NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3: The World of Numbers by PW

Class 9 Maths Chapter 3 can be challenging for children because it covers rational and irrational numbers, fractions, number line representation and decimal expansions. Understanding number properties and converting recurring decimals into fractions can become confusing, especially when children are practising different types of questions for their Maths exams.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3 by PW make revision easier by providing answers to important questions from The World of Numbers. Children can use these solutions to check their answers, understand mathematical concepts clearly and practise lecture-wise questions. Regular practice can also help improve calculation skills and confidence.

Class 9 Maths Chapter 3: The World of Numbers Solutions

The questions are organised by lecture number to help you practise the concepts covered in each set. Each lecture section includes its questions and answers, followed by a separate PDF resource for revision.

Lecture 01: Recalling Numbers and Rational Numbers

Question 1. Prove that the following rational numbers are equal.

(i) 2/3 and 4/6

(ii) 5/4 and 10/8

(iii) 3/5 and 6/10

(iv) 3/3 and 3/1

Answer:

(i) 2/3 = 4/6, since multiplying the numerator and denominator of 2/3 by 2 gives 4/6.

(ii) 5/4 = 10/8, since multiplying the numerator and denominator of 5/4 by 2 gives 10/8.

(iii) 3/5 = 6/10, since multiplying the numerator and denominator of 3/5 by 2 gives 6/10.

(iv) 3/3 = 1, whereas 3/1 = 3. Therefore, these two rational numbers are not equal.

Question 2. Find the sum.

(i) 2/5 + 3/10

(ii) 7/12 + 5/8

(iii) −4/7 + 3/14

Answer:

(i) 2/5 + 3/10 = 4/10 + 3/10 = 7/10

(ii) 7/12 + 5/8 = 14/24 + 15/24 = 29/24

(iii) −4/7 + 3/14 = −8/14 + 3/14 = −5/14

Question 3. Find the difference.

(i) 5/6 − 1/4

(ii) 11/8 − 3/4

(iii) 7/9 − (−2/3)

Answer:

(i) 5/6 − 1/4 = 10/12 − 3/12 = 7/12

(ii) 11/8 − 3/4 = 11/8 − 6/8 = 5/8

(iii) 7/9 − (−2/3) = 7/9 + 6/9 = 13/9

Question 4. Find the product.

(i) 2/3 × 3/10

(ii) 7/11 × 5/8

(iii) −4/7 × 6/14

Answer:

(i) 2/3 × 3/10 = 6/30 = 1/5

(ii) 7/11 × 5/8 = 35/88

(iii) −4/7 × 6/14 = −24/98 = −12/49

Question 5. Find the quotient.

(i) 2/3 ÷ 3/10

(ii) 7/11 ÷ 5/8

(iii) 4/7 ÷ 6/14

Answer:

(i) 2/3 ÷ 3/10 = 2/3 × 10/3 = 20/9

(ii) 7/11 ÷ 5/8 = 7/11 × 8/5 = 56/55

(iii) 4/7 ÷ 6/14 = 4/7 × 14/6 = 4/3

Question 6. State whether the following statements are True or False.

(i) Every integer is rational.

(ii) Every rational number is an integer.

(iii) Every whole number is rational.

(iv) Every natural number is rational.

(v) Every integer is a natural number.

Answer:

The World of Numbers 01: Recalling Numbers and Rational Numbers PDF

The World of Numbers 01 focuses on recalling number concepts, equivalent rational numbers and arithmetic operations involving fractions. Practise the exercises to revise addition, subtraction, multiplication, division and number properties.

Lecture 02: Plotting Rational Numbers on the Number Line

Question 1. What is the simplified value of (7/9) × (6/7 − 3/4) using the distributive property?

A) 1/12

B) 1/6

C) 5/12

D) 1/4

Answer: A) 1/12

Question 2. Solve for x: 2x + 5/3 = 9/2.

A) x = 13/12

B) x = 17/12

C) x = 17/6

D) x = 7/12

Answer: B) x = 17/12

Question 3. Where does the proper fraction 1/2 lie on the number line?

A) Between −1 and 0

B) Between 1 and 2

C) Between 0 and 1

D) Beyond 2

Answer: C) Between 0 and 1

Question 4. When expressing the improper fraction 5/4 on a number line, between which two whole numbers does it lie?

A) 0 and 1

B) 1 and 2

C) 2 and 3

D) 3 and 4

Answer: B) 1 and 2

Question 5. What is one rational number exactly halfway between 1 and 3?

A) 1.5

B) 2

C) 2.5

D) 3.5

Answer: B) 2

The World of Numbers 02: Plotting Rational Numbers on the Number Line PDF

The World of Numbers 02 explains how to represent rational numbers on a number line and locate fractions between whole numbers. Use the PDF to review the lecture concepts and practise related questions.

Lecture 03: Finding Rational Numbers Between Two Rational Numbers and Decimal Expansions

Question 1. How many rational numbers exist between 1/2 and 3/4?

A) 1

B) 10

C) 100

D) Infinitely many

Answer: D) Infinitely many

Question 2. What is the decimal expansion of 3/8?

A) 0.35

B) 0.375

C) 0.625

D) 0.125

Answer: B) 0.375

Question 3. What is the decimal expansion of −2/11?

A) −0.181818... or −0.18

B) −0.272727... or −0.27

C) −0.090909... or −0.09

D) −0.151515... or −0.15

Answer: A) −0.181818... or −0.18

Question 4. The decimal expansion of 13/6 is 2.16. What type of decimal expansion is this?

A) Terminating

B) Non-terminating repeating (NTR)

C) Non-terminating non-repeating (NTNR)

D) Irrational

Answer: B) Non-terminating repeating (NTR)

Question 5. The decimal expansion of any rational number is always:

A) Terminating or non-terminating repeating

B) Terminating or non-terminating non-repeating

C) Only non-terminating non-repeating

D) Never terminating

Answer: A) Terminating or non-terminating repeating

The World of Numbers 03: Finding Rational Numbers Between Two Rational Numbers PDF

The World of Numbers 03 covers finding rational numbers between two given values and identifying decimal expansions of rational numbers. Review the PDF to practise these concepts and understand the difference between terminating and recurring decimals.

Lecture 04: Converting Decimal Expansions into P/Q Form

Question 1. Express the terminating decimal 0.125 in the simplest p/q form.

A) 1/4

B) 1/8

C) 1/16

D) 3/8

Answer: B) 1/8

Question 2. Convert 0.7 into p/q form.

A) 7/10

B) 7/9

C) 7/99

D) 1/7

Answer: B) 7/9

Question 3. Convert 0.27 into the simplest p/q form.

A) 27/100

B) 3/11

C) 27/10

D) 9/11

Answer: B) 3/11

Question 4. Convert 1.3 into p/q form.

A) 4/3

B) 13/10

C) 13/9

D) 3/4

Answer: A) 4/3

Question 5. What is the fraction p/q representation of 0.999999... or 0.9?

A) 9/10

B) 1

C) 99/100

D) 8/9

Answer: B) 1

The World of Numbers 04: Decimal Expansion into P/Q Form PDF

The World of Numbers 04 explains how to express terminating and recurring decimals as fractions. Use the PDF to review the conversion methods and practise writing decimal values in their simplest p/q form.

Lecture 05: Irrational Numbers

The fifth lecture introduces irrational numbers and their properties. The questions cover identifying irrational values, understanding the relationship between rational and irrational numbers, and classifying the results of calculations involving square roots.

Question 1. Which of the following numbers is irrational?

A) √25

B) √19

C) 5.375

D) √121

Answer: B) √19

Question 2. What is the result when a non-zero rational number is added to an irrational number?

A) Always a rational number

B) Always an irrational number

C) Can be either rational or irrational

D) Always zero

Answer: B) Always an irrational number

Question 3. State whether the following statement is True or False: “π is irrational and 22/7 is rational.”

A) True

B) False

Answer: A) True

Question 4. Identify the nature of (2 − √2)(2 + √2).

A) Irrational

B) Rational

C) Complex

D) Undefined

Answer: B) Rational

Working: (2 − √2)(2 + √2) = 2² − (√2)² = 4 − 2 = 2.

Question 5. Which of the following statements is true?

A) Every whole number is a natural number.

B) Every real number is either rational or irrational.

C) √7 is irrational but not a real number.

D) There are finitely many rational numbers between two given rational numbers.

Answer: B) Every real number is either rational or irrational.

The World of Numbers 05: Irrational Numbers PDF

The World of Numbers 05 focuses on irrational numbers and their properties. Practise identifying irrational values and solving problems involving square roots to strengthen your understanding of rational and irrational numbers.

Why Choose NCERT Class 9 Maths Ganita Manjari Chapter 3 Solutions by PW?

The World of Numbers questions help you understand rational and irrational numbers, fractions, number lines and decimal expansions. The solutions can help reinforce these concepts through question practice.

• Revise rational numbers: Understand equivalent fractions and practise arithmetic operations.

• Represent numbers on a number line: Learn to locate rational numbers correctly.

• Understand decimal expansions: Distinguish between terminating and non-terminating recurring decimals.

• Convert decimals into fractions: Express recurring decimals in their simplest p/q form.

• Identify irrational numbers: Recognise irrational numbers and understand their properties.

• Practise different question types: Solve lecture-wise questions to strengthen your understanding.

• Check your answers: Identify calculation errors and revisit concepts that need more practice.

Practising the NCERT Class 9 Maths Ganita Manjari Chapter 3 Solutions by PW can help you revise The World of Numbers and build confidence in solving mathematical problems. Regular practice can also improve your accuracy and understanding of number concepts.