NCERT Class 9 Maths Exercise 3.1 Solutions – The World of Numbers

Short Intro

Exercise 3.1 introduces students to the origin of numbers, natural numbers, prime numbers, and properties of number systems. These questions help learners understand how mathematics developed from practical human needs such as counting, trading, and measuring. Below are the detailed step-by-step solutions in simple English for easy understanding and direct publishing on educational portals.

Quick Information Box

Topic Details
Chapter The World of Numbers
Exercise 3.1
Class Grade 9
Main Concepts Natural Numbers, Prime Numbers, Closure Property
Difficulty Level Easy to Moderate
Useful For School Exams & Foundation Preparation

Concepts Used (Topics Covered)

  • Natural Numbers
  • Prime Numbers
  • Ratio and Proportion
  • Closure Property of Numbers
  • Subtraction in Natural Numbers
  • Base-12 Counting System

Important Formulas

  1. Ratio Formula

Required Quantity=Given QuantityGiven Ratio×Required Ratio\text{Required Quantity} = \frac{\text{Given Quantity}}{\text{Given Ratio}} \times \text{Required Ratio}

  1. Prime Number
    A number greater than 1 having exactly two factors: 1 and itself.
  2. Closure Property
    A set is closed under an operation if the result always belongs to the same set.

Questions & Step-by-Step Solutions

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

Given:

  • 2 bags of spices = 15 copper ingots
  • 12 bags of spices = ?

Using proportion:152×12\frac{15}{2} \times 12=15×6= 15 \times 6=90= 90

Final Answer

The merchant will receive 90 copper ingots.

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:

11, 13, 17, 19

All these numbers are:

  • Prime numbers
  • Greater than 10 and less than 20

The next prime numbers after 19 are:

23, 29, 31

Final Answer

These are prime numbers.
The next three numbers are:23, 29, 3123,\ 29,\ 31

Question 3

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

Solution

Natural numbers are:N={1,2,3,4,…}\mathbb{N} = \{1,2,3,4,\ldots\}

To check closure under subtraction, we subtract two natural numbers.

Example 1

8−3=58 – 3 = 5

Here, 5 is a natural number.

Example 2

3−8=−53 – 8 = -5

But −5-5 is not a natural number.

Therefore, subtraction does not always produce a natural number.

Final Answer

Natural numbers are not closed under subtraction because subtraction may produce negative numbers.

Question 4

Ancient Indians used the joints of their fingers to count. 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:

  • There are 4 fingers used for counting.
  • Each finger has 3 joints.

Total joints:4×3=124 \times 3 = 12

So, one hand can count up to:1212

This is related to the ancient base-12 system because counting finger joints naturally gives a total of 12.

Final Answer

One hand can count up to 12 using finger joints. This explains the origin of the ancient base-12 counting system.

Common Mistakes

  1. Confusing prime numbers with odd numbers.
  2. Forgetting that negative numbers are not natural numbers.
  3. Using wrong proportion methods in Question 1.
  4. Counting thumb joints in Question 4 incorrectly.

Exam Tips

  • Always check whether the result belongs to the given number set.
  • Remember: Prime numbers have exactly two factors.
  • Learn closure properties carefully because they are frequently asked in exams.
  • Use step-by-step calculations for better presentation.

Practice MCQs

1. Which of the following is a prime number?

A. 15
B. 21
C. 29
D. 35

✅ Answer: 29

2. Natural numbers are closed under:

A. Subtraction
B. Division
C. Addition
D. None

✅ Answer: Addition

3. What is the result of 5−95 – 9?

A. 4
B. -4
C. 14
D. 0

✅ Answer: -4

4. How many joints are counted on one hand in the ancient counting system?

A. 8
B. 10
C. 12
D. 15

✅ Answer: 12

FAQ Section

Q1. What are natural numbers?

Natural numbers are counting numbers:1,2,3,4,…1,2,3,4,\ldots

Q2. Are natural numbers closed under division?

No. Example:3÷2=1.53 \div 2 = 1.5

which is not a natural number.

Q3. What is a prime number?

A prime number has exactly two factors: 1 and itself.

Example: 2, 3, 5, 7

Q4. Why is subtraction not closed for natural numbers?

Because subtraction can produce negative numbers which are not natural numbers.

Q5. What is the base-12 system?

A counting system based on 12 instead of 10 is called the base-12 system.

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