Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1 Solutions: The World of Numbers

Before students work with integers, rational numbers and real numbers, Chapter 3 begins with a simple question: why did humans need numbers at all? Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1 Solutions focus on this starting point of The World of Numbers, where counting, trade, tally marks and natural numbers are connected with real historical examples.

In Ganita Manjari Class 9 Chapter 3 Exercise 3.1, students solve questions based on the port city of Lothal, the Ishango bone, prime numbers, closure property of natural numbers and ancient base-12 counting. These Class 9 Maths Chapter 3 Exercise 3.1 Solutions explain each answer step by step while keeping the chapter’s historical context intact.

Key Takeaways

Counting Need: Numbers developed from real needs such as trade and record-keeping.
Natural Numbers: Counting numbers begin from 1.
Prime Numbers: Numbers with exactly two factors are prime.
Closure Property: Natural numbers are closed under addition but not under subtraction.

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1 Solutions Structure 2026

Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1 Solutions

Exercise Set 3.1 comes from the section “The Dawn of Mathematics: The Human Need to Count.” The textbook explains that early counting began with one-to-one correspondence, such as matching one pebble with one cow. This idea led to the birth of natural numbers, written as N = {1, 2, 3, 4, …}.

These Class 9 Maths natural numbers answers also connect counting with trade, tally marks and ancient number systems. The exercise checks whether students understand ratio-based counting, prime number patterns, closure under subtraction and finger-joint counting.

Exercise 3.1 Question 1

A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?

Solution:

The merchant receives 15 copper ingots for every 2 bags of spices.

So, for 2 bags:

2 bags = 15 ingots

Now, 12 bags can be grouped as:

12 / 2 = 6 groups

Each group gives 15 ingots.

So,

Total ingots = 6 × 15

Total ingots = 90

Answer: The merchant will leave with 90 copper ingots.

Exercise 3.1 Question 2

Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.

Solution:

The numbers are:

11, 13, 17, 19

All these numbers are prime numbers.

A prime number is a natural number greater than 1 that has exactly two factors: 1 and itself.

The prime numbers after 19 are:

23, 29, 31

Answer: The numbers 11, 13, 17 and 19 are prime numbers. The next three numbers in the pattern are 23, 29 and 31.

Exercise 3.1 Question 3

We know that Natural Numbers are closed under addition. Are they closed under subtraction? Provide a couple of examples to justify your answer.

Solution:

Natural numbers are:

N = {1, 2, 3, 4, …}

A set is closed under an operation if the result of that operation always remains inside the same set.

Natural numbers are closed under addition because adding any two natural numbers always gives a natural number.

Example:

3 + 5 = 8

Here, 8 is a natural number.

But natural numbers are not closed under subtraction because subtracting one natural number from another may give 0 or a negative number, which are not natural numbers.

Examples:

5 − 8 = −3

Here, −3 is not a natural number.

4 − 4 = 0

Here, 0 is also not a natural number in the set N = {1, 2, 3, 4, …}.

Answer: No, natural numbers are not closed under subtraction. For example, 5 − 8 = −3 and 4 − 4 = 0, and neither result is a natural number.

Exercise 3.1 Question 4

Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?

Solution:

On one hand, the thumb is used to count the joints of the four fingers.

Each finger has 3 joints.

There are 4 fingers to count.

So,

Total joints = 4 × 3

Total joints = 12

Therefore, one hand can be used to count up to 12 using finger joints.

This relates to the ancient base-12 counting system because 12 becomes one full counting cycle. A base-12 system groups numbers in sets of 12, just as the finger-joint method lets a person count 12 positions on one hand.

Answer: We can count 12 on one hand. This relates to the base-12 counting system because one full hand-count gives a group of 12.

Final Answers for Exercise 3.1

Concept Used in The World of Numbers Exercise 3.1

The World of Numbers Exercise 3.1 uses the earliest ideas of counting and number patterns. The exercise is not only calculation-based; it also asks students to connect mathematics with history.

Important concepts include:

• One-to-one correspondence: matching one object with one object to count.
• Natural numbers: counting numbers such as 1, 2, 3, 4, …
• Ratio-based counting: using a fixed exchange rate, such as 15 ingots for 2 bags.
• Prime numbers: numbers with exactly two factors.
• Closure property: checking whether an operation keeps the result inside the same set.
• Base-12 counting: counting in groups of 12 using finger joints.

These ideas form the base of Class 9 Ganita Manjari number system solutions and prepare students for integers, rational numbers, irrational numbers and real numbers later in the chapter.

About Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1

Ganita Manjari Class 9 Chapter 3 Exercise 3.1 belongs to the opening section of The World of Numbers. It introduces how human beings began counting before formal symbols and written number systems developed.

These Class 9 Maths Chapter 3 Exercise 3.1 Solutions prepare students for later topics such as:

• natural numbers,
• zero and integers,
• negative numbers,
• rational numbers,
• irrational numbers,
• real numbers,
• number line representation,
• decimal expansions.

NCERT Solutions for Class 9 Maths Ganita Manjari Chapter 3