Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.3 Solutions: The World of Numbers

Rational numbers help us work with fractions, negative fractions and integers in one system. Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.3 Solutions focus on this part of The World of Numbers, where students practise equivalent rational numbers and the four basic operations on fractions.

In Ganita Manjari Class 9 Chapter 3 Exercise 3.3, students solve questions on equality of fractions, sums, differences, products, quotients and the distributive property. These Class 9 Maths Chapter 3 Exercise 3.3 Solutions explain every step clearly so students can revise rational numbers Class 9 concepts in a simple, textbook-based way.

Key Takeaways

Rational Numbers: Numbers written as p/q, where q is not zero, are rational.
Equivalent Fractions: Cross multiplication checks whether two rational numbers are equal.
Operations: Addition, subtraction, multiplication and division follow fraction rules.
Distributive Property: Rational numbers also follow a(b + c) = ab + ac.

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.3 Solutions Structure 2026

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.3 Solutions

Exercise Set 3.3 comes from the section on rational numbers. A rational number is any number that can be written in the form p/q, where p and q are integers and q ≠ 0. This exercise helps students practise operations on rational numbers Class 9 using equivalent denominators, cross multiplication and fraction rules.

These Class 9 Maths rational numbers answers also revise rational number addition subtraction multiplication division, which forms an important part of Class 9 Ganita Manjari number system solutions.

Exercise 3.3 Question 1

Prove that the following rational numbers are equal:

(i) 2/3 and 4/6

Solution:

Two rational numbers a/b and c/d are equal if:

a × d = b × c

For 2/3 and 4/6:

2 × 6 = 12

3 × 4 = 12

Since both products are equal:

2/3 = 4/6

Answer: The rational numbers 2/3 and 4/6 are equal.

(ii) 5/4 and 10/8

Solution:

For 5/4 and 10/8:

5 × 8 = 40

4 × 10 = 40

Since both products are equal:

5/4 = 10/8

Answer: The rational numbers 5/4 and 10/8 are equal.

(iii) −3/5 and −6/10

Solution:

For −3/5 and −6/10:

(−3) × 10 = −30

5 × (−6) = −30

Since both products are equal:

−3/5 = −6/10

Answer: The rational numbers −3/5 and −6/10 are equal.

(iv) 9/3 and 3

Solution:

Write 3 as 3/1.

Now compare 9/3 and 3/1.

9 × 1 = 9

3 × 3 = 9

Since both products are equal:

9/3 = 3

Answer: The rational numbers 9/3 and 3 are equal.

Exercise 3.3 Question 2

Find the sum:

(i) 2/5 + 3/10

Solution:

Find a common denominator.

LCM of 5 and 10 is 10.

2/5 = 4/10

So,

2/5 + 3/10 = 4/10 + 3/10

= 7/10

Answer: 7/10

(ii) 7/12 + 5/8

Solution:

LCM of 12 and 8 is 24.

7/12 = 14/24

5/8 = 15/24

So,

7/12 + 5/8 = 14/24 + 15/24

= 29/24

Answer: 29/24

(iii) −4/7 + 3/14

Solution:

LCM of 7 and 14 is 14.

−4/7 = −8/14

So,

−4/7 + 3/14 = −8/14 + 3/14

= −5/14

Answer: −5/14

Exercise 3.3 Question 3

Find the difference:

(i) 5/6 − 1/4

Solution:

LCM of 6 and 4 is 12.

5/6 = 10/12

1/4 = 3/12

So,

5/6 − 1/4 = 10/12 − 3/12

= 7/12

Answer: 7/12

(ii) 11/8 − 3/4

Solution:

Convert 3/4 to denominator 8.

3/4 = 6/8

So,

11/8 − 3/4 = 11/8 − 6/8

= 5/8

Answer: 5/8

(iii) −7/9 − (−2/3)

Solution:

Subtracting a negative rational number is the same as adding the corresponding positive rational number.

−7/9 − (−2/3) = −7/9 + 2/3

Convert 2/3 to denominator 9.

2/3 = 6/9

So,

−7/9 + 6/9 = −1/9

Answer: −1/9

Exercise 3.3 Question 4

Find the product:

(i) 2/3 × 3/10

Solution:

Multiply numerators and denominators.

2/3 × 3/10 = (2 × 3) / (3 × 10)

Cancel the common factor 3.

= 2/10

= 1/5

Answer: 1/5

(ii) 7/11 × 5/8

Solution:

7/11 × 5/8 = (7 × 5) / (11 × 8)

= 35/88

Answer: 35/88

(iii) −4/7 × 5/14

Solution:

−4/7 × 5/14 = (−4 × 5) / (7 × 14)

= −20/98

Simplify by dividing numerator and denominator by 2.

= −10/49

Answer: −10/49

Exercise 3.3 Question 5

Find the quotient:

(i) 2/3 ÷ 3/10

Solution:

To divide by a fraction, multiply by its reciprocal.

2/3 ÷ 3/10 = 2/3 × 10/3

= 20/9

Answer: 20/9

(ii) 7/11 ÷ 5/8

Solution:

7/11 ÷ 5/8 = 7/11 × 8/5

= 56/55

Answer: 56/55

(iii) −4/7 ÷ 5/14

Solution:

−4/7 ÷ 5/14 = −4/7 × 14/5

Cancel 14 ÷ 7 = 2.

= −4 × 2 / 5

= −8/5

Answer: −8/5

Exercise 3.3 Question 6

Show that: (1/2 + 3/4) × 8/3 = 1/2 × 8/3 + 3/4 × 8/3.

Solution:

First solve the left-hand side.

LHS = (1/2 + 3/4) × 8/3

Convert 1/2 to denominator 4.

1/2 = 2/4

So,

LHS = (2/4 + 3/4) × 8/3

= 5/4 × 8/3

= 40/12

= 10/3

Now solve the right-hand side.

RHS = 1/2 × 8/3 + 3/4 × 8/3

= 8/6 + 24/12

= 4/3 + 2

= 4/3 + 6/3

= 10/3

Since:

LHS = RHS = 10/3

Answer: Hence proved.

Exercise 3.3 Question 7

Simplify the following using the distributive property: 7/9 (6/7 − 3/4).

Solution:

Use the distributive property:

a(b − c) = ab − ac

So,

7/9 (6/7 − 3/4) = 7/9 × 6/7 − 7/9 × 3/4

Now simplify each term.

7/9 × 6/7 = 6/9 = 2/3

7/9 × 3/4 = 21/36 = 7/12

So,

7/9 (6/7 − 3/4) = 2/3 − 7/12

Convert 2/3 to denominator 12.

2/3 = 8/12

Therefore:

8/12 − 7/12 = 1/12

Answer: 1/12

Exercise 3.3 Question 8

Find the rational number x such that: 5/6 (x + 3/5) = 5/6 x + 1/2.

Solution:

Given equation:

5/6 (x + 3/5) = 5/6 x + 1/2

Use the distributive property on the left-hand side.

5/6 (x + 3/5) = 5/6 × x + 5/6 × 3/5

= 5x/6 + 15/30

= 5x/6 + 1/2

So, the left-hand side becomes:

5x/6 + 1/2

This is exactly the same as the right-hand side:

5/6 x + 1/2

Therefore, the equation is true for every rational number x.

Answer: The equation is true for all rational numbers x.

Final Answers for Exercise 3.3

Concept Used in The World of Numbers Exercise 3.3

The World of Numbers Exercise 3.3 is based on rational numbers and their arithmetic operations. Students practise how rational numbers behave under addition, subtraction, multiplication and division.

Important concepts include:

• Rational numbers Class 9: numbers written as p/q, where p and q are integers and q ≠ 0.
• Equivalent rational numbers: rational numbers that represent the same value, such as 2/3 and 4/6.
• Addition and subtraction: make denominators equal before combining numerators.
• Multiplication: multiply numerators and denominators.
• Division: multiply by the reciprocal of the divisor.
• Distributive property rational numbers: a(b + c) = ab + ac and a(b − c) = ab − ac.

These ideas are important in Class 9 Maths Ganita Manjari Chapter 3 Solutions because they prepare students for number line representation, density of rational numbers, irrational numbers and real numbers.

About Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.3

Ganita Manjari Class 9 Chapter 3 Exercise 3.3 belongs to the section on fractions and rational numbers. It comes after students learn natural numbers, zero and integers.

These Class 9 Maths Chapter 3 Exercise 3.3 Solutions help students revise:

• equality of rational numbers,
• equivalent fractions,
• rational number addition,
• rational number subtraction,
• rational number multiplication,
• rational number division,
• distributive law for rational numbers,
• solving simple rational number equations.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3