Class 8 Maths Chapter 1 Rational Numbers Worksheet

The Class 8 Maths Chapter 1 Rational Numbers Worksheet is designed as per the latest CBSE and NCERT syllabus. Rational numbers are one of the most important topics for Class 8, forming the foundation for algebra, equations, and higher mathematics.

On this page, you will find a worksheet on rational numbers for Class 8 with step-by-step answers. Students can download the rational numbers Class 8 worksheet with answers PDF and practice offline. This class 8 maths rational numbers worksheet includes NCERT-based questions, word problems, and HOTS (Higher Order Thinking Skills).

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• NCERT Aligned: Every worksheet on rational numbers for Class 8 is based on NCERT and CBSE guidelines.
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Related Resources 

• NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers
• Class 8 Maths Sample Papers with Solutions
• Important Questions for Class 8 Maths Chapter 1
• Worksheet on Linear Equations in One Variable – Class 8

Class 8 Maths – Chapter 1: Rational Numbers

Q1. Convert to p/q form (denominator ≠ 0): (i) −0.375, (ii) 2.125, (iii) 0.&overline;3

Step-by-step solution

1. (i) −0.375
   1. Write as a fraction: −0.375 = −375/1000.
   2. Simplify by dividing numerator and denominator by 125: −375 ÷ 125 = −3, 1000 ÷ 125 = 8.
   3. So, −0.375 = −3/8.
2. (ii) 2.125
   1. Write as an improper fraction: 2.125 = 2125/1000.
   2. Simplify by 125: 2125 ÷ 125 = 17, 1000 ÷ 125 = 8.
   3. So, 2.125 = 17/8.
3. (iii) 0.&overline;3
   1. Let x = 0.&overline;3.
   2. Multiply by 10: 10x = 3.&overline;3.
   3. Subtract: (10x − x) = 3.&overline;3 − 0.&overline;3 = 3.
   4. Thus 9x = 3 &Rightarrow x = 1/3.

Answers: (i) −3/8, (ii) 17/8, (iii) 1/3.

Q2. Find additive and multiplicative inverses of (i) −7/9 and (ii) 0

Step-by-step solution

1. (i) −7/9
   • Additive inverse is the number which sums to 0: +7/9.
   • Multiplicative inverse is the reciprocal: −9/7.
2. (ii) 0
   • Additive inverse of 0 is 0.
   • Multiplicative inverse does not exist (division by zero is undefined).

Q3. Verify associativity of addition for 1/2, −3/4, 5/8

Step-by-step solution

1. Compute (1/2 + (−3/4)) + 5/8:
   1. 1/2 − 3/4 = 2/4 − 3/4 = −1/4.
   2. −1/4 + 5/8 = −2/8 + 5/8 = 3/8.
2. Compute 1/2 + (−3/4 + 5/8):
   1. −3/4 + 5/8 = −6/8 + 5/8 = −1/8.
   2. 1/2 − 1/8 = 4/8 − 1/8 = 3/8.

Both give 3/8. Hence, addition is associative on rational numbers.

Q4. Verify commutativity of multiplication for −2/3 and 9/4

Step-by-step solution

1. (−2/3) × (9/4) = −18/12 = −3/2.
2. (9/4) × (−2/3) = −18/12 = −3/2.

Both products are equal; multiplication is commutative on rational numbers.

Q5. Find five rational numbers between 1/3 and 1/2

Step-by-step solution

1. Equalize denominators to create room between the fractions: use denominator 60.
2. 1/3 = 20/60 and 1/2 = 30/60.
3. Numbers between are 21/60, 22/60, 23/60, 24/60, 25/60 (you may simplify).

One possible set: 21/60, 22/60, 23/60, 24/60, 25/60.

Q6. Simplify: (−3/5 + 7/10) − (2/15 − 1/6)

Step-by-step solution

1. First bracket: LCM(5,10) = 10 &Rightarrow −6/10 + 7/10 = 1/10.
2. Second bracket: LCM(15,6) = 30 &Rightarrow 4/30 − 5/30 = −1/30.
3. Expression: 1/10 − (−1/30) = 1/10 + 1/30 = 3/30 + 1/30 = 4/30 = 2/15.

Answer: 2/15.

Q7. Compute: (−4/9) ÷ (2/3) × (3/8)

Step-by-step solution

1. Division first: (−4/9) × (3/2) = −12/18 = −2/3.
2. Multiply by 3/8: (−2/3) × (3/8) = −6/24 = −1/4.

Answer: −1/4.

Q8. Word problem: A tank is filled by one pipe 3/5 in an hour and emptied by another 1/4 in an hour. What fraction is filled in 1 hour?

Step-by-step solution

1. Net effect = 3/5 − 1/4.
2. LCM(5,4) = 20; hence 12/20 − 5/20 = 7/20.

Answer: 7/20 of the tank is filled in 1 hour.

Q9. Number line: Place −7/4 accurately

Explanation

1. Convert to mixed number: −7/4 = −1 3/4.
2. It lies between −2 and −1, at three-quarters of a unit to the right of −2 (or one-quarter to the left of −1).

Q10. Solve for x: x/5 + 1/2 = 7/10

Step-by-step solution

1. Move 1/2 to the right: x/5 = 7/10 − 1/2 = 7/10 − 5/10 = 2/10 = 1/5.
2. Hence x/5 = 1/5 &Rightarrow x = 1.

Answer: x = 1.

Q11. Evaluate: −2/7 + 3 + 5/14

Step-by-step solution

1. Write 3 with denominator 14: 3 = 42/14.
2. −2/7 = −4/14; so sum is (−4/14) + (42/14) + (5/14) = 43/14.

Answer: 43/14 (or 3 1/14).

Q12. Use distributive property to compute: (5/6) × (−3/5 + 1/2)

Method 1: Simplify inside first

1. LCM(5,2) = 10; inside: −6/10 + 5/10 = −1/10.
2. (5/6) × (−1/10) = −5/60 = −1/12.

Method 2: Distribute first

1. (5/6) × (−3/5) = −1/2; (5/6) × (1/2) = 5/12.
2. Sum: −1/2 + 5/12 = −6/12 + 5/12 = −1/12.

Answer: −1/12.

Q13. Which is greater: −5/8 or −3/10?

Step-by-step solution

1. Use common denominator 40: −5/8 = −25/40, −3/10 = −12/40.
2. Among negatives, the number closer to 0 is greater: −12/40 > −25/40.

Answer: −3/10 is greater.

Q14. Evaluate: (−5/12) × (9/10) + (7/8) ÷ (−14/3)

Step-by-step solution

1. First product: (−5/12) × (9/10) = −45/120 = −3/8.
2. Division: (7/8) ÷ (−14/3) = (7/8) × (3/−14) = 21/−112 = −3/16.
3. Sum: −3/8 + (−3/16) = −6/16 − 3/16 = −9/16.

Answer: −9/16.

Q15. Show that (ab)−1 = a−1 b−1 for a = −2/3 and b = 9/5

Step-by-step solution

1. Compute ab: (−2/3) × (9/5) = −18/15 = −6/5.
2. Find (ab)−1: reciprocal of −6/5 is −5/6.
3. Compute a−1 and b−1:
   • a−1 = reciprocal of −2/3 = −3/2.
   • b−1 = reciprocal of 9/5 = 5/9.
4. Multiply a−1 b−1: (−3/2) × (5/9) = −15/18 = −5/6.
5. Since (ab)−1 = −5/6 and a−1 b−1 = −5/6, the identity holds.

Verified: (ab)−1 = a−1 b−1.

CBSE Class 8 Maths Chapter 1 – Rational Numbers Worksheet

Q1. Express the following decimals as rational numbers in the form p/q, q ≠ 0 (3 marks)

1. a) −0.6 b) 2.75 c) 0.8̄

Answer

• a) −0.6 = −6/10 = −3/5(1 mark)
• b) 2.75 = 275/100 = 11/4(1 mark)
• c) 0.8̄ = 8/9(1 mark)

Q2. Find the additive and multiplicative inverses of the following rational numbers (2 marks)

1. a) −13/7 b) 5

Answer

• a) Additive inverse: 13/7; Multiplicative inverse: −7/13(1 mark)
• b) Additive inverse: −5; Multiplicative inverse: 1/5(1 mark)

Q3. Verify closure property of addition for rational numbers −2/5 and 3/10 (2 marks)

Answer

−2/5 + 3/10 = −4/10 + 3/10 = −1/10 (a rational number). Therefore, rational numbers are closed under addition. (2 marks)

Q4. Simplify (3 marks)

−7/12 + 5/18 − 1/9

Answer

1. LCM of 12, 18, 9 is 36. (1 mark)
2. Convert: −7/12 = −21/36, 5/18 = 10/36, −1/9 = −4/36. (1 mark)
3. Sum: (−21 + 10 − 4)/36 = −15/36 = −5/12. (1 mark)

Q5. Show with an example that rational numbers are commutative under multiplication (2 marks)

Answer

Take 2/3 and −5/4:

• 2/3 × (−5/4) = −10/12 = −5/6. (1 mark)
• (−5/4) × 2/3 = −10/12 = −5/6. (1 mark)

Both products are equal; multiplication is commutative.

Q6. Write any three rational numbers between 1/4 and 1/2 (2 marks)

Answer

1. Convert to hundredths: 1/4 = 25/100 and 1/2 = 50/100. (1 mark)
2. Examples between them: 26/100, 27/100, 49/100 (any three valid rationals). (1 mark)

Q7. Find the multiplicative inverse (2 marks)

1. a) −9/11 b) 7

Answer

• a) −11/9(1 mark)
• b) 1/7(1 mark)

Q8. Solve for x (3 marks)

x/3 + 1/2 = 5/6

Answer

1. x/3 = 5/6 − 1/2 = (5 − 3)/6 = 2/6 = 1/3. (2 marks)
2. x = 1. (1 mark)

Answer: x = 1.

Q9. Compare −3/8 and −2/7 (2 marks)

Answer

1. LCM of 8 and 7 is 56. So −3/8 = −21/56 and −2/7 = −16/56. (1 mark)
2. Since −16/56 > −21/56, therefore −2/7 is greater. (1 mark)

Q10. A tank is filled 2/3 by one pipe in 1 hour and emptied 1/6 by another pipe in the same time. Find the fraction filled in 1 hour (3 marks)

Answer

1. Net fill = 2/3 − 1/6. (1 mark)
2. LCM of 3 and 6 is 6. So 2/3 = 4/6; 4/6 − 1/6 = 3/6 = 1/2. (2 marks)

Answer: 1/2 of the tank is filled in 1 hour.

Q11. Represent the following on a number line (2 marks)

1. a) −5/6 b) 7/4

Answer

• a) −5/6 lies between −1 and 0, closer to −1. (1 mark)
• b) 7/4 = 1.75 lies between 1 and 2. (1 mark)

Q12. Simplify (2 marks)

3/7 ÷ (−6/14)

Answer

1. 3/7 × 14/−6 = 42/−42. (1 mark)
2. = −1. (1 mark)

Q13. If x = −2/5 and y = 3/10, find (x + y) and (x × y) (2 marks)

Answer

• x + y = −2/5 + 3/10 = −4/10 + 3/10 = −1/10. (1 mark)
• x × y = (−2/5) × (3/10) = −6/50 = −3/25. (1 mark)

Q14. Verify distributive property (3 marks)

2/3 × (1/2 + 3/4) = 2/3 × 1/2 + 2/3 × 3/4

Answer

1. LHS: 2/3 × 5/4 = 10/12 = 5/6. (1 mark)
2. RHS: (2/3 × 1/2) + (2/3 × 3/4) = 1/3 + 1/2 = 2/6 + 3/6 = 5/6. (2 marks)

Therefore, LHS = RHS. Distributive property is verified.

Q15. Which is greater: −5/9 or −4/7? (2 marks)

Answer

1. LCM of 9 and 7 is 63. So −5/9 = −35/63 and −4/7 = −36/63. (1 mark)
2. Since −35/63 > −36/63, therefore −5/9 is greater. (1 mark)

Marking Scheme Summary

• 2-mark questions: Q2, Q3, Q5, Q6, Q7, Q9, Q11, Q12, Q13, Q15
• 3-mark questions: Q1, Q4, Q8, Q10, Q14