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.
Class 9 Maths Ganita Manjari Chapter 3 Exercise 3.1 Solutions
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.
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.
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.
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.
Related Links – Class 9 Maths Chapter 3
FAQs (Frequently Asked Questions)
They are called prime numbers because each of them has exactly two factors: 1 and the number itself. For example, 13 is divisible only by 1 and 13.
Natural numbers are not closed under subtraction because subtracting two natural numbers can give 0 or a negative number. For example, 4 − 4 = 0 and 5 − 8 = −3, which are not natural numbers in N = {1, 2, 3, …}.
Use the exchange rate. Since 2 bags give 15 ingots, 12 bags make 6 groups of 2 bags. So, the merchant gets 6 × 15 = 90 ingots.
One-to-one correspondence means matching each object in one group with exactly one object in another group. In the chapter, one cow is matched with one pebble to keep count of a herd.
The thumb can count the 3 joints on each of the 4 fingers, giving 4 × 3 = 12 positions. This makes one hand a natural way to count in groups of 12, which connects to base-12 counting.