Class 9 Maths Chapter 3 Important Questions (Ganita Manjari I): The World of Numbers - 2026-27

The NCERT Class 9 Maths Chapter 3 Important Questions PDF helps students practise different types of questions from The World of Numbers. It includes concept-based, calculation-based and reasoning-based questions to support revision before school tests and annual examinations.

Class 9 Maths Chapter 3 The World of Numbers MCQ

Q1. Which of the following numbers is a natural number but not a negative integer?

(a) −8

(b) 0

(c) 15

(d) −3

Answer: (c) 15

Explanation: Natural numbers begin from 1 and include 1, 2, 3, 4, and so on. Since 15 is a positive counting number, it belongs to the natural numbers.

Q2. Which of the following operations always gives zero for any integer a?

(a) a + 1

(b) a − a

(c) a × 1

(d) a ÷ 1

Answer: (b) a − a

Explanation: Subtracting any integer from itself gives zero. For example, 12 − 12 = 0 and −7 − (−7) = 0.

Q3. What is the result of (−12) × 5 + 18?

(a) −78

(b) −42

(c) 42

(d) 78

Answer: (b) −42

Explanation:

(−12) × 5 + 18

= −60 + 18

= −42

Q4. Which of the following numbers is irrational?

(a) 0.75

(b) −5/8

(c) √13

(d) 0.4444...

Answer: (c) √13

Explanation: √13 is irrational because 13 is not a perfect square. Its square root cannot be expressed as a ratio of two integers.

Q5. Which fraction has a terminating decimal expansion?

(a) 7/12

(b) 9/40

(c) 5/21

(d) 11/18

Answer: (b) 9/40

Explanation: The denominator 40 has only the prime factors 2 and 5.

40 = 2³ × 5

Therefore, 9/40 = 0.225, which is a terminating decimal.

Q6. What is the absolute value of −23/5?

(a) −23/5

(b) 0

(c) 23/5

(d) 5/23

Answer: (c) 23/5

Explanation: Absolute value represents the distance of a number from zero on the number line. Therefore, |−23/5| = 23/5.

Q7. How many rational numbers exist between 3/7 and 4/7?

(a) None

(b) Only one

(c) Exactly seven

(d) Infinitely many

Answer: (d) Infinitely many

Explanation: Rational numbers are dense. Between any two distinct rational numbers, infinitely many rational numbers exist.

Class 9 Maths Chapter 3 Very Short Answer Questions

Q8. What is one-to-one correspondence in counting?

Answer: One-to-one correspondence means matching each object in one collection with exactly one object in another collection.

For example, matching one pebble to each animal helps count the animals without using written numbers.

Q9. What is the additive identity of rational numbers?

Answer: Zero is the additive identity of rational numbers because adding zero to any rational number leaves its value unchanged.

For example:

7/9 + 0 = 7/9

Q10. Define a rational number and give two examples.

Answer: A rational number is a number that can be expressed as p/q, where p and q are integers and q ≠ 0.

Examples: 4/7 and −9/5.

Q11. Write the additive inverse of −11/6.

Answer: The additive inverse of −11/6 is 11/6.

This is because:

−11/6 + 11/6 = 0

Q12. Is √81 rational or irrational? Give a reason.

Answer: √81 is rational because √81 = 9, and 9 can be expressed as 9/1.

Q13. What is the difference between terminating and non-terminating repeating decimals?

Answer: A terminating decimal ends after a finite number of digits, such as 0.625.

A non-terminating repeating decimal continues indefinitely with a fixed digit or group of digits repeating, such as 0.272727…

Class 9 Chapter 3 Short Answer Questions with Explanations

Q14. A trader has ₹750 and pays ₹920 for goods. He later receives ₹460 from a customer. Represent these transactions using integers and calculate his final balance.

Answer:

Initial amount = ₹750

Payment = −₹920

Amount received = +₹460

Step 1: Write the expression.

750 − 920 + 460

Step 2: Simplify.

750 − 920 = −170

−170 + 460 = 290

Final Answer: The trader has a final balance of ₹290.

Q15. Simplify the rational number expression (−3/4) + (5/6) − (−1/3).

Answer:

Step 1: Write the expression.

−3/4 + 5/6 − (−1/3)

Step 2: Change subtraction of a negative number into addition.

= −3/4 + 5/6 + 1/3

Step 3: Find the LCM of 4, 6 and 3.

LCM = 12

Step 4: Convert the fractions.

−3/4 = −9/12

5/6 = 10/12

1/3 = 4/12

Step 5: Add the numerators.

= (−9 + 10 + 4)/12

= 5/12

Final Answer: 5/12

Q16. Explain how to represent the rational number −9/4 on a number line.

Answer:

Step 1: Convert the improper fraction into a mixed number.

−9/4 = −2¼

Step 2: Identify the location.

The number −2¼ lies between −3 and −2.

Step 3: Divide the interval between −3 and −2 into four equal parts.

Step 4: Starting from −2, move one-quarter unit towards the left.

The point reached represents −9/4.

Final Answer: −9/4 = −2.25, which lies between −3 and −2 on the number line.

Q17. Find four distinct rational numbers between 3/8 and 5/8 using equivalent fractions.

Answer:

Step 1: Convert both fractions to a common denominator.

3/8 = 15/40

5/8 = 25/40

Step 2: Select any four fractions between them.

For example:

16/40, 17/40, 18/40, 19/40

Step 3: Verify.

15/40 is smaller than all four selected fractions.

25/40 is greater than all four selected fractions.

Final Answer: Four rational numbers between 3/8 and 5/8 are 16/40, 17/40, 18/40 and 19/40.

Q18. Without performing long division, determine whether 17/80 and 11/42 have terminating or non-terminating repeating decimal expansions.

Answer:

A rational number in its lowest form has a terminating decimal expansion when its denominator contains only the prime factors 2 and 5.

Step 1: Examine 17/80.

80 = 2⁴ × 5

The denominator contains only 2 and 5.

Therefore, 17/80 has a terminating decimal expansion.

17/80 = 0.2125

Step 2: Examine 11/42.

42 = 2 × 3 × 7

The denominator contains the prime factors 3 and 7.

Therefore, 11/42 has a non-terminating repeating decimal expansion.

Final Answer:

• 17/80 has a terminating decimal expansion.

• 11/42 has a non-terminating repeating decimal expansion.

Q19. Classify the following numbers as rational or irrational and explain your answers.

(i) √100 (ii) √17 (iii) 0.181818... (iv) π (v) −13/7

Answer:

Final Answer: √100, 0.181818... and −13/7 are rational numbers, while √17 and π are irrational numbers.

Class 9 Maths Chapter 3 Important Long Answer Questions

Q20. Prove that √5 is an irrational number.

Answer:

Step 1: Assume that √5 is a rational number.

Therefore, √5 = p/q, where p and q are co-prime integers and q ≠ 0.

Step 2: Squaring both sides:

5 = p²/q²

p² = 5q²

This means p² is divisible by 5. Therefore, p is also divisible by 5.

Step 3: Let p = 5k, where k is an integer.

Substituting:

(5k)² = 5q²

25k² = 5q²

q² = 5k²

Therefore, q is also divisible by 5.

Step 4: Since both p and q are divisible by 5, they have a common factor of 5.

This contradicts our assumption that p and q are co-prime.

Hence, √5 is an irrational number.

Q21. Explain how to represent √2 on the number line using a ruler and compass.

Answer:

Step 1: Draw a number line and mark O as 0 and A as 1.

Step 2: Draw a perpendicular line at point A.

Step 3: Mark point B on the perpendicular such that AB = 1 unit.

Step 4: Join points O and B to form a right-angled triangle.

Using the Pythagoras theorem:

OB² = OA² + AB²

OB² = 1² + 1²

OB² = 2

OB = √2 units

Step 5: With O as the centre and OB as the radius, draw an arc using a compass.

Step 6: Let the arc intersect the positive number line at point P.

Therefore, point P represents √2 on the number line.

Q22. A trader has a debt of ₹850. He earns ₹1,200 and later suffers a loss of ₹450. Find his final financial position.

Answer:

Given:

Initial debt = -₹850

Profit earned = +₹1,200

Loss suffered = -₹450

Step 1: Write the financial transactions using integers.

Final amount = -850 + 1200 - 450

Step 2: Add the first two amounts.

-850 + 1200 = 350

Step 3: Subtract the loss.

350 - 450 = -100

The negative sign shows that the trader still owes money.

Therefore, the trader has a final debt of ₹100.

Q23. Explain the different types of decimal expansions with examples.

Answer:

Decimal expansions can be classified into three main types.

1. Terminating Decimal Expansion

A terminating decimal has a finite number of digits after the decimal point.

Example:

3/8 = 0.375

The decimal expansion ends after three decimal places.

2. Non-Terminating Repeating Decimal Expansion

A non-terminating repeating decimal has one or more digits that repeat endlessly.

Example:

1/3 = 0.333...

Here, the digit 3 repeats continuously.

Another example:

5/11 = 0.454545...

Here, the digits 45 repeat continuously.

3. Non-Terminating Non-Repeating Decimal Expansion

A non-terminating non-repeating decimal continues endlessly without repeating a fixed pattern.

Example:

√2 = 1.41421356...

π = 3.14159265...

These are irrational numbers.

Conclusion: Rational numbers have terminating or non-terminating repeating decimal expansions, whereas irrational numbers have non-terminating non-repeating decimal expansions.

How to Prepare for Chapter 3 Maths Class 9 Important Questions for Exams?

The World of Numbers includes numerical calculations, logical proofs and number line constructions. Students should practise each question type separately to understand the chapter and avoid common mistakes.

1. Understand the Classification of Numbers

Learn how natural numbers, integers, rational numbers, irrational numbers and real numbers are related. Practise classifying numbers based on their properties and decimal expansions.

2. Strengthen Rational Number Calculations

Revise addition, subtraction, multiplication and division of positive and negative fractions. Pay attention to negative signs, common denominators and simplification.

3. Practise Number Line Representations

Learn to locate positive and negative rational numbers on a number line. Practise dividing intervals into equal parts and identifying the correct position of a given fraction.

4. Focus on Irrational Number Proofs and Constructions

Understand the reasoning used to prove that √2 is irrational. Practise constructing square root lengths using the Baudhāyana–Pythagoras theorem, a ruler and a compass.

5. Revise Decimal Expansions Carefully

Learn how to identify terminating, repeating and non-repeating decimals. Practise converting decimals into fractions and predicting decimal types using the prime factors of denominators.

6. Attempt Reasoning-Based and HOTS Questions

Solve questions involving the density of rational numbers, absolute values, cyclic decimal patterns and mathematical proofs. Try explaining why a particular method works rather than memorising only the steps.

Following these strategies will help students prepare for important questions for class 9 maths chapter 3 and improve their understanding of number systems.

How Vedantu Helps Students Prepare for Class 9 Maths Chapter 3 Important Questions

Vedantu's Chapter 3 Maths Class 9 Important Questions provide organised practice for understanding number systems and their mathematical properties. The questions help students revise textbook concepts and improve their ability to solve different problem types.

• Concept-Based Learning: Understand the properties of natural, integer, rational, irrational and real numbers.

• Step-by-Step Explanations: Follow clear methods for calculations, decimal conversions and mathematical proofs.

• Exam-Oriented Questions: Practise MCQs, very short, short and long answer questions.

• Mathematical Reasoning: Develop logical thinking through irrationality proofs, number properties and HOTS problems.

• Self-Assessment: Identify calculation errors and concepts that need more practice before examinations.

Students can use these important questions with their Ganita Manjari textbook to revise the chapter and strengthen their preparation for Class 9 Maths assessments.

Important Study Materials for Class 9 Maths Chapter 3 The World of Numbers

CBSE Class 9 Maths Chapter-wise Important Questions

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