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📖 Exercise 3.3 Overview
In Class 9 Maths Chapter 3 Exercise 3.3 (Ganita Manjari 2026), you will learn how to perform different operations on rational numbers. This exercise helps you understand how to add, subtract, multiply, divide and compare rational numbers while applying the properties of rational numbers in a systematic way.
🎯 What You Will Learn
- Operations on Rational Numbers
- Equivalent Fractions
- Standard Form
- Distributive Property
- Finding Unknown Rational Numbers
📝 Types of Questions
- Prove equality of rational numbers
- Addition & Subtraction
- Multiplication & Division
- Properties of Rational Numbers
- Equation-based questions
🏆 Marks Importance
This exercise is important for CBSE examinations. Questions based on operations and properties of rational numbers are frequently asked in school exams, unit tests and annual examinations.
⏱ Expected Time
20–30 Minutes
Spend more time understanding each operation rather than memorising the steps.
🧠 Before You Start – Quick Revision
Revise these important concepts before solving Exercise 3.3. A quick revision will help you solve the questions faster and avoid common mistakes.
- Rational Number: A number that can be written in the form p/q, where q ≠ 0.
- Standard Form: A rational number is in standard form when the numerator and denominator are coprime and the denominator is positive.
- Equivalent Fractions: Multiply or divide the numerator and denominator by the same non-zero number to obtain equivalent fractions.
- Additive Inverse: The additive inverse of a is −a, and their sum is always zero.
- Reciprocal: The reciprocal of a/b is b/a, where a ≠ 0.
- Operations: Always simplify the final answer and write the rational number in its standard form whenever possible.
📝 Class 9 Maths Chapter 3 Exercise 3.3 Solutions
Step-by-step CBSE solutions based on the latest Ganita Manjari (2026), explained in a simple and student-friendly way.
Prove that the following rational numbers are equal.
📌 Given
Four pairs of rational numbers are given.
🎯 To Find
Prove whether each pair of rational numbers is equal or not.
💡 Important Note
A pair of rational numbers can be proved equal by using either of the following methods.
- 🟧 Method 1 – Using Equivalent Fractions (NCERT Method)
- 🟦 Method 2 – Using Cross Multiplication (Verification Method)
✍ Solution (i)
Prove that 2 3 and 4 6 are equal.
🟧 Using Equivalent Fractions Method
Multiply the numerator and denominator of 2 3 by 2.
Since both fractions become the same,
Hence, 2 3 = 4 6 Therefore, the given rational numbers are equal.
✅ Conclusion
Hence, 2 3 = 4 6 Therefore, the given rational numbers are equal.
✍ Solution (ii)
Prove that 5 4 and 10 8 are equal.
🟧 Using Equivalent Fractions Method
Multiply the numerator and denominator of 5 4 by 2.
Since both fractions become the same,
Hence, 5 4 = 10 8 Therefore, the given rational numbers are equal.
✅ Conclusion
Hence, 5 4 = 10 8 Therefore, the given rational numbers are equal.
✍ Solution (iii)
Prove that −3 5 and −6 10 are equal.
🟧 Method 1 – Using Equivalent Fractions (NCERT Method)
Multiply the numerator and denominator of −3 5 by 2.
Since both fractions become the same,
Hence, −3 5 = −6 10 Therefore, the given rational numbers are equal.
🟦 Method 2 – Using Cross Multiplication (Verification Method)
For two rational numbers a b and c d they are equal if ad = bc.
For the given rational numbers,
5 × (−6) = −30
−30 = −30
Since both cross products are equal, (−3) × 10 = 5 × (−6) Therefore, the given rational numbers are equal.
✅ Conclusion
Hence, −3 5 = −6 10 Therefore, the given rational numbers are equal.
✍ Solution (iv)
Prove that 9 3 and 3 are equal.
🟧 Method 1 – Using Equivalent Fractions (NCERT Method)
Write the whole number 3 as a rational number.
Now multiply the numerator and denominator by 3.
Therefore,
Hence, 3 = 9 3 Therefore, the given rational numbers are equal.
🟦 Method 2 – Cross Multiplication
5 × (−6) = −30
∴ −30 = −30
Hence, (−3) × 10 = 5 × (−6) Therefore, −3 5 = −6 10
📘 Key Idea: Equal fractions ⇒ Equal rational numbers.
🎯 CBSE Tip: Show every multiplication step clearly.
✅ Final Answer
All the given rational numbers are equal.
📘 Key Concept: Equivalent Fractions or Cross Multiplication (ad = bc).
⚠ Common Mistake: Multiply both numerator and denominator by the same non-zero number.
🎯 CBSE Tip: Write every step clearly while proving equality.
Find the sum.
(i) 2 5 + 3 10
(ii) 7 12 + 5 8
(iii) −4 7 + 3 14
📌 Given
Three pairs of rational numbers are given.
🎯 To Find
Find the sum of each pair of rational numbers in the simplest form.
✍ Solution (i)
Find the sum of 2 5 + 3 10
🟧 Method 1 – Using LCM and Equivalent Fractions (NCERT Method)
Since the denominators are different, first find their LCM.
1, 2
LCM = 5 × 2 = 10
Convert 2 5 into an equivalent fraction having denominator 10.
Now add the fractions.
Hence, 2 5 + 3 10 = 7 10
🟦 Method 2 – Using LCM (Shortcut Method)
First, find the LCM of 5 and 10.
1, 2
LCM = 5 × 2 = 10
Now write each numerator according to the LCM.
Hence, the required sum is 7 10
✅ Final Answer
2 5 + 3 10 = 7 10
✍ Solution (ii)
Find the sum of 7 12 + 5 8
🟧 Method 1 – Using LCM and Equivalent Fractions (NCERT Method)
Since the denominators are different, first find their LCM.
6, 4
2 ) 6, 4
3, 2
2 ) 3, 2
3, 1
3 ) 3, 1
1, 1
LCM = 2 × 2 × 2 × 3 = 24
Convert both fractions into equivalent fractions having denominator 24.
Now add the fractions.
Hence, 7 12 + 5 8 = 29 24
🟦 Method 2 – Using LCM (Shortcut Method)
First, find the LCM of 12 and 8.
6, 4
2 ) 6, 4
3, 2
2 ) 3, 2
3, 1
3 ) 3, 1
1, 1
LCM = 2 × 2 × 2 × 3 = 24
Now write each numerator according to the LCM.
Hence, the required sum is 29 24 .
✅ Final Answer
7 12 + 5 8 = 29 24
✍ Solution (iii)
Find the sum of −4 7 + 3 14
🟧 Method 1 – Using LCM and Equivalent Fractions (NCERT Method)
Since the denominators are different, first find their LCM.
1, 2
LCM = 7 × 2 = 14
Convert −4 7 into an equivalent fraction having denominator 14.
Now add the fractions.
Hence, −4 7 + 3 14 = −5 14
🟦 Method 2 – Using LCM (Shortcut Method)
First, find the LCM of 7 and 14.
1, 2
LCM = 7 × 2 = 14
Now write each numerator according to the LCM.
Hence, the required sum is −5 14 .
✅ Final Answer
−4 7 + 3 14 = −5 14
✅ Final Answer
| Part | Answer |
|---|---|
| (i) | 7 10 |
| (ii) | 29 24 |
| (iii) | −5 14 |
📘 Key Concept
- Find the LCM of the denominators.
- Make equivalent fractions.
- Add numerators and keep the denominator same.
⚠️ Common Mistake
- Do not add fractions before making denominators equal.
- Always use the correct LCM.
🎯 CBSE Tip
Always show the LCM and every calculation step to get full marks.
🧠 Memory Trick
LCM → Equivalent Fractions → Add → Simplify
Find the difference.
(i) 5 6 − 1 4
(ii) 11 8 − 3 4
(iii) −7 9 − ( 2 3 )
📌 Given
Three pairs of rational numbers are given for subtraction.
🎯 To Find
Find the difference of each pair of rational numbers in the simplest form.
✍ Solution (i)
Find the difference of 5 6 − 1 4
🟧 Method 1 – Using LCM and Equivalent Fractions (NCERT Method)
Since the denominators are different, first find their LCM.
3, 2
2 ) 3, 2
3, 1
3 ) 3, 1
1, 1
LCM = 2 × 2 × 3 = 12
Convert both fractions into equivalent fractions having denominator 12.
Now subtract the fractions.
Hence, 5 6 − 1 4 = 7 12
🟦 Method 2 – Using LCM (Shortcut Method)
First, find the LCM of 6 and 4.
3, 2
2 ) 3, 2
3, 1
3 ) 3, 1
1, 1
LCM = 2 × 2 × 3 = 12
Now write each numerator according to the LCM.
Hence, the required difference is 7 12 .
✅ Final Answer
5 6 − 1 4 = 7 12
✍ Solution (ii)
Find the difference of 11 8 − 3 4
🟧 Method 1 – Using LCM and Equivalent Fractions (NCERT Method)
Since the denominators are different, first find their LCM.
4, 2
2 ) 4, 2
2, 1
2 ) 2, 1
1, 1
LCM = 2 × 2 × 2 = 8
Convert 3 4 into an equivalent fraction having denominator 8.
Now subtract the fractions.
Hence, 11 8 − 3 4 = 5 8
🟦 Method 2 – Using LCM (Shortcut Method)
First, find the LCM of 8 and 4.
4, 2
2 ) 4, 2
2, 1
2 ) 2, 1
1, 1
LCM = 2 × 2 × 2 = 8
Now write each numerator according to the LCM.
Hence, the required difference is 5 8 .
✅ Final Answer
11 8 − 3 4 = 5 8
✍ Solution (iii)
Find the difference of −7 9 − 2 3
🟧 Method 1 – Using LCM and Equivalent Fractions (NCERT Method)
Since the denominators are different, first find their LCM.
3, 1
3 ) 3, 1
1, 1
LCM = 3 × 3 = 9
Convert 2 3 into an equivalent fraction having denominator 9.
Now subtract the fractions.
Hence, −7 9 − 2 3 = −13 9
🟦 Method 2 – Using LCM (Shortcut Method)
First, find the LCM of 9 and 3.
3, 1
3 ) 3, 1
1, 1
LCM = 3 × 3 = 9
Now write each numerator according to the LCM.
Hence, the required difference is −13 9 .
✅ Final Answer
−7 9 − 2 3 = −13 9
✅ Final Answers
| Part | Answer |
|---|---|
| (i) | 7 12 |
| (ii) | 5 8 |
| (iii) | −13 9 |
📘 Key Concept
LCM → Equivalent Fractions → Subtract Numerators → Simplify
⚠️ Common Mistake
Do not subtract fractions before making their denominators equal.
🎯 CBSE Tip
Always show the LCM and every calculation step.
🧠 Memory Trick
LCM → Equivalent Fractions → Subtract → Simplify
Find the product.
(i) 2 3 × 3 10
(ii) 7 11 × 5 8
(iii) −4 7 × 5 14
📌 Given
Three pairs of rational numbers are given.
🎯 To Find
Find the product of each pair of rational numbers and write the answer in the simplest form.
Find the product.
(i) 2 3 × 3 10
(ii) 7 11 × 5 8
(iii) −4 7 × 5 14
📌 Given
Three pairs of rational numbers are given.
🎯 To Find
Find the product of each pair of rational numbers and write the answer in the simplest form.
✍ Solution (i)
Find the product of 2 3 × 3 10
✍ Solution
To multiply two rational numbers, multiply their numerators and multiply their denominators.
The fraction 6 30 is simplified by dividing the numerator and denominator by their HCF, 6.
Hence, 2 3 × 3 10 = 1 5
✅ Final Answer
2 3 × 3 10 = 1 5
✍ Solution (ii)
Find the product of 7 11 × 5 8
✍ Solution
To multiply two rational numbers, multiply their numerators and multiply their denominators.
The numerator 35 and denominator 88 have no common factor other than 1. Therefore, the fraction is already in its lowest (standard) form.
Hence, 7 11 × 5 8 = 35 88
✅ Final Answer
7 11 × 5 8 = 35 88
Since 35 and 88 have no common factor other than 1, the answer is already in its simplest form.
✍ Solution (iii)
Find the product of −4 7 × 5 14
✍ Solution
To multiply two rational numbers, multiply their numerators and multiply their denominators.
The numerator −20 and denominator 98 have the common factor 2. So, divide both the numerator and denominator by 2.
Since 10 and 49 have no common factor other than 1, the fraction is already in its lowest (standard) form.
Hence, −4 7 × 5 14 = −10 49
✅ Final Answer
−4 7 × 5 14 = −10 49
After simplifying, the product is in its lowest (standard) form because 10 and 49 have no common factor other than 1.
✅ Final Answers
| Part | Answer |
|---|---|
| (i) | 1 5 |
| (ii) | 35 88 |
| (iii) | −10 49 |
📘 Key Concept
Multiply the numerators, multiply the denominators, then simplify the answer to its lowest (standard) form.
⚠️ Common Mistake
Do not forget to simplify the product and apply the correct sign rule while multiplying positive and negative rational numbers.
🎯 CBSE Exam Tip
Always show the multiplication of numerators and denominators clearly before simplifying the final answer.
🧠 Memory Trick
Numerator × Numerator → Denominator × Denominator → Simplify
Find the quotient.
(i) 2 3 ÷ 3 10
(ii) 7 11 ÷ 5 8
(iii) −4 7 ÷ 5 14
📌 Given
Three pairs of rational numbers are given.
🎯 To Find
Find the quotient of each pair of rational numbers and write the answer in the simplest (standard) form.
✍ Solution (i)
Find the quotient of 2 3 ÷ 3 10
✍ Solution
To divide one rational number by another, multiply the first rational number by the reciprocal of the second rational number.
Now multiply the numerators and denominators.
Since 20 and 9 have no common factor other than 1, the fraction is already in its lowest (standard) form.
Hence, 2 3 ÷ 3 10 = 20 9
✅ Final Answer
2 3 ÷ 3 10 = 20 9
The quotient is already in its lowest (standard) form because 20 and 9 have no common factor other than 1.
✍ Solution (ii)
Find the quotient of 7 11 ÷ 5 8
✍ Solution
To divide one rational number by another, multiply the first rational number by the reciprocal of the second rational number.
Now multiply the numerators and denominators.
Since 56 and 55 have no common factor other than 1, the fraction is already in its lowest (standard) form.
Hence, 7 11 ÷ 5 8 = 56 55
✅ Final Answer
7 11 ÷ 5 8 = 56 55
The quotient is already in its lowest (standard) form because 56 and 55 have no common factor other than 1.
✍ Solution (iii)
Find the quotient of −4 7 ÷ 5 14
✍ Solution
To divide one rational number by another, multiply the first rational number by the reciprocal of the second rational number.
Now multiply the numerators and denominators.
The numerator and denominator have the common factor 7. Divide both by 7.
Since 8 and 5 have no common factor other than 1, the fraction is already in its lowest (standard) form.
Hence, −4 7 ÷ 5 14 = −8 5
✅ Final Answer
−4 7 ÷ 5 14 = −8 5
After simplifying, the quotient is in its lowest (standard) form because 8 and 5 have no common factor other than 1.
✅ Final Answers
| Part | Answer |
|---|---|
| (i) | 20 9 |
| (ii) | 56 55 |
| (iii) | −8 5 |
📘 Key Concept
Divide by multiplying with the reciprocal, then simplify.
⚠️ Common Mistake
Do not forget to take the reciprocal before multiplying.
🎯 CBSE Exam Tip
Show the reciprocal step to score full marks.
🧠 Memory Trick
Divide → Flip → Multiply → Simplify
Show that
📌 Given
We are given the equation
🎯 To Find
Verify that the Left-Hand Side (LHS) is equal to the Right-Hand Side (RHS).
🟧 Solution (L.H.S.)
Simplify the Left-Hand Side step by step.
= ( 1 2 + 3 4 ) × 8 3
= ( 2 4 + 3 4 ) × 8 3
= 5 4 × 8 3
= 5 × 8 4 × 3
= 40 12
= 10 3
Therefore, L.H.S. = 10 3
🟦 Solution (R.H.S.)
Now simplify the Right-Hand Side.
= 1 2 × 8 3 + 3 4 × 8 3
= 1 × 8 2 × 3 + 3 × 8 4 × 3
= 8 6 + 24 12
= 16 12 + 24 12
= 40 12
= 10 3
Therefore, R.H.S. = 10 3
✅ Final Answer
Since LHS = RHS = 10 3 , the given statement is verified (proved).
📘 Key Concept: This question verifies the Distributive Property of Multiplication over Addition for rational numbers.
⚠️ Common Mistake: Adding the fractions after multiplication without first simplifying the intermediate steps.
🎯 CBSE Exam Tip: Show the complete LHS and RHS separately. Never skip steps while proving an identity.
🧠 Memory Trick: “Multiply Each, Then Add” → (a + b) × c = (a × c) + (b × c).
Simplify the following using the Distributive Property.
📌 Given
The expression is
🎯 To Find
Simplify the given expression using the Distributive Property and write the answer in the lowest (standard) form.
✍ Solution
Using the Distributive Property,
= 7 9 × 6 7 − 7 9 × 3 4
= 7 × 6 9 × 7 − 7 × 3 9 × 4
= 42 63 − 21 36
= 2 3 − 7 12
= 8 12 − 7 12
= 1 12
✅ Final Answer
The simplified value is 1 12 .
📘 Key Concept: Apply the Distributive Property first, then simplify each product before performing subtraction.
⚠️ Common Mistake: Multiplying the outside fraction with only one term inside the bracket.
🎯 CBSE Exam Tip: Expand the brackets completely, reduce each fraction to its lowest form, and then perform subtraction.
🧠 Memory Trick: “Multiply Every Term, Then Simplify.”
Find the rational number x such that
📌 Given
The equation is
🎯 To Find
Find the value of the rational number x that satisfies the given equation.
✍ Solution
Apply the Distributive Property to the left-hand side.
= 5 6 ( x + 3 5 )
= 5 6 x + 5 × 3 6 × 5
= 5 6 x + 15 30
= 5 6 x + 1 2
RHS = 5 6 x + 1 2
✅ Final Answer
After applying the distributive property,
LHS = RHS.
Hence, the given equation is true for every rational number.
∴ x can be any rational number.
📚 Quick Summary
Before moving to the next exercise, quickly revise the most important concepts of Exercise 3.3 – Rational Numbers. This summary is perfect for last-minute revision before school tests and CBSE examinations.
🔄 Equivalent Fractions
Multiply or divide the numerator and denominator by the same non-zero number to obtain equivalent fractions.
📘 Standard Form
A rational number is in standard form when the numerator and denominator are coprime and the denominator is positive.
➕ Addition & ➖ Subtraction
Always make the denominators the same (LCM), then add or subtract the numerators.
✖ Multiplication
Multiply the numerators together and the denominators together. Simplify the answer whenever possible.
➗ Division
Division of rational numbers means multiply by the reciprocal of the second rational number.
🔁 Reciprocal
The reciprocal of a rational number is obtained by interchanging the numerator and denominator. It exists only when the numerator is not zero.
🧠 Remember These 5 Rules
- ✔ Make the denominator same before addition and subtraction.
- ✔ Multiply numerator with numerator and denominator with denominator.
- ✔ For division, always take the reciprocal first.
- ✔ Simplify every rational number into its standard form.
- ✔ Check your final answer carefully before moving to the next question.
⚠️ Common Mistakes to Avoid
Many students lose marks in Exercise 3.3 because of small calculation mistakes. Avoid these common errors to improve your accuracy in school and CBSE examinations.
❌ Forgetting to Simplify
Always reduce the final rational number to its lowest (standard) form. Unsimplified answers may lose marks.
❌ Taking the Wrong LCM
While adding or subtracting rational numbers, first find the correct LCM of the denominators before performing the operation.
❌ Forgetting the Reciprocal
During division, many students divide directly. Always change the division into multiplication by taking the reciprocal of the second rational number.
❌ Keeping the Denominator Negative
The denominator should always be positive. If it is negative, multiply both numerator and denominator by −1.
❌ Not Writing Standard Form
A rational number should always be written in its standard form, where the numerator and denominator are coprime and the denominator is positive.
✅ Teacher’s Advice
Read the question carefully, solve each step neatly, simplify the final answer, and check your calculations before moving to the next question.
❓ Frequently Asked Questions (FAQs)
Here are the most frequently asked questions related to Class 9 Maths Chapter 3 Exercise 3.3 Solutions (Ganita Manjari 2026). These answers will help you quickly revise the important concepts of rational numbers.
1. What is Exercise 3.3 about?
Exercise 3.3 focuses on operations on rational numbers. It includes questions on equivalent fractions, addition, subtraction, multiplication, division, equality of rational numbers, reciprocal, and solving simple expressions involving rational numbers.
2. How do you add rational numbers?
To add rational numbers, first make their denominators the same by taking the LCM. Then add the numerators and simplify the final answer.
3. How do you subtract rational numbers?
Find the LCM of the denominators, convert the fractions into equivalent fractions, subtract the numerators, and reduce the answer to its lowest form.
4. How do you multiply rational numbers?
Multiply the numerators together and the denominators together. Simplify the resulting fraction and write it in standard form.
5. How do you divide rational numbers?
Division of rational numbers is done by multiplying the first rational number by the reciprocal of the second rational number. Simplify the final answer if possible.
6. What is the reciprocal of a rational number?
The reciprocal of a rational number is obtained by interchanging its numerator and denominator. It exists only when the numerator is not zero.
7. Are these solutions based on Ganita Manjari 2026?
Yes. All solutions are prepared according to the latest NCERT Ganita Manjari (2026) textbook and follow the current CBSE curriculum.
8. Are these solutions suitable for CBSE examinations?
Yes. Every solution follows the CBSE answer-writing pattern with step-by-step explanations, making them ideal for school exams, unit tests, periodic tests, and annual examinations.
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👨🏫 Reviewed & Prepared By
Rakesh Kumar Singh
Founder, Newton Study Point & Maths Gurukulam | Mathematics Educator Since 2006
Rakesh Kumar Singh has been teaching CBSE Mathematics since 2006 and has guided hundreds of students from Classes 8 to 12. His teaching approach emphasizes conceptual understanding, logical thinking, step-by-step problem solving, and exam-oriented preparation, enabling students to build confidence and achieve excellent results in Mathematics.
These Class 9 Maths Chapter 3 Exercise 3.3 Solutions (Ganita Manjari 2026) are independently prepared and carefully reviewed according to the latest NCERT Ganita Manjari (2026) textbook and the current CBSE curriculum. Every solution follows a student-friendly format with Given, To Find, Step-by-Step Solution, Final Answer, Key Concept, Common Mistake, Exam Tip, and Memory Trick to help students understand the concepts of Rational Numbers. This exercise covers equivalent rational numbers, standard form, addition, subtraction, multiplication, division, reciprocal, and the properties of rational numbers through easy explanations designed for school examinations and self-study.