Mathematics solution NCERT
Class 9 - Chapter 3: The World of Numbers
Q1: 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 = ? copper ingots
Using the unitary method:
Number of groups of 2 bags in 12 bags
= 12 ÷ 2
= 6
Each group gives 15 copper ingots.
Total copper ingots = 6 × 15
= 90
Answer: The merchant will receive 90 copper ingots.
Q2: 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 of these numbers are prime numbers.
A prime number has exactly two factors:
- 1
- The number itself
The next prime numbers after 19 are:
23, 29, 31
| Prime Numbers in Sequence |
|---|
| 11 |
| 13 |
| 17 |
| 19 |
| 23 |
| 29 |
| 31 |
Answer: These numbers are prime numbers. The next three numbers are 23, 29 and 31.
Q3: 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:
A set is said to be closed under an operation if performing that operation on any two elements of the set always gives another element of the same set.
Natural numbers are closed under addition because:
5 + 7 = 12
12 is also a natural number.
Now consider subtraction:
Example 1:
8 − 3 = 5
5 is a natural number.
Example 2:
3 − 8 = -5
-5 is not a natural number.
Example 3:
4 − 4 = 0
0 is generally not included in the set of natural numbers in many school-level definitions.
Since subtraction does not always produce a natural number, natural numbers are not closed under subtraction.
Answer: No, natural numbers are not closed under subtraction.
Q4: 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 system?
Solution:
On one hand, the thumb is used as a pointer.
The other four fingers each have 3 joints.
| Finger | Number of Joints |
|---|---|
| Index Finger | 3 |
| Middle Finger | 3 |
| Ring Finger | 3 |
| Little Finger | 3 |
| Total | 12 |
Total joints = 4 × 3 = 12
Using the thumb to touch each joint, a person can count from 1 to 12 on one hand.
This counting method is believed to have contributed to the development of the base-12 (duodecimal) number system.
In the base-12 system:
- 12 is treated as an important counting unit.
- After counting 12 joints on one hand, the other hand can be used to keep track of how many groups of 12 have been counted.
Some modern examples related to base-12 are:
- 12 months in a year
- 12 inches in a foot
- 12 items in a dozen
Answer: A person can count up to 12 on one hand using finger joints. This counting method is closely related to the ancient base-12 counting system.