ii) x = -13/17

Step 1: Find -x = -(-13/17) = 13/17 Step 2: Find -(-x) = -(13/17) = -13/17 Step 3: Compare with original x = -13/17

Verification: -(-x) = -13/17 = x 

Question 4: Fill in the blanks:

(i) Zero has no reciprocal.

(ii) The numbers 1 and -1 are their own reciprocals.

(iii) The reciprocal of -5 is -1/5.

(iv) Reciprocal of 1/x, where x ≠ 0, is x.

(v) The product of two rational numbers is always a rational number.

(vi) The reciprocal of a positive rational number is positive.

Question 5: State whether the following statements are true or false:

(a) Every integer is a rational number.

Answer: True Explanation: Every integer n can be written as n/1, which is in the form p/q where q ≠ 0.

(b) Zero is a positive rational number.

Answer: False Correction: Zero is neither positive nor negative. It is a neutral rational number.

(c) The additive inverse of a rational number is always negative.

Answer: False Correction: The additive inverse of a positive rational number is negative, but the additive inverse of a negative rational number is positive.

Question 6: Multiply 2/3 by the reciprocal of 3/4

Step 1: Find the reciprocal of 3/4 = 4/3 Step 2: Multiply 2/3 × 4/3 Step 3: Multiply numerators: 2 × 4 = 8 Step 4: Multiply denominators: 3 × 3 = 9 Step 5: Result = 8/9

Answer: 8/9

Question 7: Find the multiplicative inverse of:

(i) -13/19

Step 1: The multiplicative inverse (reciprocal) is obtained by flipping the fraction Step 2: Multiplicative inverse = -19/13

Answer: -19/13

(ii) -7

Step 1: Write -7 as a fraction: -7/1 Step 2: Multiplicative inverse = -1/7

Answer: -1/7

(iii) 0

Step 1: Zero has no multiplicative inverse Step 2: Division by zero is undefined

Answer: Does not exist

(iv) 1

Step 1: The multiplicative inverse of 1 is 1 Step 2: Because 1 × 1 = 1

Answer: 1

(v) -1

Step 1: The multiplicative inverse of -1 is -1 Step 2: Because (-1) × (-1) = 1

Answer: -1

Question 8: Is the sum of two rational numbers always a rational number?

Answer: Yes

Justification: Let's take two rational numbers a/b and c/d, where b ≠ 0 and d ≠ 0.

Step 1: Sum = a/b + c/d Step 2: Find common denominator = (ad + bc)/(bd) Step 3: Since a, b, c, d are integers and bd ≠ 0, the sum is also a rational number

Example: 3/4 + 1/6 = (3×6 + 1×4)/(4×6) = (18 + 4)/24 = 22/24 = 11/12

The result 11/12 is also a rational number.

Question 9: Represent the following rational numbers on the number line:

(a) -3/4

Step 1: -3/4 = -0.75 Step 2: Divide the unit length between -1 and 0 into 4 equal parts Step 3: Mark the point that is 3 parts from 0 towards -1 Step 4: This point represents -3/4

(b) 5/8

Step 1: 5/8 = 0.625 Step 2: Divide the unit length between 0 and 1 into 8 equal parts Step 3: Mark the point that is 5 parts from 0 towards 1 Step 4: This point represents 5/8

Question 10: Find five rational numbers between -3/4 and 1/4

Step 1: Convert to decimal form: -3/4 = -0.75 and 1/4 = 0.25 Step 2: Convert to equivalent fractions with common denominator: -3/4 = -12/16 and 1/4 = 4/16 Step 3: Five rational numbers between -12/16 and 4/16 are:

1. -11/16
2. -10/16 = -5/8
3. -8/16 = -1/2
4. -4/16 = -1/4
5. 0/16 = 0

Answer: -11/16, -5/8, -1/2, -1/4, 0

Question 11: Simplify and express in simplest form:

(i) 2/3 + (-5/6)

Step 1: 2/3 + (-5/6) = 2/3 - 5/6 Step 2: Find LCM of 3 and 6 = 6 Step 3: Convert to equivalent fractions: 4/6 - 5/6 Step 4: Subtract: (4-5)/6 = -1/6

Answer: -1/6

(ii) 5/8 - (-7/8)

Step 1: 5/8 - (-7/8) = 5/8 + 7/8 Step 2: Add numerators: (5+7)/8 = 12/8 Step 3: Simplify: 12/8 = 3/2

Answer: 3/2

(iii) -2/5 × 3 1/4

Step 1: Convert mixed number: 3 1/4 = 13/4 Step 2: Multiply: -2/5 × 13/4 Step 3: Multiply numerators and denominators: (-2×13)/(5×4) = -26/20 Step 4: Simplify: -26/20 = -13/10

Answer: -13/10

Question 12: Show that subtraction is not commutative for rational numbers

Definition: Subtraction is commutative if a - b = b - a for all rational numbers a and b.

Example to disprove: Let a = 3/4 and b = 1/2

Step 1: Calculate a - b = 3/4 - 1/2 = 3/4 - 2/4 = 1/4 Step 2: Calculate b - a = 1/2 - 3/4 = 2/4 - 3/4 = -1/4 Step 3: Compare: 1/4 ≠ -1/4

Conclusion: Since a - b ≠ b - a, subtraction is not commutative for rational numbers.

Question 13: Find one rational number between 2/7 and 3/7

Method 1 (Average): Step 1: Add the two numbers: 2/7 + 3/7 = 5/7 Step 2: Divide by 2: (5/7) ÷ 2 = 5/7 × 1/2 = 5/14 Step 3: Verify: 2/7 = 4/14 and 3/7 = 6/14 Step 4: Check: 4/14 < 5/14 < 6/14 

Answer: 5/14

Question 14: If 1/7 × x = 2, find the value of x

Step 1: Given equation: (1/7) × x = 2 Step 2: Multiply both sides by 7: x = 2 × 7 Step 3: Calculate: x = 14

Verification: 1/7 × 14 = 14/7 = 2 

Answer: x = 14

Question 15: Explain why zero has no reciprocal

Step 1: The reciprocal (multiplicative inverse) of a number x is a number y such that x × y = 1.

Step 2: If zero had a reciprocal, say k, then 0 × k = 1.

Step 3: But we know that 0 × k = 0 for any number k.

Step 4: Since 0 ≠ 1, there is no number k that satisfies 0 × k = 1.

Step 5: Also, finding the reciprocal involves division: reciprocal of x = 1/x.

Step 6: For zero, this would be 1/0, which is undefined in mathematics.

Conclusion: Zero has no reciprocal because division by zero is undefined and no number when multiplied by zero gives 1.