Short Introduction

The End-of-Chapter Exercises of Chapter 3 “The World of Numbers” help students strengthen their understanding of rational numbers, irrational numbers, decimal expansions, and number line representation. In this article, we provide detailed step-by-step solutions for Questions 1 to 6 in a simple and exam-oriented manner.

Quick Information Box

Chapter: The World of Numbers

Class: Grade 9 Mathematics

Exercise: End-of-Chapter Exercises (Part 1)

Questions Covered: Q.1 to Q.6

Main Topics:

• Terminating and Repeating Decimals
• Irrational Numbers
• Conversion of Decimals into Fractions
• Number Line Representation
• Rational Numbers Between Two Numbers

Concepts Used (Topics Covered)

• Rational Numbers
• Irrational Numbers
• Decimal Expansion
• Long Division Method
• Fraction Conversion
• Number Line Representation
• Finding Rational Numbers Between Two Numbers

Important Formulas

1. Rational Number

A number of the form:

p/q, where q ≠ 0

2. Terminating Decimal Test

If the denominator (in lowest form) contains only prime factors 2 and/or 5, then the decimal expansion terminates.

3. Repeating Decimal Test

If the denominator contains any prime factor other than 2 or 5, the decimal expansion is non-terminating repeating.

4. Converting Decimal into Fraction

Decimal × 10ⁿ / 10ⁿ

where n = number of decimal places.

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Question 1

Convert the following rational numbers into terminating or non-terminating repeating decimals by long division:

Solution

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Question 2

Prove that √5 is an irrational number.

Solution

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Question 3

Convert the following decimal numbers into p/q form.

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Question 4

Locate the following rational numbers on the number line.

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Question 5

Find six rational numbers between 3 and 4.

Solution

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Question 6

Find five rational numbers between 2/5 and 3/5.

Solution

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Question 7

Find 5 rational numbers between1/6 and 2/5 .

Solution :

Question 8

Question 9

Let a and b be two non-zero rational numbers such that a + (1/ b) = 0. Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Common Mistakes

• Forgetting to simplify fractions.
• Confusing repeating decimals with irrational numbers.
• Writing √5 as a rational number.
• Incorrect placement of decimals on the number line.
• Missing the lowest form while converting decimals into fractions.

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Exam Tips

✅ Reduce every fraction to its simplest form.

✅ Check denominator factors before deciding whether a decimal terminates.

✅ Use proof by contradiction for irrationality proofs.

✅ Draw neat number lines with equal intervals.

✅ For finding rational numbers between two fractions, first make their denominators equal.

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Practice MCQs

1. Which of the following is a terminating decimal?

A. 2/9

B. 3/50

C. 5/12

D. 7/18

Answer: B

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2. √5 is:

A. Natural

B. Rational

C. Irrational

D. Integer

Answer: C

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3. 2.125 in fraction form is:

A. 15/8

B. 17/8

C. 19/8

D. 21/8

Answer: B

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4. 3.3 lies between:

A. 2 and 3

B. 3 and 4

C. 4 and 5

D. 1 and 2

Answer: B

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5. Which is a rational number between 2/5 and 3/5?

A. 13/30

B. 19/30

C. 1/5

D. 4/5

Answer: A

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FAQ Section

Q1. What is a rational number?

A number that can be written as p/q, where q ≠ 0.

Q2. What is an irrational number?

A number that cannot be expressed as p/q.

Q3. Is every terminating decimal rational?

Yes.

Q4. Can a repeating decimal be rational?

Yes. Every repeating decimal is rational.

Q5. Why is √5 irrational?

Because assuming it to be rational leads to a contradiction.

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