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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.
📒 Formula Sheet – Exercise 3.3
Revise these important formulas before solving Exercise 3.3.
➕ Addition of Rational Numbers
| a b | + | c d | = | ad + bc bd |
Find the LCM first, then add the numerators.
➖ Subtraction of Rational Numbers
| a b | − | c d | = | ad − bc bd |
Find the LCM first, then subtract the numerators.
✖ Multiplication of Rational Numbers
| a b | × | c d | = | ac bd |
Multiply numerator with numerator and denominator with denominator.
➗ Division of Rational Numbers
| a b | ÷ | c d | = | a b | × | d c | = | ad bc |
Division means multiply by the reciprocal.
⚖ Equality of Rational Numbers
| a b | = | c d | ⇔ | ad = bc |
Two rational numbers are equal when their cross products are equal.
📝 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.
(i) 2 3 and 4 6
(ii) 5 4 and 10 8
(iii) −3 5 and −6 10
(iv) 9 3 and 3
📌 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.
🟧 Method 1 – Using Equivalent Fractions (NCERT Method)
Multiply the numerator and denominator of 2 3 by 2.
Since both fractions become the same,
Hence, the two 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.
Since 2 × 6 = 3 × 4, the given rational numbers are equal.
✅ Conclusion
Therefore, the given rational numbers are equal.
✍ Solution (ii)
Prove that 5 4 and 10 8 are equal.
🟧 Method 1 – Using Equivalent Fractions (NCERT Method)
Multiply the numerator and denominator of 5 4 by 2.
Since both fractions become the same,
Hence, the two 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.
Since 5 × 8 = 4 × 10, the given rational numbers are equal.
✅ Conclusion
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, the two 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.
Since (−3) × 10 = 5 × (−6), the given rational numbers are equal.
✅ Conclusion
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, the two rational numbers are equal.
🟦 Method 2 – Using Cross Multiplication (Verification Method)
Write the whole number 3 as
Now compare
Cross multiply the numerator of each fraction with the denominator of the other.
Since 9 × 1 = 3 × 3, the given rational numbers are equal.
✅ Conclusion
Therefore, the given rational numbers are equal.
✅ Final Answer
| Part | Result |
|---|---|
| (i) | 2 3 = 4 6 ✔ Equal |
| (ii) | 5 4 = 10 8 ✔ Equal |
| (iii) | −3 5 = −6 10 ✔ Equal |
| (iv) | 9 3 = 3 ✔ Equal |
📘 Key Concept Used
- Two rational numbers are equal if one can be converted into the other by multiplying or dividing both the numerator and denominator by the same non-zero number.
- Equivalent fractions represent the same rational number.
- Two rational numbers are also equal if their cross products are equal, that is, ad = bc.
⚠️ Common Mistakes
- Multiplying only the numerator or only the denominator while finding equivalent fractions.
- Using different numbers to multiply the numerator and denominator.
- Making mistakes in cross multiplication.
- Not writing the whole number as a rational number before comparing it with a fraction.
🎯 CBSE Exam Tip
In CBSE examinations, you may use either the Equivalent Fraction Method or the Cross Multiplication Method. However, when a question asks to “prove”, always show every step clearly instead of writing only the final answer.
🧠 Memory Trick
Equal Fractions → Same Value
🟧 Learn: Make equivalent fractions.
🟦 Check: Cross multiply (ad = bc).
Understand first, verify later!
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.
💡 Important Note
- If the denominators are the same, add only the numerators and keep the denominator unchanged.
- If the denominators are different, first find the LCM, convert the fractions into equivalent fractions, then add the numerators.
- Always write the final answer in its lowest (standard) 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,
🟦 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 .
✍ 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,
🟦 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 .
✍ 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,
🟦 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
| Part | Answer |
|---|---|
| (i) | 7 10 |
| (ii) | 29 24 |
| (iii) | −5 14 |
📘 Key Concept Used
- Before adding rational numbers, make their denominators the same.
- Find the LCM of the denominators and convert the fractions into equivalent fractions.
- After the denominators become equal, add only the numerators and keep the denominator unchanged.
- Always reduce the final answer to the lowest (standard) form, if possible.
⚠️ Common Mistakes
- Adding numerators without making the denominators equal.
- Finding the wrong LCM.
- Multiplying only the numerator or only the denominator while making equivalent fractions.
- Making mistakes while performing addition of positive and negative numbers.
- Forgetting to simplify the final answer.
🎯 CBSE Exam Tip
In CBSE examinations, you can solve addition of rational numbers by using either the Equivalent Fraction Method or the LCM Shortcut Method. However, always show the LCM and all calculation steps to score full marks.
🧠 Memory Trick
Remember:
✅ Different Denominators → Find LCM → Make Equivalent Fractions → Add Numerators → Simplify
A simple way to remember is:
“LCM First, Addition Next!”
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.
🎯 To Find
Find the difference of each pair of rational numbers in the simplest form.
💡 Important Note
- If the denominators are the same, subtract only the numerators and keep the denominator unchanged.
- If the denominators are different, first find the LCM, convert the fractions into equivalent fractions, and then subtract the numerators.
- Subtracting a negative rational number is the same as adding its additive inverse.
- Always write the final answer in its lowest (standard) 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,
🟦 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 .
✍ 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,
🟦 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
| Part | Answer |
|---|---|
| (i) | 7 12 |
| (ii) | 5 8 |
| (iii) | −1 9 |
📘 Key Concept Used
- Before subtracting rational numbers with different denominators, first find their LCM.
- Convert the given fractions into equivalent fractions having the same denominator.
- Subtract only the numerators and keep the denominator unchanged.
- Subtracting a negative rational number means adding its additive inverse.
- Write the answer in the lowest (standard) form.
⚠️ Common Mistakes
- Subtracting numerators directly without making the denominators equal.
- Finding the wrong LCM.
- Forgetting to change subtraction of a negative fraction into addition.
- Making sign errors while subtracting positive and negative numbers.
- Not simplifying the final answer.
🎯 CBSE Exam Tip
To score full marks in CBSE examinations, always show the LCM, convert the fractions into equivalent fractions (or use the LCM shortcut method correctly), and write every calculation step clearly. Do not skip intermediate steps.
🧠 Memory Trick
Different Denominators?
✅ Find LCM → Make Equivalent Fractions → Subtract Numerators → Simplify
Special Rule:
Subtracting a negative number = Adding a positive number.
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, the required product is 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, the required product is 35 88 .
✍ 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 in its lowest (standard) form.
Hence, the required product is −10 49 .
✅ Final Answer
| Part | Answer |
|---|---|
| (i) | 1 5 |
| (ii) | 35 88 |
| (iii) | −10 49 |
📘 Key Concept Used
- To multiply two rational numbers, multiply their numerators and multiply their denominators.
- If one rational number is negative and the other is positive, the product is negative.
- After multiplication, always reduce the fraction to its lowest (standard) form.
- If the numerator and denominator have a common factor, divide both by their HCF.
⚠️ Common Mistakes
- Multiplying the numerator of one fraction with the denominator of the other.
- Forgetting the sign rule while multiplying positive and negative rational numbers.
- Not simplifying the final answer.
- Making calculation mistakes during multiplication.
- Leaving the answer in a non-standard form.
🎯 CBSE Exam Tip
In CBSE examinations, always write the multiplication step clearly by showing the multiplication of numerators and denominators separately. Finally, simplify the answer completely to score full marks.
🧠 Memory Trick
Remember the rule:
✅ Numerator × Numerator
✅ Denominator × Denominator
✅ Simplify the Answer
A simple formula to remember is:
“Multiply → Simplify → Standard Form”
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, the required quotient is 20 9 .
✍ 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, the required quotient is 56 55 .
✍ 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.
Multiply the numerators and denominators.
The numerator and denominator have a 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, the required quotient is −8 5 .
✅ Final Answer
| Part | Answer |
|---|---|
| (i) | 20 9 |
| (ii) | 56 55 |
| (iii) | −8 5 |
📘 Key Concept
- Division of rational numbers is done by multiplying with the reciprocal.
- Always simplify the final answer to the lowest form.
⚠️ Common Mistake
- Forgetting to take the reciprocal of the divisor.
- Not simplifying the final answer.
🎯 CBSE Exam Tip
Always write the reciprocal first, then multiply and simplify. Showing these steps helps you score full marks.
🧠 Memory Trick
Division = Change ÷ into × and take the Reciprocal.
Remember: Divide → Flip → Multiply → Simplify.
Show that
📌 Given
The equation is
🎯 To Find
Verify that the values of the Left-Hand Side (LHS) and the Right-Hand Side (RHS) are equal.
🟧 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
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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.