Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions: The World of Numbers

Rational numbers are not limited to whole-number positions on the number line; they can also lie between integers. Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions focus on representing rational numbers, finding numbers between two given values, simplifying fractions and using the idea that rational numbers are dense.

In Ganita Manjari Class 9 Chapter 3 Exercise 3.4, students move from calculating with rational numbers to placing them correctly on a number line. The exercise includes values like 2/3, −5/4 and 11/2, along with questions on finding rational numbers between two given numbers. These Class 9 Maths Chapter 3 Exercise 3.4 Solutions also cover close decimal values such as 3.1415 and 3.1416, helping students understand that rational numbers are densely spread across the number line.

Key Takeaways

Number Line: Rational numbers can be placed between integers on a number line.
Between Numbers: Common denominators help find rational numbers between two values.
Density: Infinitely many rational numbers lie between any two rational numbers.
Mixed Fractions: Improper fractions help solve measurement-based division problems.

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions Structure 2026

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4 Solutions

Exercise Set 3.4 comes from the section on representation of rational numbers on the number line and density of rational numbers. A rational number can be written as p/q, where p and q are integers and q ≠ 0.

These Class 9 Maths rational numbers on number line answers help students practise locating positive and negative fractions, simplifying rational expressions, dividing mixed fractions and finding rational numbers between two numbers.

Exercise 3.4 Question 1

Represent the rational numbers 2/3, −5/4 and 11/2 on a single number line.

Solution:

We need to locate:

2/3, −5/4, 11/2

First understand their positions.

2/3 lies between 0 and 1.

To represent 2/3, divide the distance between 0 and 1 into 3 equal parts. Mark the second part from 0 on the right side.

−5/4 = −1 1/4

So, −5/4 lies between −2 and −1.

To represent −5/4, divide the distance between −1 and −2 into 4 equal parts. Mark the first part to the left of −1.

11/2 = 5 1/2

So, 11/2 lies between 5 and 6.

To represent 11/2, divide the distance between 5 and 6 into 2 equal parts. Mark the midpoint between 5 and 6.

Answer: On the number line, −5/4 lies between −2 and −1, 2/3 lies between 0 and 1, and 11/2 lies between 5 and 6.

Exercise 3.4 Question 2

Find three distinct rational numbers that lie strictly between −1/2 and 1/4.

Solution:

We need rational numbers strictly between:

−1/2 and 1/4

Convert them to a common denominator.

−1/2 = −4/8

1/4 = 2/8

Now choose rational numbers between −4/8 and 2/8.

Examples:

−3/8, −1/8, 1/8

Check:

−4/8 < −3/8 < −1/8 < 1/8 < 2/8

So, all three lie strictly between −1/2 and 1/4.

Answer: Three rational numbers between −1/2 and 1/4 are −3/8, −1/8 and 1/8.

Exercise 3.4 Question 3

Simplify the expression: −1/4 + 5/12.

Solution:

Given expression:

−1/4 + 5/12

Find the LCM of 4 and 12.

LCM = 12

Convert −1/4 to denominator 12.

−1/4 = −3/12

Now add:

−1/4 + 5/12 = −3/12 + 5/12

= 2/12

= 1/6

Answer: 1/6

Exercise 3.4 Question 4

A tailor has 15 3/4 metres of fine silk. If making one kurta requires 2 1/4 metres of silk, exactly how many kurtas can he make?

Solution:

Total silk = 15 3/4 metres

Silk required for one kurta = 2 1/4 metres

Convert mixed numbers into improper fractions.

15 3/4 = 63/4

2 1/4 = 9/4

Number of kurtas:

63/4 ÷ 9/4

To divide by a fraction, multiply by its reciprocal.

63/4 × 4/9

Cancel 4 from numerator and denominator.

= 63/9

= 7

Answer: The tailor can make exactly 7 kurtas.

Exercise 3.4 Question 5

Find three rational numbers between 3.1415 and 3.1416.

Solution:

We need three rational numbers between:

3.1415 and 3.1416

Terminating decimals are rational numbers. So, we can choose any three terminating decimals between these two values.

Examples:

3.14151, 3.14155, 3.14159

Check:

3.1415 < 3.14151 < 3.14155 < 3.14159 < 3.1416

Answer: Three rational numbers between 3.1415 and 3.1416 are 3.14151, 3.14155 and 3.14159.

Exercise 3.4 Question 6

Can you think of other way(s) to find a rational number between any two rational numbers?

Solution:

Yes. There are many ways to find a rational number between any two rational numbers.

Method 1: Taking the average

If a and b are two rational numbers, then:

(a + b) / 2

lies between a and b.

For example, between 1/2 and 3/4:

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

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

= (5/4) / 2

= 5/8

So, 5/8 lies between 1/2 and 3/4.

Method 2: Using a common denominator

Convert both rational numbers to the same denominator and choose a numerator between them.

For example:

1/2 = 4/8

3/4 = 6/8

A rational number between them is:

5/8

Method 3: Using decimal form

Write the numbers as decimals and choose a decimal between them.

For example, between 0.5 and 0.75, we can choose 0.6, which is rational.

Answer: We can find a rational number between two rational numbers by taking their average, converting them to a common denominator, or using decimal form.

Final Answers for Exercise 3.4

Concept Used in The World of Numbers Exercise 3.4

The World of Numbers Exercise 3.4 focuses on rational numbers, their representation and their density on the number line.

Important concepts include:

• Rational numbers Class 9: numbers that can be written as p/q, where p and q are integers and q ≠ 0.
• Number line rational numbers Class 9: rational numbers can lie between two integers, not just at integer points.
• Rational numbers between two numbers: infinitely many rational numbers lie between any two rational numbers.
• Density of rational numbers: no matter how close two rational numbers are, another rational number can be found between them.
• Rational number operations Class 9: simplification, addition and division of rational numbers are used in the exercise.
• Mixed numbers to improper fractions: needed for questions such as the tailor and silk problem.

These ideas are important in Class 9 Maths Ganita Manjari Chapter 3 Solutions because they prepare students for irrational numbers, decimal expansions and real numbers.

About Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.4

Ganita Manjari Class 9 Chapter 3 Exercise 3.4 comes after students learn rational number operations in Exercise 3.3. This exercise moves the focus from calculation to representation and density.

These Class 9 Maths Chapter 3 Exercise 3.4 Solutions help students revise:

• locating rational numbers on the number line,
• finding rational numbers between two given numbers,
• simplifying rational expressions,
• dividing fractions and mixed numbers,
• understanding why rational numbers are dense.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3