Welcome to Sid Classes. In this session, we explore the foundations of arithmetic with complete, step-by-step solutions for NCERT Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1 from the unit titled "The World of Numbers". These problems dive into ratio applications, historical number properties, closure laws, and non-decimal base counting tracking systems.
Our solutions use clean standard mathematical notation (\(\text{LaTeX}\)) and breakdown logic to ensure you score 100% marks in your school tests and board examinations.
🔑 Key Concepts to Remember:
- Proportional Exchange rates: Solve using unitary operations where total yield equals \(\text{Quantity} \times \text{Rate per Unit}\).
- Prime Sequences: Prime numbers are integers strictly greater than 1 that possess no positive divisors other than 1 and themselves.
- Closure Property: A set is closed under an operation if applying that operation to any two elements in the set always results in an element that belongs to the same set.
NCERT Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1 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:
We can solve this problem using the unitary method or direct proportion ratios:
Given that:
$$\text{Number of ingots received for 2 bags of spices} = 15$$
Therefore, the number of copper ingots received for 1 bag of spices is:
$$\text{Rate} = \frac{15}{2} \text{ ingots per bag}$$
The merchant brings 12 bags of spices to the market. We multiply the rate by the total quantity:
$$\text{Total Copper Ingots} = 12 \times \frac{15}{2}$$
Final Answer: The merchant will leave the market with 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:
Property in common:
The given numbers \(11, 13, 17, \text{ and } 19\) are all **consecutive prime numbers**. They cannot be divided evenly by any numbers except 1 and themselves.
Finding the next numbers:
To extend this exact sequence, we look for the next three consecutive prime numbers that follow 19:
- 20 is composite (\(2 \times 10\))
- 21 is composite (\(3 \times 7\))
- 22 is composite (\(2 \times 11\))
- 23 is prime
- 24, 25, 26, 27, 28 are all composite
- 29 is prime
- 30 is composite
- 31 is prime
Final Answer: The numbers are all prime numbers. The next three numbers fitting this pattern are 23, 29, and 31.
Question 3
We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.
Solution:
Answer: No, Natural Numbers (\(\mathbb{N} = \{1, 2, 3, \dots\}\)) are **not closed under subtraction**.
Justification & Examples:
For a set to be closed under subtraction, subtracting any natural number from another must yield a result that is also a natural number. Let's test this with two counter-examples:
Example 1:
Let's choose the natural numbers 4 and 7.
$$4 - 7 = \mathbf{-3}$$
Since \(-3\) is an integer (\(\mathbb{Z}\)) and not a natural number (\(-3 \notin \mathbb{N}\)), closure fails.
Example 2:
Let's choose the natural numbers 5 and 5.
$$5 - 5 = \mathbf{0}$$
Since \(0\) is a whole number but not a natural number (\(0 \notin \mathbb{N}\)), closure fails again.
Conclusion: Because subtraction can lead to zero or negative values, the set of natural numbers is not closed under this operation.
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:
Part 1: Counting on one hand
A single human hand has 4 fingers (index, middle, ring, and little finger), excluding the thumb. Each of these 4 fingers contains 3 distinct skeletal joints (phalanxes). Using the thumb as a pointer to touch each joint, we can count:
$$4 \text{ fingers} \times 3 \text{ joints per finger} = \mathbf{12 \text{ units}}$$
Part 2: Relation to base-12 system
This anatomical structure is the direct historical origin of the **duodecimal (base-12) system**. Instead of counting up to 10 using individual fingertips, ancient civilizations could count up to 12 on just a single hand. By using the five fingers of the opposite hand to keep track of completed cycles of 12, a person could easily count up to \(12 \times 5 = 60\) items, which forms the basis of ancient sexagesimal systems used for tracking time and angles today.
Frequently Asked Questions (FAQs)
Q1: What is the difference between natural numbers and whole numbers?
A: Natural numbers start from 1 (\(1, 2, 3, \dots\)), whereas whole numbers include zero along with the natural numbers (\(0, 1, 2, 3, \dots\)).
Q2: What is the Ishango bone?
A: It is an ancient mathematical artifact discovered in Africa, featuring grouped scratch marks that suggest early humans tracked prime number sequences and basic multiplication patterns.
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